 Research
 Open access
 Published:
Reproduction number projection for the COVID19 pandemic
Advances in Continuous and Discrete Models volume 2023, Article number: 46 (2023)
Abstract
The recently derived HybridIncidence SusceptibleTransmissibleRemoved (HISTR) prototype is a deterministic compartment model for epidemics and an alternative to the SusceptibleInfectedRemoved (SIR) model. The HISTR predicts that pathogen transmission depends on host population characteristics including population size, population density and social behaviour common within that population.
The HISTR prototype is applied to the ancestral Severe Acute Respiratory Syndrome Coronavirus 2 (SARSCoV2) to show that the original estimates of the Coronavirus Disease 2019 (COVID19) basic reproduction number \(\mathcal{R}_{0}\) for the United Kingdom (UK) could have been projected onto the individual states of the United States of America (USA) prior to being detected in the USA.
The Imperial College London (ICL) group’s estimate of \(\mathcal{R}_{0}\) for the UK is projected onto each USA state. The difference between these projections and the ICL’s estimates for USA states is either not statistically significant on the paired Student ttest or not epidemiologically significant.
The SARSCoV2 Delta variant’s \(\mathcal{R}_{0}\) is also projected from the UK to the USA to prove that projection can be applied to a Variant of Concern (VOC). Projection provides both a localised baseline for evaluating the implementation of an intervention policy and a mechanism for anticipating the impact of a VOC before local manifestation.
1 Introduction
Motivation
The COVID19 pandemic highlighted the need to anticipate the impact of a novel pathogen on healthcare [1–4] or the economy [5, 6]. One of the impact factors is the basic reproduction number \(\mathcal{R}_{0}\) [7], a demographic concept that has been repurposed for infectious disease epidemiology [8–12]. \(\mathcal{R}_{0}\) represents the average number of susceptible people a host infects in a completely susceptible population whilst that host is in its infected state [13, 14]. Based on estimates of \(\mathcal{R}_{0}\) for COVID19’s causative agent SARSCoV2, various categories of predictive [1, 15–18], forecast [19–21] and regression [22–25] models have been constructed to anticipate healthcare system demand.
The COVID19 pandemic’s infections have been periodic [24–28]. Continuous feedback control loops like prevalencedependent contact rates [29] and intervention fatigue [30] may contribute. Equally, irregular events like relaxation of previous restrictions, superspreader events and migration [30] result in perturbations in the rate of new infections or the active infections [26]. Cyclical events like seasonal host behaviour, pathogen biology, migration or waning immunity [28, 30] result in periodic infection perturbations [25]. The superposition of these perturbations manifest as pandemic waves [31].
For the COVID19 pandemic, some subsequent waves of infection have been associated with mutations to the ancestral (wildtype) SARSCoV2 in some countries [30, 32–36]. Paradoxically, these distinct variant waves may be a consequence of SARSCoV2’s slow virion mutation rate [35, 37–39]. Even if the virion mutation rate is constant, the time to accumulate the appropriate number of mutations at the appropriate loci of a virion’s genome in a sufficiently gregarious individual and sufficiently connected geographical location to collectively constitute a VOC may vary [35, 37, 38]. Thus the timing and the impact of these events will be treated as random [35], and prediction requires manifestation in at least one region. This paper projects the impact of a random event like a novel VOC from a region in which it has manifested to one in which it has not [32]. We propose that the distinct collection and distribution of clinical manifestations, pathologies and mortality of each of the SARSCoV2 VOCs [40–42] can be treated as a novel pandemic and that COVID19 is the collective manifestation of this family of overlapping pandemics [43]. These VOCs’ distinct transmission dynamics [32, 44–47] provide additional justification for this approach.
Implicitly, each VOC contender is a potential new pandemic [30, 39, 48–50]. For the particular case where the contender and incumbent VOC do not form a mixture and, instead, the contender rapidly replaces the incumbent, the reproduction number \(\mathcal{R}(t)\) for the SARSCoV2 variant family at the time of transition \(t_{0}\) represents the challenger VOC’s \(\mathcal{R}_{0}\). This \(\mathcal{R}_{0}\) represents an upperbound of the challenger VOC’s impact in anticipation of it outcompeting and supplanting the incumbent [51]. It is an upperbound because by definition the \(\mathcal{R}_{0}\) assumes complete susceptibility to the new variant.
The HISTR[52] model is a deterministic, compartment model prototype constructed to replace two assumptions of Kermack–McKendrick’s SIR prototype [53–55]. It replaces the assumption that the removal rate from a compartment is proportional to the size of that compartment with the more biologically appropriate assumption that the transmissible period is fixed and, consequently, the removal rate is the same as the entrance rate one transmissible period ago [52]. It also replaces Hamer’s mass action law with its chemistry precursor – the law of mass action [56]. The latter allows the derivation of a population densitydependent \(\mathcal{R}_{0}\) [22, 23, 52].
The HISTR model differs from existing compartment models by predicting that \(\mathcal{R}_{0}\) is not only a pathogen property but also depends on the host population’s characteristics, including population size N, density \(\rho _{n}\) [23] and social behaviour [30, 57–59] common to that population. This paper describes a novel method of foretelling local \(\mathcal{R}_{0}\) in isolated populations with similar social behaviour. The method is designated projection. It proposes that if an estimate of \(\mathcal{R}_{0}\) exists for an isolated population \(y \; ({}^{y}\widehat{\mathcal{R}}_{0} )\), then the projection of \({}^{y}\widehat{\mathcal{R}}_{0}\) onto an isolated population z \(({}^{z}\widetilde{\mathcal{R}}_{0} )\) with similar social behaviour is
where \(\mathfrak{B}\) is specific for that pathogen variant’s transmission dynamics in those populations with similar social behaviour. The symbol ˆ represents an estimate.
Background
The omnipresent SIR compartment model for the temporal evolution of an infectious disease proposes that the individuals of an homogenous population can be grouped into three compartments: susceptible, infected and removed [53–55]. Susceptible implies capable of contracting a pathogen, infected implies capable of replicating and spreading the pathogen, and removed refers to either recovery (expulsion of the pathogen and immunity) or death. Additional compartments [1, 60] and stratified or heterogenous populations [61–64] result in sophisticated deterministic, compartment models.
An infectious epidemiology modelling taxonomy is proposed (Fig. 1) to distinguish between foretell’s common synonyms [65] in mathematical epidemiology. This taxonomy proposes that deterministic, compartment models are a subcategory of Differential Equation (DE), orthodox, predictive models. Predictive (mechanistic [66]) models presuppose that phenomena can be explained and that these explanations can be simulated. The orthodox predictive models consist of a three or fourstep process of explanation, abstraction into mathematics, the application of a numerical method and in silico simulation of the abstraction. The pioneering categories of orthodox models are stochastic and deterministic.
The deterministic compartment models are DE models. The DEs assume a homogenous population and simulate averaged phenomena. The Ordinary Differential Equation (ODE) models only simulate the rate of change of the compartment sizes. Historically, the Delay Differential Equation (DDE) compartment models [67, 68] are an alternative to the Exposed (E) compartment of the SusceptibleExposedInfectiousRemoved (SEIR) ODE model [69–71]. Both the traditional delay term and the E compartment incorporate an incubation period into the SIR prototype. The HISTR is a DDE model that reduces to an ODE for periodic phenomena [52]. The HISTR’s delay is not due to incubation; it is intended to simulate a constant transmissible period. The transmissible period is another subtle difference from traditional ODE models. Similar to the infectious period, it is the period of time that a host can transmit the disease but can be limited biologically (e.g. the incubation period), behaviourally (e.g. isolation, quarantine [72] or hospitalisation) or technologically (e.g. face mask or pharmacy). An example of pharmacological restriction to a transmissible period is Human Immunodeficiency Virus (HIV) control where antiretrovirals (ARVs) substantially reduce viral load and therefore transmissibility. Thus transmissibility may be idealised as a step function under appropriate circumstances [73]. Implicitly, transmissibility is a population characteristic, whereas infectivity is an individual characteristic. Partial Differential Equation (PDE) models typically model spatial spread as diffusion [74–76]. Algebraic formulae for thresholds like \(\mathcal{R}_{0}\) and proportion to vaccinate are a consequence of deterministic models.
Stochastic, orthodox models translate to Binomial Chain Models (BCMs) [77–79] or Stochastic Differential Equation (SDE) models that superimpose uncertainty on ODE models [80–82]. They complement deterministic models with their ability to assign probabilities to outlier events [83] like pathogen extinction, provide confidence intervals to their predictions, incorporate noise and their applicability and utility in small samples. The distribution of the uncertainty is an assumption [84]. Note that the forecasting models (to be described) are also statistical. The distinction is that like the deterministic models, the stochastic models simulate a theory to prophesize the future, whereas the forecasting models extrapolate the past into the future.
Graph based epidemiological models can be interpreted as an abstraction of an explanation (or a translation) to a branch of mathematics, graph theory [85, 86], before in silico simulation [87–90]. The latter interpretation provides the flexibility of graph theory or the heritage of an established application like social network theory [91–94]. Here graph or networkbased methods are therefore classified as orthodox predictive methods and ODE alternatives.
The unorthodox predictive methods also presume that phenomena can be explained but the explanation is not translated into mathematics before simulation. Rather, direct in silico simulation of the explanation is performed. Thus some graphbased implementations can be interpreted as unorthodox [18, 95–98]. Graphs consist of vertices and edges, where (for infectious diseases and social networks) the vertices represent individuals, and the edges represent relationships or interactions. Traditionally, the vertices have no geometric interpretation and do not simulate spatial spread [99], but the vertices can be mapped to location [97].
Agent Based Models (ABMs) [6, 17, 100, 101] and Cellular Automata (CA) [102–104] are spatial, unorthodox, predictive models and PDE alternatives. CA are constructed on a regular lattice, and this restriction is removed for ABMs [105]. As examples of Artificial Life [106], an agent (or node) acts independently subject to simple rules on the local environment. The collective can prophesize complex phenomena that other predictive methods cannot [107]. These models simulate heterogeneity and mixing [108], but the PDEs that they represent are not apparent [52, 109]. CA can reduce to ODEs [110], and for at least one application (computational fluid dynamics), the PDEs that they represent have been derived [111]. Lattice Gas Cellular Automata (LGCA) [110] and Probabilistic Cellular Automata (PCA) or Stochastic Cellular Automata (SCA) [103, 104] are subclassifications of CA [52, 109].
The author considers Monte Carlo simulations of COVID19 to be an unorthodox predictive method [112–115]. Note that here the BCMs have been classified as stochastic orthodox predictive. One may require reclassification [116]. Other less traditional, in the infectious epidemiology context, stochastic models [113] include multivariate stochastic processes [117] and Brownian motion [118].
Forecasting presumes that phenomena have a recognisable and reproducible pattern. Forecasting fits a curve to a historical pattern and extrapolates the pattern into the foreseeable future [66, 119–124]. The Fourier theorem states that any curve can be reproduced by an infinite series of superimposed sinusoidal waves [24, 125–129]. Filtering refers to the attenuation (or omission) of frequencies that do not substantively contribute to the signal [24, 125, 130], resulting in a finite series. In electrical engineering, signal noise is presumed to have high frequency. A lowpass filter (allowing low frequencies to pass) attenuates the noise and smooths the resultant signal [125]. Generally, smoothing is a subset of filtering [125] that attenuates highfrequency signals.
The Box–Jenkins forecasting models [121, 122, 131–133] also fit curves. The prototype is the Autoregressive Moving Average (ARMA) model that forecasts weakly stationary behaviour. The Autoregressive Integrated Moving Average (ARIMA) includes trend by differencing to transform the ARIMA model into a stationary model [121, 131–136]. Seasonality (periodicity) can be incorporated into these time series models [132, 137]. The term autoregression refers to historical data points of a curve being used to estimate the model parameters that predict future values of that same curve [138]. Autocorrelation is a metric of how well past results may foretell future results.
ARIMA models fit a linear combination of a finite number of earlier observations and their differences – parsimonious models [131, 133, 137, 139–142]. See Mills [138, Chaps. 6 and 11] for an introduction to nonlinear functions. The curve fitting distinguishes the traditional timeseries models [19, 122, 133, 135, 137, 140–145] from the Artificial Intelligence (AI) timeseries models [143, 146–148]. Traditional, statistical estimation methods include the maximum likelihood method, the conditional sum of least squares and the ordinary sum of least squares [131–133, 135, 145, 149, 150]. AI is an expanding topic with an evolving definition [151–156]. For the examples in this paper, AI is defined mechanistically as a collection of methods (techniques) that searches a space for an adequate solution [152, 153, 155]. For COVID19 forecasting, the AI methods search for a set of parameters that fits a curve adequately [20, 21, 146, 147, 157]. Although the parameters are not necessarily optimal, AI excels at nonlinear models with or without a priori knowledge or understanding of the system’s behaviour [158]. The search techniques used in COVID19 are biologically inspired, modern AI methods [153, 155, 156]. These include swarminginspired Particle Swarm Optimisation (PSO)[159], evolutioninspired Genetic Algorithms (GA)[160], neurologically inspired Artificial Neural Network (ANN) [161] and deep learning methods like Long Shortterm Memory (LSTM) [161].
Orthodox, predictive models also generate timeseries that can be compared to field epidemiology timeseries [162, 163]. Machine Learning (ML) [143, 164–166], ANN [167–169] and GA [168, 169] have found orthodox, predictive model parameters that adequately replicate COVID19 field timeseries. Deep learning is a neurologically inspired ML technique [146, 154, 156, 164] that has also been used in orthodox, predictive model parameter estimation [170, 171]. Forecasting provides a framework (e.g. ARIMA) and the orthodox, predictive models provide context (e.g. SIR) that constrain the AI search. AI also finds “blackbox” associations [147, 161, 172] – associations found without a priori knowledge of mechanism (transmission dynamics in the infectious epidemiology context) and not providing a posteriori understanding of causation (pathophysiology here). ANN [173–175] and ML techniques like RandomForest [176–179], decision tree [178, 180, 181], support vector machine [136, 178, 179, 182–184] and LSTM [136, 185, 186] have found such unbiased COVID19 associations [147, 156]. A time series can also be an association [136, 186–189]. Unorthodox, predictive models like CA and ABMs are swarminginspired AI [153, 155], which are not search methods. Figure 2 classifies the AI methods discussed in citations of this paper.
Statespace models are a subset of signalplusnoise problems [130] and are introduced as a form of forecasting [132, 138]. Briefly, an observation (space) equation and a state equation are coupled. Each of these equations has a superimposed uncertainty that is assumed Gaussian [190]. The observation (measurement) equation’s independent variable is the signal. In infectious epidemiology, reported new cases, disease mortality [191], waste water serology [192] and combinations thereof are examples of signals. The signal can be a proxy [193, 194] that can be affected by both testing strategy and implementation [36, 47, 191]. For example, South Korea’s strategy of significantly increasing access to testing [195] in COVID19 may have affected the signal quality. Conceivably, universal testing is more effective [196, 197] but less efficient [198] than opportunistic, symptomatic testing [199–202]. Nevertheless, the effectiveness of these strategies should converge when asymptomatic infection is rare. Conceivably, a wellimplemented trackandtrace policy can outperform a poorly promoted/implemented universal testing policy [198, 203, 204].
The unobserved state function is based on a priori knowledge of a system’s behaviour and can be deterministic [205, 206] or empiric [207, 208]. The measured observation/signal is coupled to an unknown state. Backward and forward recursion approximates a state that corresponds to the signal [130]. The Kalman filter is a popular recursion method for implementing statespace models [192, 205–210].
The above models require local, diseasespecific data for prophesy. The local estimate requirement causes delay. Cardoso and Gonçalves [22] propose a form for a universal population size or densitydependent scaling law and use regression [119] to determine the parameters for COVID19. Their approach potentially circumvents the need to estimate local modelling parameters locally. Rather, parameters from other centres can be projected – adjusted for local conditions. Figure 3 illustrates the one week delay [211] in the stage of spread of the ancestral SARSCoV2 between the UK and the USA [212, 213]. Given the time dependence of intervention, the universal scaling law may prove more beneficial to regions less connected to the epicentre like India in Fig. 3’s COVID19 example.
Hu et al. [214] potentially circumvent Cardoso and Gonçalves’ [22] regression’s requirement for multiple preexisting disease centres by repurposing formulae from the kinetic theory of ideal gases to derive population densitydependent contact rates. Hu et al.’s contact rates are an alternative to the HISTR model’s law of mass action. The HISTR prototypes’s contact rate is populationsize and density dependent [52].
The orthodox, predictive models generally assume that the rate of change of the infectious compartment is directly proportional to the size of the infectious and susceptible compartments:
where \(S\ast I\) is a state where members of the two compartments are sufficiently close to transmit the pathogen (or react in the chemistry analogue to follow). This assumption is Hamer’s mass action law. As its name suggests, it is modified from chemistry’s (empiric) law of mass action, which states that the reaction rate is proportional to the concentration of the reagents:
where s and i are the concentration in the chemistry analogue and the density (per unit surface area) in the epidemiology analogue. Superficially, if the volumes (or surfaces) remain constant, then these equations are the same. Intuitively, S and I molecules occupying 1 m^{3} are more likely to interact than the same number of molecules occupying 1000 m^{3} because of their proximity to each other.
The HISTR [52] proposes that a probability density function exists for the likelihood that an infected (or transmissioncable) individual in compartment I can be sufficiently physically close to an individual in compartment S to transmit enough pathogen for the individual in S to become infected and a member of I – transmission. It further proposes that this probability density function is proportional to the population densities in the two compartments:
For population \(N \gg 1\), the possible interactions are \(\approxeq \frac{N^{2}}{2}\) [52], and the transmission rate Ṫ is
which resembles the law of mass action – the rate of the reaction is proportional to the concentration of the reactants.
The HISTR further proposes that there are locationspecific behavioural differences that either retard or promote the transmission of the pathogen [59] such that the rate at which individuals enter the transmissible compartment is
where \(\kappa (\mathbf{x})\) is a metric reflecting behavioural difference with respect to the transmission of disease.
Given that experimentally determined \(\kappa (\mathbf{x})\) are unavailable, HISTR projection can only be applied across populations with the same behaviour. This paper creates the intuitive concept of Sufficiently Similar Social Bahaviour (SSSB) with respect to pathogen transmission. Two populations have SSSB with respect to a disease’s transmission if transposing the behaviour of group A to group B does not result in an appreciable change in the transmission dynamics for that disease in group B. A possible formalism could be let \(\varepsilon (x)\) represent uncertainty in x. Then the uncertainty in the measurement of disease transmission metric ζ for behaviour A is \(\varepsilon (\widehat{\zeta}(A))\). After fitting a prediction curve, let the predicted \(\zeta (A)\) be \(\breve{\zeta}(A)\). The prediction uncertainty is
Let SSSB exist if
The rate at which individuals leave the transmissible compartment is not proportional to the size of the compartment but to the rate at which they leave the susceptible compartment one transmissible period (Δτ̅) ago. Thus
The resultant rate of change in the T compartment (\(\dot{T}_{\mathrm{in}} + \dot{T}_{\mathrm{out}}\)) is a DDE,
The population densities \(s(t)\) and \(i(t)\) emphasize that (2) has been constructed for an homogenous population on a surface. However, (2) is not a PDE and can predict neither spatial spread nor a spatial gradient.
The HISTR creates another intuitive concept – the Sufficiently Isolated Population (SIP) – to circumvent the problem that (2) is not a PDE. The SIP is related to the flow of information – in this instance, SARSCoV2 Ribonucleic Acid (RNA) – onto and over a surface [215]. There are two requirements for a SIP to exist. For a completely isolated population, information never flows across the boundary of the SIP’s surface. For a completely connected region, information flows instantaneously across the boundary between regions A and B as soon as it appears in one. Clearly, each of these extremes is an idealisation. Incompletely isolated (or connected) regions are characterised by a delay in the transmission of information. This first SIP criterion can be identified retrospectively and phenomenologically as illustrated in Fig. 3, where the daily per capita cases for six countries follow the same trajectory, but there is s delay. In Fig. 3 the UK is well connected to Germany, and the USA is well connected to Canada. In Fig. 3, it is also evident that India is more isolated from Italy than the UK is isolated from Italy. In the opposite direction, it is not obvious that the UK is more isolated from India than the USA is isolated from India. Restated, isolation may be directional. The sufficiency in SIP depends on objective. For example, India may take longer to convert 10% of ward beds to intensive care unit (ICU) beds than Germany. In this context the time to readiness may define the delay that equates to sufficient isolation. The second SIP criterion avoids a spatial gradient on the surface on which a SIP exists by requiring information to flow instantly over the surface. Effectively, this criterion is recursive – SIPs cannot exist within a SIP – and the probability of transmission is the same for all points on the surface. For HISTR projection, the delay is of less significance. Potential SIPs are recognised by demonstrable distinct regional pandemic characteristics [23, 216–219] instead.
The much broader topic of \(\mathcal{R}_{0}\) estimation is beyond the scope of this paper [7, 59, 220]. Let it suffice that Böckh’s original concept of \(\mathcal{R}_{0}\) – the average number of girls produced by a female during her reproductive years – was calculated by counting births in the public record [8, 12, 220, 221]. This statistical approach to the estimation of \(\mathcal{R}_{0}\) is distinct from the stochastic models presented above, in part, because they make no a priori assumptions about the mechanism by which pathogens propagate and spread – in the epidemiology analogue. Here the (noun) estimate of a parameter refers to the metrics that quantify that parameter (e.g. \(\mathcal{R}_{0}\)) and the uncertainty in the quantification of that parameter. The verb estimate refers to the experimental or field epidemiology techniques used to generate an estimate. Implicitly, estimating does not prophesize. The latter also distinguishes estimating from stochastic modelling and forecasting. This paper reviews models that prophesize the proliferation of diseased individuals, contextualises the recently derived HISTR model within a classification of these models, uses \(\mathcal{R}_{0}\) estimates \(\widehat{\mathcal{R}}_{0}\) to validate the HISTR’s prediction of population size and densitydependent \(\mathcal{R}_{0}\) and demonstrates the HISTR’s ability to project \(\mathcal{R}_{0}\) (\(\widetilde{\mathcal{R}}_{0}\)).
Restated, a spectrum of methods exist for estimating \(\mathcal{R}_{0}\). Direct counting makes no a priori assumptions about the mechanism of disease propagation nor does it prophesize disease proliferation [222]. The most obvious and direct counting method would involve frequently, regularly, efficiently and effectively recording the state of every individual in a population with a perfect test of state (infection) [223]. At the other extreme, \(\mathcal{R}_{0}\) estimation assumes a model of disease proliferation and then uses curvefitting to estimate parameters [215, 224–227]. In principle, the predictive models reviewed in this paper (and summarised in Fig. 1) constitute the a priori assumptions of the curvefitting extreme. The forecasting models described above can appear as methods at either extreme depending upon whether they make a priori assumptions about the mechanism of propagation. The method of curvefitting can either be traditional (a collection of statistical methods) or AI as described above.
Between the above estimation extremes are at least two types of indirect counting methods. The first counts the relevant parameter (e.g. \(\mathcal{R}_{0}\)) more efficiently than direct counting by assuming a recognised statistical distribution for a sample of that parameter (\(\mathcal{R}_{0}\)) [223, 228]. It assumes neither a mechanism of propagation nor a form of proliferation. The second indirect method assumes a model of propagation to relate the parameter being estimated to a proxy and then indirectly counts the proxy [211, 216, 219, 223, 226, 228–231]. For example, \(\mathcal{R}(t)\), the number of secondary cases directly generated by a primary case, is mathematically related to the generation interval \(T_{c}\) by assuming a form to the proliferation. \(T_{c}\) is counted indirectly and efficiently by assuming a statistical distribution for the \(T_{c}\) sample [229, 232]. Finally, \(\mathcal{R}\) is inferred from the mathematical relationship between \(T_{c}\) and \(\mathcal{R}\). Several relationships \(T_{c} \sim \mathcal{R}\) exist [9, 225, 233, 234]. Two such relationships are [229]
where r is the rate of exponential growth, a consequence of the Lotka–Euler equation [9, 229]). Other proxies include final epidemic size, equilibrium conditions like ageindependent prevalence data and agespecific prevalence data [8].
Superimposed on biases introduced by different estimation methods, differences in the definition of \(\mathcal{R}_{0}\) and their formalisation can exist. The Sharpe and Lotka [235] formalisation of Böckh’s demographic calculation [8, 12, 229] is
In the demographic analogy, \(p(a)\) is the probability of a woman surviving to age a, and \(\beta (a)\) is the rate at a which women of age a give birth to girls. In the epidemic analogue, a is the time from becoming infected and is designated the age of infection [222], \(p(a)\) is the probability that a host remains a host at time a, and \(\beta (a)\) is the rate at which new hosts are generated at time a. The formalism can, in principle, be applied directly early in the epidemic when contact tracing is practical. In this context, (3) is the average number of secondary cases that a host would infect in a completely susceptible population – MacDonald’s epidemiological definition of \(\mathcal{R}_{0}\) [8, 12, 236]. The author has not seen McDonald’s original paper [236], only quotations [8, 12].
For ODE compartment models, several formalisations of \(\mathcal{R}_{0}\) exist. For the SIR model [13, 237],
where \(\xi (t)\) is the contact rate [13] (the average number of adequate contacts that an infective makes per day [237]) or the horizontal transmission incidence [238] (infection rate of susceptible individuals through their contacts with infectives), α is the removal rate [13, 237], and \(1/{\alpha}\) is the duration of infection. This formalism and similar (e.g. the equivalent for the SEIR model) are sufficiently intuitive that the parameters can be estimated and, in turn, be used to calculate \(\widehat{\mathcal{R}}_{0}\) [8, 230, 237, 238].
In an homogenous population, one may anticipate a unimodal (and possibly symmetric) distribution of secondary infections from a sample of hosts. In such a population, a direct application of formalism (3) may calculate \(\widehat{\mathcal{R}}_{0}\). In a heterogenous population, host samples with such unimodal distributions may be anticipated in carefully selected subpopulations, but this assumption may be implausible for the whole population [239]. To apply (3) directly would require both appropriately defined subpopulations and the appropriate formula for averaging over the subpopulations [7, 59]. Diekmann et al. [240] define \(\mathcal{R}_{0}\) as the average number of new cases of an infection caused by one typical infected individual in a population consisting of susceptibles only [241]. A typical host is a distributed individual – a composite of the homogenous subpopulations that form the heterogenous population [240]. This \(\mathcal{R}_{0}\) is formalised as the spectral radius (dominant eigenvalue) of the Next Generation Matrix (NGM) [240]. Neither the definition nor the formalism is intuitive and does not necessarily produce the same value as MacDonald’s definition and corresponding formalisms [8, 240]. However, it identifies the average \(\mathcal{R}_{0}\) as the geometric mean of the subpopulations’ \(\mathcal{R}_{0}\)s [8, 239, 240]; it provides the formula for relating the fitted parameters to \(\widehat{\mathcal{R}}_{0}\) in curvefitting estimates [163, 227, 241] and retains the threshold (\(\mathcal{R}_{0} =1\)) between selflimitation and endemicity. Although Diekmann et al.’s definition is interpreted as applying to heterogenous populations occupying the same geographic space [242] (like the 8 population gonorrhea model [239]), it can also be interpreted as applying to composite generations – supergenerations consisting of subgenerations containing unique hosts in the same spatial region. Examples include the twogender gonorrhea model [239] and hostvector models [240].
For the HISTR[52], two timescales are constructed. The transmissible timescale is constructed to reduce the DDE model to a more familiar ODE model. A rhythmic timescale is constructed to obviate the periodicity of the contact rate (e.g. diurnal variation). \(\mathcal{R}_{0}\) is defined as the average number of secondary cases that a host produces in a completely susceptible population while that primary host is in a transmissible state. It differs from MacDonald’s definition because the primary host does not necessarily directly infect secondary hosts. Restated, Benjamin [52]’s definition of \(\mathcal{R}_{0}\) allows (incomplete) tertiary and quarternary generations of hosts to be included in the sum, provided that the infections occur whilst the primary host is still in the transmissible state. This definition was selected to be consistent with the \(\mathcal{R}_{0}\) formalism derived from the HISTR’s system of ODEs in the transmissible timescale. Like MacDonald’s definition, it is for a homogenous population. Nevertheless, given that the spectral radius of the NGM reduces to ODE compartment models’ formulae for \(\mathcal{R}_{0}\) in homogenous populations, the NGM represents an alternative formalism for the HISTR’s \(\mathcal{R}_{0}\) in the transmissible times scale. A generation in the transmissible timescale is a compound generation consisting of \(\mathfrak{B}\) generations in the rhythmic timescale. A compound generation is a supergeneration consisting of subgenerations having the same characteristics. The \(\mathcal{R}_{0}\) in the rhythmic time scale is the geometric mean of the \(\mathcal{R}_{0}\) in the transmissible timescale [52] – similar to the NGM’s \(\mathcal{R}_{0}\) for composite generations.
This paper neither endeavours to compare \(\mathcal{R}_{0}\) definitions and formalisms nor the most efficient and effective \(\mathcal{R}_{0}\) estimation methods. It accepts that every combination of definition, formalism and method is biased. Of relevance is that the same definitionformalismmethod combination is used across regions to validate HISTR projection for COVID19.
2 Methods
The HISTR prototype is based on the SIR model but replaces two assumptions and is formulated for an isolated population on a surface [52]. Thus

1.
it is explicit that the model only applies to SIPs,

2.
results in a populationdensitydependent contact rate, and

3.
the PDE problem is replaced by SIPs and SSSB recognition problems.
Hamer’s mass action law [243] assumption is replaced with the law of mass action [52], its chemistry precursor [244, 245]. A probability density function for a single successful transmission is constructed, which reflects the proximity of transmission capable and susceptible:
where η is an infectious diseasespecific variable that reflects avidity, μ is a function of mode of transmission, \(\kappa (\mathbf{x})\) is a function of social behaviour, \(s(t)\) is the density of susceptible individuals, and \(\tau (t)\) is the density of hosts capable of transmitting the pathogen [52]. The total transmission (including those of secondary hosts) in a population of size \(N \gg 1\) and population density \(\rho _{n}\) over the period that the primary host is transmission capable (Δτ̅) is shown to be
where \(S(t)\) is the size of the susceptible population, \(T(t)\) is the size of the transmissioncapable population, and \(\beta _{A} =\frac{\eta \mu \kappa}{2}\) [52].
The SIR model’s exponential infectious period assumption is replaced with the HISTR prototype’s more biologically appropriate constant transmission period. This results in the SIRlike DDE system of equations [52]
The delay term reflects that the rate at which individuals leave a compartment is the same as that at which they entered one transmissible period (Δτ̅) ago.
Selecting a timescale (the transmissible timescale) in which a unit of time Δt equates to Δτ̅ (\(1:\Delta t = 1: \Delta \overline{\tau}\)) renders the delay negligible, reducing the above DDE model to an ODE model [52]:
where the _{τ} identifies the timescale being used. The transmissible timescale is a short timescale with less temporal detail – remaining consistent with the spatial scaling analogy’s terminology.
The shortterm periodicity of infection opportunity in (4) – e.g. diurnal variation in contact rate in (5) – is obviated by constructing a second timescale (the rhythmic timescale) in which the infection opportunity can be treated as constant. The rhythmic timescale is \(1:\Delta t = 1: \delta t\), where δt is the period of the infection opportunity cycle for Equation (4). For respiratory infectious diseases, δt is the host’s sleepwake cycle (i.e. 1 day). A unit of time in the rhythmic timescale is necessarily shorter than the equivalent in the transmissible timescale because at least one cycle of infection opportunity must exist within the period that the host is infectious for an infectious disease to spread. The rhythmic timescale is a long timescale with greater temporal detail to remain consistent with the spatial scaling analogy’s terminology.
Let there be \(\mathfrak{B}\in \mathbb{N}\) units of δt in Δτ̅. For temporal continuity between the timescales, the number of new transmission after one compound generation in the transmissible timescale must equal the new transmissions after \(\mathfrak{B}\) generations in the rhythmic timescale. Alternatively, the new infections after \(\mathfrak{B}\delta t\) in the rhythmic timescale are the same as after Δτ̅ in the transmissible timescale [52]. The temporal continuity condition can be formalised as
where _{ρ} identifies the rhythmic timescale. Using a binomial expansion to enforce continuity, the transmissible timescale ODE system (5) reduces to
in the rhythmic timescale [52], where _{τ}α is the infection frequency in the transmissible timescale. The HISTR model’s rhythmic timescale basic reproduction number for SIP z is then [52, 246]
Both \(\beta _{A}\) and _{τ}α are dependent on social behaviour, which may be cultural [47, 57, 247]. These are assumed constant for populations with SSSB. There is a subtle difference between Böckh’s \(\mathcal{R}_{0}\) and its rhythmic timescale equivalent \({}_{\rho}\mathcal{R}_{0}\). The former only counts new infections in the second generation, whereas the latter also counts new infections in subsequent generations, provided that they are infected in the primary host’s transmissible period. Nevertheless, these are used interchangeably here [52]. Equation (6) is recognised as the geometric mean of the \(\mathcal{R}_{0}\)s for the \(\mathfrak{B}\) generations in the rhythmic timescale that constitute a compound generation in the transmissible timescale.
Dividing Equation (6) for SIP z by the same for SIP y with SSSB derives
– the origin of Equation (1). It is assumed that the anglophone UK and USA have similar concepts of personal space and familiarity with an associated hierarchy of physical interaction rituals [248] such that Equation (1) applies. A metric for SSSB was not identified. Conceivably, host social behaviour could be sufficiently similar across all SIPs. In this case, Equation (1) is a universal scaling law (independent of social behaviour) and should be compared with Cardoso and Gonçalves’ universal scaling law [22] obtained by regression. The UK and USA were selected to increase the likelihood of a successful validation. From Fig. 3, projection should provide the states most connected to the UK an additional week to prepare based on \(\widetilde{\mathcal{R}}_{0}\). The ancestral SARSCoV2 pathogen was selected because transmission dynamics data were available; there was no interference from VOCs, and the ICL group used the same \(\mathcal{R}_{0}\) estimation method for the ancestral SARSCoV2 in both the USA and the UK.
The ICL group’s statistical estimates of \(\mathcal{R}_{0}\) for the wildtype SARSCoV2 in the UK and the states of the USA are used to validate the HISTR’s projection from the UK to these USA states. The ICL group estimated the ancestral SARSCoV2’s \(\widehat{\mathcal{R}}_{0}\) for the UK [211] and the individual states of the USA [219] by counting a proxy – reported mortality. For the UK, estimates counted reported COVID19 deaths from February 2020 to 4 May 2020 [211]. The USA estimates extend the UK method by counting \(100{,}506\) deaths and \(479{,}422\) cases due to COVID19 from 11 May 2020 to 1 June 2020 for each state [219]. Their semimechanistic Bayesian hierarchical model is sensitive to the generation interval [211], but a gamma distribution with mode 6.5 days was used in their \(\widehat{\mathcal{R}}_{0}\) for both the UK and the USA’s states. Consequently, biases introduced by different field and estimation methods are avoided.
From Equation (1),
The paired Student ttest is used to compare the USA basic reproduction number estimate for state z \(( {}^{z}\widehat{\mathcal{R}}_{0} )\) [219] to the UK’s projection on z \(( {}^{z}\widetilde{\mathcal{R}}_{0} )\).
Fortuitously, the UK and USA subsequently embarked on similar COVID19 interventions [247]. Thus, having demonstrated that projection can be performed between the UK and the USA’s states, a contender VOC – common to the UK and USA – that rapidly replaced an incumbent was selected to demonstrate that projection could be applied to a VOC. For this implementation of projection, rapid replacement (minimal mixing) is required to ensure that the SARSCoV2 family’s reproduction number \(\widehat{\mathcal{R}}(t)\) estimated when the VOC contender emerges at \(t = t_{0}\) represents that VOC’s \(\widehat{\mathcal{R}}_{0}\). The Delta (B.1.617.2) variant was selected because it replaced the incumbent Alpha (B.1.1.7) variant in both the UK [32, 249] and the USA [35, 47, 250] within a month as described in Appendix D.
Vaccination protects against symptomatic SARSCoV2 Delta infection [251] but is not as effective against Delta infection [252–255] depending on the time from vaccination [256, 257]. This finding supports the argument that \(\mathcal{R}(t_{0})\) is equivalent to \(\mathcal{R}_{0}\) for the SARSCoV2 Delta variant when it emerged.
A private organisation’s variant sequencing data [250] was used to establish \(t_{0}\) – the date of the transition to Delta – for each state of the USA. They were selected because they provided state level variant sequencing data for the USA. Variant prevalence data can be found at https://public.tableau.com/ [258]. The threshold prevalence for \(t_{0}\) was arbitrarily set as the Delta variant representing 20% of the sequenced SARSCoV2 genomes. 20% was considered small enough to satisfy the condition that the population is completely susceptible to the Delta variant. Conversely, given that most time series treat COVID19 as one disease with one time series for all SARSCoV2 variants, 20% was considered large enough for a substantial portion of a combined \(\widehat{\mathcal{R}}(t)\) to be due to the Delta variant. It is recognised that these criteria introduce bias.
Counting a proxy estimates \(\mathcal{R}_{0}\) for the Delta variant. These estimates validated the HISTR’s VOC \(\widehat{\mathcal{R}}_{0}\) projection from the UK to the USA. Yap and Yong [216] smooth reported case time series with a moving average. A Poisson distribution then constructs a symptom onset time series based on the reported case time series. A second Poisson distribution constructs a date infected time series (\(k_{t}\)). A third Poisson distribution is fitted to the date infected time series to estimate \(\lambda _{t}\) – the expected infections for the day. For the previous day’s estimated new infections (\(k_{t1}\)), the formula relating \(\mathcal{R} \sim \lambda _{t}, k_{t1}\) is
where \(\alpha ^{1}\) is the infectious period. The https://cv19.one [216, 259] data repository was selected to provide \(\widehat{\mathcal{R}}_{0}(\Delta )\) – the Delta variant’s \(\widehat{\mathcal{R}}_{0}\). This selection ensures that the same method can be used to estimate the UK’s \(\widehat{\mathcal{R}}(t,\Delta )\) and state level USA \(\widehat{\mathcal{R}}(t,\Delta ) \) [216, 259]; \(\widehat{\mathcal{R}}(t_{0},\Delta ) \approx \widehat{\mathcal{R}}_{0}( \Delta )\).
Independent \({}^{UK}\widehat{\mathcal{R}}_{0}(\Delta ) = 1.441.5\) [249, 260] with 95% CI [1.21.75] [260] is provided for comparison. The statistical analysis was conducted in the opensource R Project for statistical computing (https://www.rproject.org).
The \(\mathfrak{B}\) for the ancestral SARSCoV2 and the Delta variant are estimated in Appendices A and D, respectively. The transmissible periods are estimated as the median of other estimates with neither weighting nor confidence intervals.
3 Results
This parameter estimations performed for both the wildtype SARSCoV2 and for the SARSCoV2 Delta variant assume no inherent immunity to COVID19. In immunology, innate immunity refers to a static, generic immune system that nonspecifically targets any pathogen invasion. The adaptive immune system generates variations that specifically target a particular invasive pathogen. Natural immunity to a pathogen is due to either previous exposure to that pathogen or crossreactivity from exposure to another pathogen – heterologous immunity [261]. To avoid confusion with these terms, inherent immunity refers to a genetic (inborn) resistance to or attenuation of certain diseases. Examples of inherent immunity are mutations to the CCR5 receptor on cells that renders some individuals immune to infection by HIV [262], the haemoglobinopathymalaria hypothesis that proposes that the high haemoglobinopathy carrier prevalence in some population may exist because this carrier state protects against malaria [263] and cystic fibrosis that may offer carriers protection against cholera [264]. Immunity can be natural, inherent or due to vaccination. Vaccination can be heterologous immunity.
3.1 The ancestral SARSCoV2
The transmission dynamics, distribution of pathology, case fatality rate and other clinical, pathological and epidemiological characteristics associated with the ancestral (wildtype) SARSCoV2 variant are collectively designated COVID19(wt). Appendix A demonstrates that the transmissible timescale for COVID19(wt) is \(1:9\) days, and the rhythmic timescale is \(1:1\) day. The scaling factor between these timescales (\(\mathfrak{B}(wt)\)) is 9.
The \({}^{UK}\widehat{\mathcal{R}}_{0}(wt) = 3.8 \) [3.0–4.5] [211], the \({}^{UK}\widehat{N} = 67{,}886{,}011\) and the \({}^{UK}\hat{\rho}_{n} = 280.6~\mathrm{km^{2}}\) in 2020 [265]. Equation (7) projects this \({}^{UK}\widehat{\mathcal{R}}_{0}(wt)\) onto the states of the USA. Appendix B removes any outliers among these projections – reducing the sample size to 40 states.
Figure 4 compares the estimated basic reproduction number \(\widehat{\mathcal{R}}_{0}(wt)\) density distribution [219] of the remaining 40 states to those projected from the UK’s wild type SARSCoV2 estimate \({}^{UK}\widehat{\mathcal{R}}_{0}(wt)\) [211]. Figure 4(a) projects the median UK estimate \(({}^{UK}\widehat{\mathcal{R}}_{0}(wt) = 3.8 )\) for the wildtype SARSCoV2, whereas Fig. 4(b) projects \({}^{UK}\widehat{\mathcal{R}}_{0}(wt) = 4.2\). The latter remains within the uncertainty of the UK’s \(\widehat{\mathcal{R}}_{0}(wt)\) estimate [211].
Table 1 summarises the results of the paired Student ttest comparing the \(\widehat{\mathcal{R}}_{0}(wt)\)s [219] of the 40 USA states \(({}^{z}\widehat{\mathcal{R}}_{0}(wt):1\leq z\leq 40, z \in \mathbb{N} )\) to the UK’s projections \((\widetilde{\mathcal{R}}_{0}(wt) )\) for \(3.0 \leq {}^{UK}\widehat{\mathcal{R}}_{0}(wt) \leq 4.5 \) [211] onto those 40 states. For \(4.2 \leq {}^{UK}\widehat{\mathcal{R}}_{0}(wt) \leq 4.5\), a statistically significant difference between the \({}^{z}\widehat{\mathcal{R}}_{0}(wt)\) and \({}^{z}\widetilde{\mathcal{R}}_{0}(wt)\) samples does not exist for those 40 states. For \(3.0 \leq {}^{UK}\widehat{\mathcal{R}}_{0}(wt) \leq 4.1\), although there is a statistically significant difference between the \(\widehat{\mathcal{R}}_{0}(wt)\) and \(\widetilde{\mathcal{R}}_{0}(wt)\) samples, this difference is not epidemiologically significant when compared to the uncertainty in \({}^{z}\widehat{\mathcal{R}}_{0}(wt) \) [219]. An epidemiologically significant change is one greater than the uncertainty in the estimate.
The parameter estimates for \(\mathfrak{B}(wt)\) have considerable variation (Tables 2 and 3). Appendix C is a sensitivity analysis demonstrating that, up to an inherently immune fraction of 50%, the change in the \({}^{z}\mathcal{R}_{0}(wt)\) projections are not epidemiologically significant – Fig. 5(b). Similarly, changing the symptomatic fraction causes no significant change in \(\mathcal{R}_{0}(wt)\) relative to the uncertainty in the estimate – Fig. 5(a). The infectious and transmissible periods are varied in Figs. 5(c) and (d), respectively.
Despite symptomatic fraction and inherent immune fraction not having significant effects on \(\widetilde{\mathcal{R}}_{0}(wt)\) relative to the uncertainty in the ICL’s \(\widehat{\mathcal{R}}_{0}(wt)\) for the UK and the states of the USA, the HISTR predicts that increasing the symptomatic fraction decreases \(\mathfrak{B}(wt)\) and, consequently, \(\mathcal{R}_{0}(wt)\) by increasing the contribution of those with a shorter transmissible period (Fig. 5(a)). It confirms that increasing the inherently immune fraction reduces \(\mathcal{R}_{0}(wt)\) (Fig. 5(b)). As expected, increasing the infectious or transmissible periods increase \(\mathfrak{B}(wt)\) and therefore \(\mathcal{R}_{0}(wt)\) (Figs. 5(c) and (d)).
3.2 The SARSCoV2 Delta variant
Appendix D calculates that \(\mathfrak{B}(\Delta )=6\) when 40% of the UK and USA populations were fully vaccinated against COVID19.
The ‘COVID19 HeatMap’ dashboard [216, 259] was used to lookup \(\widehat{\mathcal{R}}_{0}(\Delta )\). This database relates reported cases to infection dates via Poisson distributions. The infection date time series is related to \(\mathcal{R}(t,\Delta )\). This database was selected because it provides both state and country level \(\widehat{\mathcal{R}}(t,\Delta )\) using the same method – avoiding method biases in \(\mathcal{R}(t,\Delta )\) estimation.
The Delta variant transition date for a SIP was arbitrarily set as the date when the Delta variant represents 20% of the SARSCoV2 Reverse Transcriptase Polymerase Chain Reaction (RTPCR) sequences for that SIP on that date. From Fig. 10(a) in Sect. D, the UK’s transition date (\(t_{0}\)) is May 15, 2021 [249, 260]. On May 15, 2021, the ‘COVID19 HeatMap’ dashboard [216, 259] finds that UK’s \(\widehat{\mathcal{R}}(t_{0}) = 1.5 \approx {}^{UK} \widehat{\mathcal{R}}_{0}(\Delta )\). The ‘our world in data’ dashboard [212] and the ‘COVID19 HeatMap’ dashboard [216, 259] estimates in May 2021 are within the \({}^{UK}\widehat{\mathcal{R}}_{0}(\Delta )\)’s 95% confidence interval [249, 260]. The population size and density estimates for the UK and the USA states used for the ancestral SARSCoV2 were reused for the SARSCoV2 Delta variant’s projection.
The date of the transition to the Delta variant could be established for 11 states of the USA. This small sample is because the private organisation’s [250] variant sequencing is opportunistic (not from a screening program) and several states had no data for periods that included the transition prevalence of 20%. \({}^{UK}\widehat{\mathcal{R}}_{0}(\Delta )\) was projected onto these states using (7) with \({}^{UK}\widehat{\mathcal{R}}_{0}(\Delta )\) as the independent variable. These projections are compared to the ‘COVID19 HeatMap’ estimates [216, 259] in Fig. 6.
4 Discussion
COVID19 is the collective pathological manifestation of the ancestral SARSCoV2 and its variants. New variants have the potential to supplant preexisting variants. Projection provides an efficient method to prophesize location and variantspecific resource requirements. The HISTR has demonstrated that projection can foretell the impact of a pathogen variant (the ancestral SARSCoV2) on the individual states of the USA, provided that an estimate exists for the UK. This was possible because the HISTR accounts for the effect of population characteristics on the basic reproduction number \(\mathcal{R}_{0}\). These regions were selected because it is assumed that the individual states of the USA can be approximated as SIPs and because it is assumed that these anglophone regions possess SSSB.
It is noted that the HISTR prototype does not include the effect of demography [266] on \(\mathcal{R}_{0}\) estimation and projection but agestratified SIR models can be adapted for the HISTR. Genetically or behaviourally predisposed individuals also represent subpopulations that affect the average transmission period. For COVID19, diabetics are a subpopulation that are at increased risk of severe disease and death [267, 268]. The opportunity exists to extend the HISTR to heterogenous populations. Clearly, neither the UK nor the individual states of the USA are homogenous, but an accepted collection of public social behaviourial norms must exist in each instance.
Hawaii, Montana, Alaska and Wyoming are among the outliers. The HISTR model’s \(\widetilde{\mathcal{R}}_{0}\) is an overestimate for these states. For Hawaii, the sea acts as a natural barrier between SIPs. Because the HISTR is nonlinear, these regions cannot be combined. Combining SIPs results in \(\widetilde{\mathcal{R}}_{0}\) overestimates. Communities within Alaska, Montana and Wyoming may be sufficiently isolated for them to be treated as SIPs. Conversely, states like New York and Washington, DC, may be insufficiently isolated. Neither the method for averaging \(\widehat{\mathcal{R}}_{0}\) across SIPs nor the determination of \(\widehat{\mathcal{R}}_{0}\) across SIPs is obvious.
The transition to the SARSCoV2 Delta variant could only be identified for 11 States of the USA. The significance of the comparison between the estimated and projected basic reproduction numbers for the 11 states of the USA could not be determined. Nevertheless, it has been demonstrated that for the minimalmixing VOC projection, the SARSCoV2 Delta variant’s \(\widehat{\mathcal{R}}_{0}\) for the UK could have been projected onto at least 11 states in the USA.
This paper’s motivation is the anticipation and preparation for the local impact of novel pathogen or new VOC. Implicitly, each variant is being treated as a new pathogen to which the local population is completely susceptible. The SARSCoV2 variants are sufficiently closely related that both vaccination and previous infection by the incumbent may confer immunity to the new variant in some individuals. Thus the projection represents an upperbound in which the challenger VOC replaces the incumbent [51]. This model does not address equilibrium states where VOCs form a mixture [39]. Intuitively and theoretically, the inherently and naturally immune individuals should affect the transmission dynamics of the variant and the transmissible period. Given the uncertainty in the wildtype SARSCoV2 \(\widehat{\mathcal{R}}_{0}\), an epidemiologically significant impact could not be demonstrated here.
Intuitively, asymptomatic carriers increase the reproduction number [269]. Uniquely, the HISTR predicts this phenomenon (see Appendix C and Fig. 5(a)) but, given the uncertainty in the ancestral SARSCoV2’s \(\mathcal{R}_{0}\) estimates, epidemiological significance could not be demonstrated. For diseases where a correlation exists between symptoms and mortality, an intervention that only converts symptomatic individuals into asymptomatic individuals may reduce mortality. Ironically, the theory predicts that such an intervention will increase \(\mathcal{R}_{0}\) [270]. Conversely, the HISTR model explains why the SARSCoV2 Delta’s higher symptomatic ratio [270] is associated with lower reproductive numbers.
5 Conclusion
When confronting a novel pathogen, the impact of the disease has to be foretold to prepare accordingly. Some of these impacts are the basic reproduction numbers (a proxy for how fast the disease will spread), mortality and morbidity. Some of the impacts that are beyond the scope of this document are the economic and sociopolitical instabilities caused by the disease and interventions, as well as pathogen evolution.
The HISTR prototype is a deterministic alternative to the SIR prototype. In principle, it has two advantages over the more mature deterministic compartment models – it incorporates the population size and density in the model, and it acknowledges and includes the impact of social behaviour. The latter is controversial, but it should be noted that the HISTR has the flexibility to include social behaviour. It may be that physical interaction across regions and cultures is sufficiently similar (from an infectious disease perspective), so that this variable can be treated as a constant. In the latter case, the HISTR derives a population size and densitydependent universal scaling law for \(\mathcal{R}_{0}\).
Projection allows regionspecific planning and preemptive resource allocation. This regionspecific \(\mathcal{R}_{0}\) provides a baseline to compare interventions. This paper demonstrates that the HISTR model can project the UK’s ancestral SARSCoV2’s \(\widehat{\mathcal{R}}_{0}\) and the SARSCoV2 Delta variant’s \(\widehat{\mathcal{R}}_{0}\) onto the states of the USA. Applicability in other anglophone and nonanglophone regions remains to be demonstrated.
The longterm success of an intervention depends both on the policy [271, 272] and implementation [273, 274]. Policies and strategies can only be evaluated retrospectively [275–278] because of unforeseen longterm risks [279–283], lowprobability highimpact events [284–287] and unintended consequences [288–294]. Nevertheless, projection provides timeous local baselines for the comparison of the implementation of similar policies across SIPs with SSSB.
The HISTR model, like other ODE compartment models, does not predict COVID19’s waves. COVID19 has demonstrated that some of these waves may be due to new variants outcompeting incumbents [30]. Although the HISTR model does not incorporate pathogen evolution and random events like VOCs, here (for the SARSCoV2 Delta variant) it has been demonstrated that an impact of such a random event can still be projected timeously.
Availability of data and materials
The data used in this paper is publicly available. The basic reproduction number estimates for the projection of the wildtype SARSCoV2 is available at ICL (https://www.imperial.ac.uk/mrcglobalinfectiousdiseaseanalysis/covid19/). The Delta variant data is available at https://public.tableau.com and https://cv19.one. Population sizes and densities are available at https://worldpopulationreview.com.
Abbreviations
 HISTR:

HybridIncidence SusceptibleTransmissibleRemoved
 SIR:

SusceptibleInfectedRemoved
 SARSCoV2:

Severe Acute Respiratory Syndrome Coronavirus 2
 COVID19:

Coronavirus Disease 2019
 UK:

United Kingdom
 USA:

United States of America
 ICL:

Imperial College London
 VOC:

Variant of Concern
 DE:

Differential Equation
 ODE:

Ordinary Differential Equation
 DDE:

Delay Differential Equation
 E:

Exposed
 SEIR:

SusceptibleExposedInfectiousRemoved
 HIV:

Human Immunodeficiency Virus
 ARVs:

Antiretrovirals
 PDE:

Partial Differential Equation
 BCMs:

Binomial Chain Models
 SDE:

Stochastic Differential Equation
 ABMs:

Agent Based Models
 CA:

Cellular Automata
 LGCA:

Lattice Gas Cellular Automata
 PCA:

Probabilistic Cellular Automata
 SCA:

Stochastic Cellular Automata
 ARMA:

Autoregressive Moving Average
 ARIMA:

Autoregressive Integrated Moving Average
 AI:

Artificial Intelligence
 ML:

Machine Learning
 PSO:

Particle Swarm Optimisation
 GA:

Genetic Algorithms
 ANN:

Artificial Neural Network
 LSTM:

Long Shortterm Memory
 SIP:

Sufficiently Isolated Population
 SSSB:

Sufficiently Similar Social Bahaviour
 NGM:

Next Generation Matrix
 RNA:

Ribonucleic Acid
 RTPCR:

Reverse Transcriptase Polymerase Chain Reaction
 Ct:

Cycle Count Threshold
 \(\mathcal{R}_{0}\) :

the theoretical basic reproduction number – only primary infections
 \(\widehat{\mathcal{R}}_{0}\) :

the experimentally estimated basic reproduction number
 \(\widetilde{\mathcal{R}}_{0}\) :

the basic reproduction number projected from an estimate
 \(\bar{\bar{\mathcal{R}}}_{0}\) :

basic reproduction number obtained by applying regression to the theory
 \(\mathcal{R}_{e}\) :

the effective reproduction number – partial susceptibility
 \({}_{\rho}\mathcal{R}_{0}\) :

number of primary and secondary infections due to a host over its transmissible period
 N :

theoretical population size
 N̂ :

experimentally determined estimate of population size
 \(\rho _{n}\) :

theoretical population density
 \(\hat{\rho}_{n}\) :

experimentally determined estimate population density
 \(\mathfrak{B}\) :

temporal scaling factor between transmissible and rhythmic timescales
 \(P(t)\) :

probability density function for the infection of a potential host
 η :

pathogen antigen host receptor avidity
 μ :

variable associated with the pathogen’s mode of transmission
 \(\kappa (\mathbf{x})\) :

variable associated with host social behaviour
 \(s(t)\) :

density of susceptible individuals – potential hosts
 \(S(t)\) :

number of susceptible individuals – potential hosts
 \(\tau (t)\) :

density of transmission cable individuals or hosts
 \(T(t)\) :

number of transmission cable individuals or hosts
 \(R(t)\) :

number of individuals removed due to immunity or death
 _{τ} α :

transmission frequency in the transmissible timescale
 ΔI :

the period of time that a host is infectious
 Δτ :

the period of time a host is both infectious and interacting socially (with close physical proximity) with potential hosts
 Δτ̅ :

the average time that a group of hosts is both infectious and interacting socially (with close physical) contact with potential hosts
 δt :

the period of the infection opportunity cycle
 σ :

inherently susceptible proportion of the population
 \(1\sigma \) :

inherently immune proportion of the population
 ψ :

symptomatic proportion of those infected
 \(1\psi \) :

asymptomatic proportion of those infected
 M :

predicted proportionality constant relating \(\mathcal{R}_{0}\) to N and \(\rho _{n}\)
 E :

the relative difference between the experimental \(\mathcal{R}_{0}\) and the equivalent obtained by regression based on the theory
 p :

statistical significance level
 \(\mu _{d}\) :

mean of the differences
 \(CI_{d}\) :

95% confidence interval of the difference
 IQR :

interquartile range
References
Giordano, G., Blanchini, F., Bruno, R., Colaneri, P., Di Filippo, A., Di Matteo, A., Colaneri, M.: Modelling the COVID19 epidemic and implementation of populationwide interventions in Italy. Nat. Med. 26(6), 855–860 (2020). https://doi.org/10.1038/s4159102008837
Nikolopoulos, K., Punia, S., Schäfers, A., Tsinopoulos, C., Vasilakis, C.: Forecasting and planning during a pandemic: COVID19 growth rates, supply chain disruptions, and governmental decisions. Eur. J. Oper. Res. 290(1), 99–115 (2021). https://doi.org/10.1016/j.ejor.2020.08.001
CampilloFunollet, E., Van Yperen, J., Allman, P., Bell, M., Beresford, W., Clay, J., Dorey, M., Evans, G., Gilchrist, K., Memon, A., Pannu, G., Walkley, R., Watson, M., Madzvamuse, A.: Predicting and forecasting the impact of local outbreaks of COVID19: use of SEIRD quantitative epidemiological modelling for healthcare demand and capacity. Int. J. Epidemiol. 50(4), 1103–1113 (2021). https://doi.org/10.1093/ije/dyab106
Ioannidis, J.P.A., Cripps, S., Tanner, M.A.: Forecasting for COVID19 has failed. Int. J. Forecast. (2020). https://doi.org/10.1016/j.ijforecast.2020.08.004
Lazebnik, T., Shami, L., BunimovichMendrazitsky, S.: Spatiotemporal influence of nonpharmaceutical interventions policies on pandemic dynamics and the economy: the case of COVID19. Ekon. Istraž. 0(0), 1–29 (2021). https://doi.org/10.1080/1331677X.2021.1925573
Silva, P.C.L., Batista, P.V.C., Lima, H.S., Alves, M.A., Guimarães, F.G., Silva, R.C.P.: COVIDABS: an agentbased model of COVID19 epidemic to simulate health and economic effects of social distancing interventions. Chaos Solitons Fractals 139, 110088 (2020). https://doi.org/10.1016/j.chaos.2020.110088
Ahammed, T., Anjum, A., Rahman, M.M., Haider, N., Kock, R., Uddin, M.J.: Estimation of novel coronavirus (COVID19) reproduction number and case fatality rate: a systematic review and metaanalysis. Health Sci. Rep. 4(2), 274 (2021). https://doi.org/10.1002/hsr2.274
Dietz, K.: The estimation of the basic reproduction number for infectious diseases. Stat. Methods Med. Res. 2(1), 23–41 (1993). https://doi.org/10.1177/096228029300200103
Dublin, L.I., Lotka, A.J.: On the true rate of natural increase. J. Am. Stat. Assoc. 20(151), 305–339 (1925). https://doi.org/10.2307/2965517. Full publication date: Sep., 1925
Lotka, A.J.: The measure of net fertility. J. Wash. Acad. Sci. 15(21), 469–472 (1925)
Heesterbeek, J.A.P., Dietz, K.: The concept of \(\mathcal{R}_{0}\) in epidemic theory. Stat. Neerl. 50, 89–110 (1993)
Heesterbeek, J.A.P.: A brief history of \(\mathcal{R}_{0}\) and a recipe for its calculation. Acta Biotheor. 50, 189–204 (2002). https://doi.org/10.1023/A:1016599411804
Dietz, K.: Transmission and control of arbovirus diseases. In: In Proceedings of the Society for Industrial and Applied Mathematics, pp. 104–121. Philadelphia, Epidemiology (1975)
Delamater, P.L., Street, E.J., Leslie, T.F., Yang, Y.T., Jacobsen, K.H.: Complexity of the basic reproduction number (\(\mathcal{R}_{0}\)). Emerg. Infect. Dis. 25(1), 1–4 (2019). https://doi.org/10.3201/eid2501.171901
Zhang, Y., You, C., Cai, Z., Sun, J., Hu, W., Zhou, X.H.: Prediction of the COVID19 outbreak in China based on a new stochastic dynamic model. Sci. Rep. 10(1), 21522 (2020). https://doi.org/10.1038/s41598020766300
Podolski, P., Nguyen, H.S.: Cellular automata in COVID19 prediction. Proc. Comput. Sci. 192, 3370–3379 (2021). https://doi.org/10.1016/j.procs.2021.09.110
Hoertel, N., Blachier, M., Blanco, C., Olfson, M., Massetti, M., Rico, M.S., Limosin, F., Leleu, H.: A stochastic agentbased model of the SARSCoV2 epidemic in France. Nat. Med. 26(9), 1417–1421 (2020). https://doi.org/10.1038/s4159102010016
Firth, J.A., Hellewell, J., Klepac, P., Kissler, S., Jit, M., Atkins, K.E., Clifford, S., VillabonaArenas, C.J., Meakin, S.R., Diamond, C., Bosse, N.I., Munday, J.D., Prem, K., Foss, A.M., Nightingale, E.S., van Zandvoort, K., Davies, N.G., Gibbs, H.P., Medley, G., Gimma, A., Flasche, S., Simons, D., Auzenbergs, M., Russell, T.W., Quilty, B.J., Rees, E.M., Leclerc, Q.J., Edmunds, W.J., Funk, S., Houben, R.M.G.J., Knight, G.M., Abbott, S., Sun, F.Y., Lowe, R., Tully, D.C., Procter, S.R., Jarvis, C.I., Endo, A., O’Reilly, K., Emery, J.C., Jombart, T., Rosello, A., Deol, A.K., Quaife, M., Hué, S., Liu, Y., Eggo, R.M., Pearson, C.A.B., Kucharski, A.J., Spurgin, L.G., Group, C.C..W.: Using a realworld network to model localized COVID19 control strategies. Nat. Med. 26(10), 1616–1622 (2020). https://doi.org/10.1038/s4159102010368
Maurya, S., Singh, S.: Time series analysis of the COVID19 datasets. In: 2020 IEEE International Conference for Innovation in Technology (INOCON), pp. 1–6 (2020). https://doi.org/10.1109/INOCON50539.2020.9298390
Shaikh, S., Gala, J., Jain, A., Advani, S., Jaidhara, S., Roja Edinburgh, M.: Analysis and prediction of COVID19 using regression models and time series forecasting. In: 2021 11th International Conference on Cloud Computing, Data Science Engineering (Confluence), pp. 989–995 (2021). https://doi.org/10.1109/Confluence51648.2021.9377137
Zeroual, A., Harrou, F., Dairi, A., Sun, Y.: Deep learning methods for forecasting COVID19 timeseries data: a comparative study. Chaos Solitons Fractals 140, 110121 (2020). https://doi.org/10.1016/j.chaos.2020.110121
Cardoso, B.H.F., Gonçalves, S.: Universal scaling law for humantohuman transmission diseases. Europhys. Lett. 133(5), 58001 (2021). https://doi.org/10.1209/02955075/133/58001
Sy, K.T.L., White, L.F., Nichols, B.E.: Population density and basic reproductive number of COVID19 across United States counties. PLoS ONE 16(4), 1–11 (2021). https://doi.org/10.1371/journal.pone.0249271
Derakhshan, M., Ansarian, H.R., Ghomshei, M.: Temporal variations in COVID19: an epidemiological discussion with a practical application. J. Int. Med. Res. 49(8), 3000605211033208 (2021). https://doi.org/10.1177/03000605211033208
Cherednik, I.: Modeling the waves of Covid19. Acta Biotheor. 70(1), 8 (2021). https://doi.org/10.1007/s1044102109428w
Cacciapaglia, G., Cot, C., Sannino, F.: Multiwave pandemic dynamics explained: how to tame the next wave of infectious diseases. Sci. Rep. 11(1), 6638 (2021). https://doi.org/10.1038/s41598021858752
Hethcote, H.W., Levin, S.A.: Periodicity in epidemiological models. In: Levin, S.A., Hallam, T.G., Gross, L.J. (eds.) Applied Mathematical Ecology. Biomathematics, vol. 18, pp. 193–211. Springer, Berlin Heidelberg (1989)
Buonomo, B., Chitnis, N., d’Onofrio, A.: Seasonality in epidemic models: a literature review. Ric. Mat. 67(1), 7–25 (2018). https://doi.org/10.1007/s1158701703486
Ochab, M., Manfredi, P., Puszynski, K., d’Onofrio, A.: Multiple epidemic waves as the outcome of stochastic SIR epidemics with behavioral responses: a hybrid modeling approach. Nonlinear Dyn. (2022). https://doi.org/10.1007/s11071022073176
Brand, S.P.C., Ojal, J., Aziza, R., Were, V., Okiro, E.A., Kombe, I.K., Mburu, C., Ogero, M., Agweyu, A., Warimwe, G.M., Nyagwange, J., Karanja, H., Gitonga, J.N., Mugo, D., Uyoga, S., Adetifa, I.M.O., Scott, J.A.G., Otieno, E., Murunga, N., Otiende, M., OcholaOyier, L.I., Agoti, C.N., Githinji, G., Kasera, K., Amoth, P., Mwangangi, M., Aman, R., Ng’ang’a, W., Tsofa, B., Bejon, P., Keeling, M.J., Nokes, D.J., Barasa, E.: COVID19 transmission dynamics underlying epidemic waves in Kenya. Science 374(6570), 989–994 (2021). https://doi.org/10.1126/science.abk0414
Tkachenko, A.V., Maslov, S., Elbanna, A., Wong, G.N., Weiner, Z.J., Goldenfeld, N.: Timedependent heterogeneity leads to transient suppression of the COVID19 epidemic, not herd immunity. Proc. Natl. Acad. Sci. USA 118(17) (2021). https://doi.org/10.1073/pnas.2015972118
Campbell, F., Archer, B., LaurensonSchafer, H., Jinnai, Y., Konings, F., Batra, N., Pavlin, B., Vandemaele, K., Van Kerkhove, M.D., Jombart, T., Morgan, O.: le Polain de Waroux, O.: Increased transmissibility and global spread of SARSCoV2 variants of concern as at June 2021. Euro Surveill. 26(24) (2021). https://doi.org/10.2807/15607917.ES.2021.26.24.2100509
Lauring, A.S., Tenforde, M.W., Chappell, J.D., Gaglani, M., Ginde, A.A., McNeal, T., Ghamande, S., Douin, D.J., Talbot, H.K., Casey, J.D., Mohr, N.M., Zepeski, A., Shapiro, N.I., Gibbs, K.W., Files, D.C., Hager, D.N., Shehu, A., Prekker, M.E., Erickson, H.L., Exline, M.C., Gong, M.N., Mohamed, A., Johnson, N.J., Srinivasan, V., Steingrub, J.S., Peltan, I.D., Brown, S.M., Martin, E.T., Monto, A.S., Khan, A., Hough, C.L., Busse, L.W., ten Lohuis, C.C., Duggal, A., Wilson, J.G., Gordon, A.J., Qadir, N., Chang, S.Y., Mallow, C., Rivas, C., Babcock, H.M., Kwon, J.H., Halasa, N., Grijalva, C.G., Rice, T.W., Stubblefield, W.B., Baughman, A., Womack, K.N., Rhoads, J.P., Lindsell, C.J., Hart, K.W., Zhu, Y., Adams, K., Schrag, S.J., Olson, S.M., Kobayashi, M., Verani, J.R., Patel, M.M., Self, W.H.: Clinical severity of, and effectiveness of mRNA vaccines against, Covid19 from omicron, delta, and alpha SARSCoV2 variants in the United States: prospective observational study. BMJ 376 (2022). https://doi.org/10.1136/bmj2021069761. https://www.bmj.com/content/376/bmj2021069761.full.pdf
Saha, S., Tanmoy, A.M., Tanni, A.A., Goswami, S., Sium, S.M.A., Saha, S., Islam, S., Hooda, Y., Malaker, A.R., Anik, A.M., Haq, M.S., Jabin, T., Hossain, M.M., Tabassum, N., Rahman, H., Hossain, M.J., Islam, M.S., Saha, S.K.: New waves, new variants, old inequity: a continuing COVID19 crisis. BMJ Glob. Health 6(8) (2021). https://doi.org/10.1136/bmjgh2021007031. https://gh.bmj.com/content/6/8/e007031.full.pdf
Callaway, E.: Beyond Omicron: what’s next for COVID’s viral evolution. Nature 600(7888), 204–207 (2021). https://doi.org/10.1038/d41586021036198
Yang, W., Shaman, J.L.: COVID19 pandemic dynamics in South Africa and epidemiological characteristics of three variants of concern (beta, delta, and omicron). eLife 11, 78933 (2022). https://doi.org/10.7554/eLife.78933
Callaway, E.: The coronavirus is mutating  does it matter? Nature 585(7824), 174–177 (2020). https://doi.org/10.1038/d41586020025446
Kupferschmidt, K.: The pandemic virus is slowly mutating. But does it matter? Science 369(6501), 238–239 (2020). https://doi.org/10.1126/science.369.6501.238
Arribas, M., Aguirre, J., Manrubia, S., Lázaro, E.: Differences in adaptive dynamics determine the success of virus variants that propagate together. Virus Evol. 4(1), 043 (2018). https://doi.org/10.1093/ve/vex043
LunaMuschi, A., Borges, I.C., de Faria, E., Barboza, A.S., Maia, F.L., Leme, M.D., Guedes, A.R., MendesCorrea, M.C., Kallas, E.G., Segurado, A.C., Duarte, A.J.S., Lazari, C.S., Andrade, P.S., Sales, F.C.S., Claro, I.M., Sabino, E.C., Levin, A.S., Costa, S.F.: Clinical features of COVID19 by SARSCoV2 Gamma variant: a prospective cohort study of vaccinated and unvaccinated healthcare workers. J. Infect. 84(2), 248–288 (2022). https://doi.org/10.1016/j.jinf.2021.09.005
Nyberg, T., Ferguson, N.M., Nash, S.G., Webster, H.H., Flaxman, S., Andrews, N., Hinsley, W., Bernal, J.L., Kall, M., Bhatt, S., Blomquist, P., Zaidi, A., Volz, E., Aziz, N.A., Harman, K., Funk, S., Abbott, S., Nyberg, T., Ferguson, N.M., Nash, S.G., Webster, H.H., Flaxman, S., Andrews, N., Hinsley, W., Lopez Bernal, J., Kall, M., Bhatt, S., Blomquist, P., Zaidi, A., Volz, E., Abdul Aziz, N., Harman, K., Funk, S., Abbott, S., Hope, R., Charlett, A., Chand, M., Ghani, A.C., Seaman, S.R., Dabrera, G., De Angelis, D., Presanis, A.M., Thelwall, S., Hope, R., Charlett, A., Chand, M., Ghani, A.C., Seaman, S.R., Dabrera, G., De Angelis, D., Presanis, A.M., Thelwall, S.: Comparative analysis of the risks of hospitalisation and death associated with SARSCoV2 omicron (B.1.1.529) and delta (B.1.617.2) variants in England: a cohort study. Lancet 399(10332), 1303–1312 (2022). https://doi.org/10.1016/S01406736(22)004627
Hu, Z., Huang, X., Zhang, J., Fu, S., Ding, D., Tao, Z.: Differences in clinical characteristics between Delta variant and wildtype SARSCoV2 infected patients. Front. Med. 8 (2022). https://doi.org/10.3389/fmed.2021.792135
Boehm, E., Kronig, I., Neher, R.A., Eckerle, I., Vetter, P., Kaiser, L.: Novel SARSCoV2 variants: the pandemics within the pandemic. Clin. Microbiol. Infect. 27(8), 1109–1117 (2021). https://doi.org/10.1016/j.cmi.2021.05.022
Hart, W.S., Miller, E., Andrews, N.J., Waight, P., Maini, P.K., Funk, S., Thompson, R.N.: Generation time of the alpha and delta SARSCoV2 variants: an epidemiological analysis. Lancet Infect. Dis. 22(5), 603–610 (2022). https://doi.org/10.1016/S14733099(22)000019
Davies, N.G., Abbott, S., Barnard, R.C., Jarvis, C.I., Kucharski, A.J., Munday, J.D., Pearson, C.A.B., Russell, T.W., Tully, D.C., Washburne, A.D., Wenseleers, T., Gimma, A., Waites, W., Wong, K.L.M., van Zandvoort, K., Silverman, J.D., DiazOrdaz, K., Keogh, R., Eggo, R.M., Funk, S., Jit, M., Atkins, K.E., Edmunds, W.J.: Estimated transmissibility and impact of SARSCoV2 lineage B.1.1.7 in England. Science 372(6538), 3055 (2021). https://doi.org/10.1126/science.abg3055
Gräf, T., Bello, G., Naveca, F.G., Gomes, M., Cardoso, V.L.O., da Silva, A.F., Dezordi, F.Z., dos Santos, M.C., dos Santos, K.C., Batista, É.L.R., Magalhães, A.L.Á., Vinhal, F., Miyajima, F., Faoro, H., Khouri, R., Wallau, G.L., Delatorre, E., Siqueira, M.M., Resende, P.C., Mattos, T.P., Nascimento, V.A., Souza, V., Corado, A.L.G., Nascimento, F., Silva, G., Mejía, M., Brandão, M.J., Costa, Á., Pessoa, K., Jesus, M., Gonçalves, L.F., Fernandes, C., Andrade, V., Barbagelata, L., Cruz, A.C.R., Costa, A., Silva, L.d.A., Galvão, J.D., Leite, A.B., Pereira, F.M., Costa, T.O., Sousa Jr., J.C., Neto, L.G.L., Barroso, H., Teixeira, D.L.F., Bezerra, J.F., Docena, C., de Lima, R.E., Silva, L.C.A., de Lima, G.B., Machado, L.C., Bezerra, M.F., Paiva, M.H.S., Dantas, M.E.P.L., Pereira, R.L.D.N., Araújo, J., Santos, C.A., RibeiroRodrigues, R., Bernardes, A.L., Iani, F.C.d.M., Grinsztejn, B., Veloso, V.G., Brasil, P., da Paixão, A.C.D., Appolinario, L.R., Lopes, R.S., Motta, F.d.C., Rocha, A.S., Venas, T.M.M., Pereira, E.C., Cavalcanti, A.C., Bastos, L.S., Brigido, L.F.d.M., Oliveira, M.d.M., Schemberger, M.O., Suzukawa, A.A., Riediger, I., Debur, M.d.C., Salvato, R.S., Gregianini, T.S., Rovaris, D.B., Fernandes, S.B.: Phylogeneticbased inference reveals distinct transmission dynamics of SARSCoV2 lineages Gamma and P.2 in Brazil. iScience 25(4) (2022). https://doi.org/10.1016/j.isci.2022.104156
Earnest, R., Uddin, R., Matluk, N., Renzette, N., Turbett, S.E., Siddle, K.J., Loreth, C., Adams, G., TomkinsTinch, C.H., Petrone, M.E., Rothman, J.E., Breban, M.I., Koch, R.T., Billig, K., Fauver, J.R., Vogels, C.B.F., Bilguvar, K., De Kumar, B., Landry, M.L., Peaper, D.R., Kelly, K., Omerza, G., Grieser, H., Meak, S., Martha, J., Dewey, H.B., Kales, S., Berenzy, D., CarpenterAzevedo, K., King, E., Huard, R.C., Novitsky, V., Howison, M., Darpolor, J., Manne, A., Kantor, R., Smole, S.C., Brown, C.M., Fink, T., Lang, A.S., Gallagher, G.R., Pitzer, V.E., Sabeti, P.C., Gabriel, S., MacInnis, B.L., Altajar, A., DeJesus, A., Brito, A., Watkins, A.E., Muyombwe, A., Blumenstiel, B.S., Neal, C., Kalinich, C.C., Liu, C., Loreth, C., Castaldi, C., Pearson, C., Bernard, C., Nolet, C.M., Ferguson, D., Buzby, E., Laszlo, E., Reagan, F.L., Vicente, G., Rooke, H.M., Munger, H., Johnson, H., Tikhonova, I.R., Ott, I.M., Razeq, J., Meldrim, J.C., Brown, J., Wang, J., Vostok, J., Beauchamp, J.P., Grimsby, J.L., Hall, J., Messer, K.S., Larkin, K.L., Vernest, K., Madoff, L.C., Green, L.M., Webber, L., Gagne, L., Ulcena, M.A., Ray, M.C., Fisher, M.E., Barter, M., Lee, M.D., DeFelice, M.T., Cipicchio, M.C., Smith, N.L., Lennon, N.J., Fitzgerald, N.A., Kerantzas, N., Hui, P., Harrington, R., Downing, R., Haye, R., Lynch, R., Anderson, S.E., Hennigan, S., English, S., Cofsky, S., Clancy, S., Mane, S., Ash, S., Baez, S., Fleming, S., Murphy, S., Chaluvadi, S., Alpert, T., Rivard, T., Schulz, W., Mandese, Z.M., Tewhey, R., Adams, M.D., Park, D.J., Lemieux, J.E., Grubaugh, N.D.: Comparative transmissibility of SARSCoV2 variants Delta and Alpha in new. Cell Rep. Med. 3(4), 100583 (2022). https://doi.org/10.1016/j.xcrm.2022.100583
VargasHerrera, N., AraujoCastillo, R.V., Mestanza, O., Galarza, M., RojasSerrano, N., SolariZerpa, L.: SARSCoV2 Lambda and Gamma variants competition in Peru, a country with high seroprevalence. Lancet Regional Health. Americas 6, 100112 (2022). https://doi.org/10.1016/j.lana.2021.100112
Roberts, M., Andreasen, V., Lloyd, A., Pellis, L.: Nine challenges for deterministic epidemic models. Epidemics 10, 49–53 (2015). https://doi.org/10.1016/j.epidem.2014.09.006
Bessière, P., Volmer, R.: From one to many: the withinhost rise of viral variants. PLoS Pathog. 17(9), 1–5 (2021). https://doi.org/10.1371/journal.ppat.1009811
Korber, B., Fischer, W.M., Gnanakaran, S., Yoon, H., Theiler, J., Abfalterer, W., Hengartner, N., Giorgi, E.E., Bhattacharya, T., Foley, B., Hastie, K.M., Parker, M.D., Partridge, D.G., Evans, C.M., Freeman, T.M., de Silva, T.I., Angyal, A., Brown, R.L., Carrilero, L., Green, L.R., Groves, D.C., Johnson, K.J., Keeley, A.J., Lindsey, B.B., Parsons, P.J., Raza, M., RowlandJones, S., Smith, N., Tucker, R.M., Wang, D., Wyles, M.D., McDanal, C., Perez, L.G., Tang, H., MoonWalker, A., Whelan, S.P., LaBranche, C.C., Saphire, E.O., Montefiori, D.C.: Tracking changes in SARSCoV2 spike: evidence that D614G increases infectivity of the COVID19 virus. Cell 182(4), 812–82719 (2020). https://doi.org/10.1016/j.cell.2020.06.043
Benjamin, R.L.: The hybrid incidence susceptibletransmissibleremoved model for pandemics. Acta Biotheor. 70(1), 10 (2022). https://doi.org/10.1007/s10441021094311
Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics – I. 1927. Bull. Math. Biol. 53(1–2), 33–55 (1991)
Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics – II. The problem of endemicity. 1932. Bull. Math. Biol. 53(1–2), 57–87 (1991)
Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics – III. Further studies of the problem of endemicity. 1933. Bull. Math. Biol. 53(1–2), 89–118 (1991)
Hamer, W.H.: The Milroy lectures on epidemic disease in England—the evidence of variability and of persistency of type. Lancet 167(4305), 569–574 (1906). https://doi.org/10.1016/S01406736(01)801872. Originally published as Volume 1, Issue 4305
Moore, C.A., Ruisch, B.C., Granados Samayoa, J.A., Boggs, S.T., Ladanyi, J.T., Fazio, R.H.: Contracting COVID19: a longitudinal investigation of the impact of beliefs and knowledge. Sci. Rep. 11(1), 20460 (2021). https://doi.org/10.1038/s41598021999818
Albrecht, D.: Vaccination, politics and COVID19 impacts. BMC Public Health 22(1), 96 (2022). https://doi.org/10.1186/s1288902112432x
Vegvari, C., Abbott, S., Ball, F., BrooksPollock, E., Challen, R., Collyer, B.S., Dangerfield, C., Gog, J.R., Gostic, K.M., Heffernan, J.M., Hollingsworth, T.D., Isham, V., Kenah, E., Mollison, D., PanovskaGriffiths, J., Pellis, L., Roberts, M.G., Tomba, G.S., Thompson, R.N., Trapman, P.: Commentary on the use of the reproduction number \(\mathcal{R}\) during the COVID19 pandemic. Stat. Methods Med. Res. 31(9), 1675–1685 (2022). https://doi.org/10.1177/09622802211037079
Varghese, A., Kolamban, S., Sherimon, V., Lacap, E.M., Ahmed, S.S., Sreedhar, J.P., Al Harthi, H., Al Shuaily, H.S.: SEAMHCRD deterministic compartmental model based on clinical stages of infection for COVID19 pandemic in Sultanate of Oman. Sci. Rep. 11(1), 11984 (2021). https://doi.org/10.1038/s41598021911145
Lyra, W., do Nascimento, J.D. Jr., Belkhiria, J., de Almeida, L., Chrispim, P.P.M., de Andrade, I.: COVID19 pandemics modeling with modified determinist SEIR, social distancing, and age stratification. The effect of vertical confinement and release in Brazil. PLoS ONE 15(9), 1–17 (2020). https://doi.org/10.1371/journal.pone.0237627
Rǎdulescu, A., Williams, C., Cavanagh, K.: Management strategies in a SEIRtype model of COVID 19 community spread. Sci. Rep. 10(1), 21256 (2020). https://doi.org/10.1038/s41598020776284
Ram, V., Schaposnik, L.P.: A modified agestructured SIR model for COVID19 type viruses. Sci. Rep. 11(1), 15194 (2021). https://doi.org/10.1038/s41598021946093
Balabdaoui, F., Mohr, D.: Agestratified discrete compartment model of the COVID19 epidemic with application to Switzerland. Sci. Rep. 10(1), 21306 (2020). https://doi.org/10.1038/s41598020774204
Jewell, N.P., Lewnard, J.A., Jewell, B.L.: Predictive mathematical models of the COVID19 pandemic: underlying principles and value of projections. JAMA 323(19), 1893–1894 (2020). https://doi.org/10.1001/jama.2020.6585
Holmdahl, I., Buckee, C.: Wrong but useful – what COVID19 epidemiologic models can and cannot tell us. N. Engl. J. Med. 383(4), 303–305 (2020). https://doi.org/10.1056/NEJMp2016822
Dell’Anna, L.: Solvable delay model for epidemic spreading: the case of COVID19 in Italy. Sci. Rep. 10(1), 15763 (2020). https://doi.org/10.1038/s4159802072529y
Devipriya, R., Dhamodharavadhani, s., Selvi, S.: SEIR model for COVID19 epidemic using delay differential equation. J. Phys. Conf. Ser. 1767, 012005 (2021). https://doi.org/10.1088/17426596/1767/1/012005
Feng, S., Feng, Z., Ling, C., Chang, C., Feng, Z.: Prediction of the COVID19 epidemic trends based on SEIR and AI models. PLoS ONE 16(1), 1–15 (2021). https://doi.org/10.1371/journal.pone.0245101
Carcione, J.M., Santos, J.E., Bagaini, C., Ba, J.: A simulation of a COVID19 epidemic based on a deterministic SEIR model. Front. Public Health 8, 230 (2020). https://doi.org/10.3389/fpubh.2020.00230
Khedher, N.b., Kolsi, L., Alsaif, H.: A multistage SEIR model to predict the potential of a new COVID19 wave in KSA after lifting all travel restrictions. Alex. Eng. J. 60(4), 3965–3974 (2021). https://doi.org/10.1016/j.aej.2021.02.058
Walsh, K.A., Spillane, S., Comber, L., Cardwell, K., Harrington, P., Connell, J., Teljeur, C., Broderick, N., de Gascun, C.F., Smith, S.M., Ryan, M., O’Neill, M.: The duration of infectiousness of individuals infected with SARSCoV2. J. Infect. 81(6), 847–856 (2020). https://doi.org/10.1016/j.jinf.2020.10.009
Lythgoe, K.A., Fraser, C.: New insights into the evolutionary rate of HIV1 at the withinhost and epidemiological levels. Proc. R. Soc. Lond. B, Biol. Sci. 279(1741), 3367–3375 (2012). https://doi.org/10.1098/rspb.2012.0595
Wang, H.Y., Yamamoto, N.: Using a partial differential equation with Google Mobility data to predict COVID19 in arizona. Math. Biosci. Eng. 17(5), 4891–4904 (2020). https://doi.org/10.3934/mbe.2020266
Viguerie, A., Lorenzo, G., Auricchio, F., Baroli, D., Hughes, T.J.R., Patton, A., Reali, A., Yankeelov, T.E., Veneziani, A.: Simulating the spread of COVID19 via a spatiallyresolved susceptibleexposedinfectedrecovereddeceased (SEIRD) model with heterogeneous diffusion. Appl. Math. Lett. 111, 106617 (2021). https://doi.org/10.1016/j.aml.2020.106617
Chalub, F.A.C.C., Souza, M.O.: The SIR epidemic model from a PDE point of view. Math. Comput. Model. 53(7), 1568–1574 (2011). https://doi.org/10.1016/j.mcm.2010.05.036. Mathematical Methods and Modelling of Biophysical Phenomena
Lefèvre, C., Picard, P., Simon, M., Utev, S.: A chain binomial epidemic with asymptomatics motivated by COVID19 modelling. J. Math. Biol. 83(5), 54 (2021). https://doi.org/10.1007/s00285021016805
Hellewell, J., Abbott, S., Gimma, A., Bosse, N.I., Jarvis, C.I., Russell, T.W., Munday, J.D., Kucharski, A.J., Edmunds, W.J.: The lancet. Glob. Health 8(4), 488–496 (2020). https://doi.org/10.1016/S2214109X(20)300747
Levesque, J., Maybury, D.W., Shaw, R.H.A.D.: A model of COVID19 propagation based on a gamma subordinated negative binomial branching process. J. Theor. Biol. 512, 110536 (2021). https://doi.org/10.1016/j.jtbi.2020.110536
Niu, R., Chan, Y.C., Wong, E.W.M., van Wyk, M.A., Chen, G.: A stochastic SEIHR model for COVID19 data fluctuations. Nonlinear Dyn. 1(13) (2021). https://doi.org/10.1007/s11071021066319
Zhang, Z., Zeb, A., Hussain, S., Alzahrani, E.: Dynamics of COVID19 mathematical model with stochastic perturbation. Adv. Differ. Equ. 2020(1), 451 (2020). https://doi.org/10.1186/s13662020029091
Dordevic, J., Papic, I., Suvak, N.: A two diffusion stochastic model for the spread of the new corona virus SARSCoV2. Chaos Solitons Fractals 148, 110991 (2021). https://doi.org/10.1016/j.chaos.2021.110991
Allen, L.J.S., Burgin, A.M.: Comparison of deterministic and stochastic SIS and SIR models in discrete time. Math. Biosci. 163(1), 1–33 (2000). https://doi.org/10.1016/S00255564(99)000474
Osthus, D., Hickmann, K.S., Caragea, P.C., Higdon, D., Del Valle, S.Y.: Forecasting seasonal influenza with a statespace SIR model. Ann. Appl. Stat. 11(1), 202–224 (2017). https://doi.org/10.1214/16AOAS1000
Newman, M.E.J.: Spread of epidemic disease on networks. Phys. Rev. E 66, 016128 (2002). https://doi.org/10.1103/PhysRevE.66.016128
Moreno, Y., PastorSatorras, R., Vespignani, A.: Epidemic outbreaks in complex heterogeneous networks. Eur. Phys. J. B, Condens. Matter Complex Syst. 26(4), 521–529 (2002). https://doi.org/10.1140/epjb/e20020122
Schimit, P.H.T., Pereira, F.H.: Disease spreading in complex networks: a numerical study with principal component analysis. Expert Syst. Appl. 97, 41–50 (2018). https://doi.org/10.1016/j.eswa.2017.12.021
PastorSatorras, R., Castellano, C., Van Mieghem, P., Vespignani, A.: Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979 (2015). https://doi.org/10.1103/RevModPhys.87.925
Rorres, C., Romano, M., Miller, J.A., Mossey, J.M., Grubesic, T.H., Zellner, D.E., Smith, G.: Contact tracing for the control of infectious disease epidemics: Chronic wasting disease in deer farms. Epidemics 23, 71–75 (2018). https://doi.org/10.1016/j.epidem.2017.12.006
Pavlopoulos, G.A., Secrier, M., Moschopoulos, C.N., Soldatos, T.G., Kossida, S., Aerts, J., Schneider, R., Bagos, P.G.: Using graph theory to analyze biological networks. BioData Min. 4, 10 (2011). https://doi.org/10.1186/17560381410
Read, J.M., Eames, K.T.D., Edmunds, W.J.: Dynamic social networks and the implications for the spread of infectious disease. J. R. Soc. Interface 5(26), 1001–1007 (2008). https://doi.org/10.1098/rsif.2008.0013
Jo, W., Chang, D., You, M., Ghim, G.H.: A social network analysis of the spread of COVID19 in South Korea and policy implications. Sci. Rep. 11(1), 8581 (2021). https://doi.org/10.1038/s41598021878370
Block, P., Hoffman, M., Raabe, I.J., Dowd, J.B., Rahal, C., Kashyap, R., Mills, M.C.: Social networkbased distancing strategies to flatten the COVID19 curve in a postlockdown world. Nat. Hum. Behav. 4(6), 588–596 (2020). https://doi.org/10.1038/s4156202008986
Del Valle, S.Y., Hyman, J.M., Hethcote, H.W., Eubank, S.G.: Mixing patterns between age groups in social networks. Soc. Netw. 29(4), 539–554 (2007). https://doi.org/10.1016/j.socnet.2007.04.005
Silva, C.J., Cantin, G., Cruz, C., FonsecaPinto, R., Passadouro, R., Soares Dos Santos, E., Torres, D.F.M.: Complex network model for COVID19: Human behavior, pseudoperiodic solutions and multiple epidemic waves. J. Math. Anal. Appl. 514, 125171 (2021). https://doi.org/10.1016/j.jmaa.2021.125171
Kucharski, R., Cats, O., Sienkiewicz, J.: Modelling virus spreading in ridepooling networks. Sci. Rep. 11(1), 7201 (2021). https://doi.org/10.1038/s41598021867042
Lazebnik, T., BunimovichMendrazitsky, S., Shami, L.: Pandemic management by a spatiotemporal mathematical model. Int. J. Nonlinear Sci. Numer. Simul. 000010151520210063 (2021). https://doi.org/10.1515/ijnsns20210063
Thurner, S., Klimek, P., Hanel, R.: A networkbased explanation of why most COVID19 infection curves are linear. Proc. Natl. Acad. Sci. 117(37), 22684–22689 (2020). https://doi.org/10.1073/pnas.2010398117. https://www.pnas.org/content/117/37/22684.full.pdf
Fall, A., Fortin, M.J., Manseau, M., O’Brien, D.: Spatial graphs: principles and applications for habitat connectivity. Ecosystems 10(3), 448–461 (2007). https://doi.org/10.1007/s1002100790387
Kerr, C.C., Stuart, R.M., Mistry, D., Abeysuriya, R.G., Rosenfeld, K., Hart, G.R., Núñez, R.C., Cohen, J.A., Selvaraj, P., Hagedorn, B., George, L., Jastrzebski, M., Izzo, A.S., Fowler, G., Palmer, A., Delport, D., Scott, N., Kelly, S.L., Bennette, C.S., Wagner, B.G., Chang, S.T., Oron, A.P., Wenger, E.A., PanovskaGriffiths, J., Famulare, M., Klein, D.J.: Covasim: an agentbased model of COVID19 dynamics and interventions. PLoS Comput. Biol. 17(7), 1009149 (2021). https://doi.org/10.1371/journal.pcbi.1009149
Najmi, A., Nazari, S., Safarighouzhdi, F., Miller, E.J., MacIntyre, R., Rashidi, T.H.: Easing or tightening control strategies: determination of COVID19 parameters for an agentbased model. Transportation (2021). https://doi.org/10.1007/s11116021102107
Medrek, M., Pastuszak, Z.: Numerical simulation of the novel coronavirus spreading. Expert Syst. Appl. 166, 114109 (2021). https://doi.org/10.1016/j.eswa.2020.114109
Zhou, Y., Wang, L., Zhang, L., Shi, L., Yang, K., He, J., Zhao, B., Overton, W., Purkayastha, S., Song, P.: A spatiotemporal epidemiological prediction model to inform countylevel COVID19 risk in the United States. Harv. Data Sci. Rev. (2020). https://doi.org/10.1162/99608f92.79e1f45e. https://hdsr.mitpress.mit.edu/pub/qqg19a0r
Schimit, P.H.T.: A model based on cellular automata to estimate the social isolation impact on COVID19 spreading in Brazil. Comput. Methods Programs Biomed. 200, 105832 (2021). https://doi.org/10.1016/j.cmpb.2020.105832
Almeida Simoes, J.: An agentbased approach to spatial epidemics through GIS. PhD thesis, University of London (2007) http://ethos.bl.uk/ProcessSearch.do?query=444185
Aguilar, W., SantamaríaBonfil, G., Froese, T., Gershenson, C.: The past, present, and future of artificial life. Front. Neurorobot. 1, 8 (2014). https://doi.org/10.3389/frobt.2014.00008
Bonabeau, E.: Agentbased modeling: methods and techniques for simulating human systems. Proc. Natl. Acad. Sci. 99(suppl 3), 7280–7287 (2002). https://doi.org/10.1073/pnas.082080899. https://www.pnas.org/content/99/suppl_3/7280.full.pdf
Jin, Z., Liu, Q.X.: A cellular automata model of epidemics of a heterogeneous susceptibility. Chin. Phys. 15, 1248 (2006). https://doi.org/10.1088/10091963/15/6/019
Schneckenreither, G., Popper, N., Zauner, G., Breitenecker, F.: Modelling SIRtype epidemics by ODEs, PDEs, difference equations and cellular automata – a comparative study. Simul. Model. Pract. Theory 16(8), 1014–1023 (2008). https://doi.org/10.1016/j.simpat.2008.05.015
Boccara, N., Cheong, K.: Automata network SIR models for the spread of infectious diseases in populations of moving individuals. J. Phys. A, Math. Gen. 25(9), 2447–2461 (1992). https://doi.org/10.1088/03054470/25/9/018
Guo, Z., Shi, B., Wang, N.: Lattice bgk model for incompressible Navier–Stokes equation. J. Comput. Phys. 165(1), 288–306 (2000). https://doi.org/10.1006/jcph.2000.6616
Ciufolini, I., Paolozzi, A.: Mathematical prediction of the time evolution of the COVID19 pandemic in Italy by a Gauss error function and Monte Carlo simulations. Eur. Phys. J. Plus 135(4), 355 (2020). https://doi.org/10.1140/epjp/s1336002000383y
Amaro, J., Orce, J.: Monte Carlo simulation of COVID19 pandemic using Planck’s probability distribution. Biosystems 218, 104708 (2022). https://doi.org/10.1016/j.biosystems.2022.104708
Triambak, S., Mahapatra, D.P.: A random walk Monte Carlo simulation study of COVID19like infection spread. Phys. A, Stat. Mech. Appl. 574, 126014 (2021). https://doi.org/10.1016/j.physa.2021.126014
Xie, G.: A novel Monte Carlo simulation procedure for modelling COVID19 spread over time. Sci. Rep. 10(1), 13120 (2020). https://doi.org/10.1038/s41598020700911
Olabode, D., Culp, J., Fisher, A., Tower, A., HullNye, D., Wang, X.: Deterministic and stochastic models for the epidemic dynamics of COVID19 in Wuhan, China. Math. Biosci. Eng. 18(1), 950–967 (2021). https://doi.org/10.3934/mbe.2021050
Kim, M., Paini, D., Jurdak, R.: Modeling stochastic processes in disease spread across a heterogeneous social system. Proc. Natl. Acad. Sci. USA 116(2), 401–406 (2019). https://doi.org/10.1073/pnas.1801429116
Ming, R.X., Liu, J.M., Cheung, W.K.W., Wan, X.: Stochastic modelling of infectious diseases for heterogeneous populations. Infect. Dis. Poverty 5(1), 107 (2016). https://doi.org/10.1186/s4024901601995
Montgomery, D., Jennings, C., Kulahci, M.: Introduction to Time Series Analysis and Forecasting, 2nd edn. Wiley, Hoboken, New Jersey (2015)
Papastefanopoulos, V., Linardatos, P., Kotsiantis, S.: COVID19: a comparison of time series methods to forecast percentage of active cases per population. Appl. Sci. 10(11) (2020). https://doi.org/10.3390/app10113880
Wang, Y., Yan, Z., Wang, D., Yang, M., Li, Z., Gong, X., Wu, D., Zhai, L., Zhang, W., Wang, Y.: Prediction and analysis of COVID19 daily new cases and cumulative cases: times series forecasting and machine learning models. BMC Infect. Dis. 22(1), 495 (2022). https://doi.org/10.1186/s12879022074726
Gecili, E., Ziady, A., Szczesniak, R.D.: Forecasting COVID19 confirmed cases, deaths and recoveries: revisiting established time series modeling through novel applications for the USA and Italy. PLoS ONE 16(1), 1–11 (2021). https://doi.org/10.1371/journal.pone.0244173
Roosa, K., Lee, Y., Luo, R., Kirpich, A., Rothenberg, R., Hyman, J.M., Yan, P., Chowell, G.: Realtime forecasts of the COVID19 epidemic in China from February 5th to February 24th, 2020. Infect. Dis. Model. 5, 256–263 (2020). https://doi.org/10.1016/j.idm.2020.02.002
Koczkodaj, W.W., Mansournia, M.A., Pedrycz, W., WolnyDominiak, A., Zabrodskii, P.F., Strzalka, D., Armstrong, T., Zolfaghari, A.H., Debski, M., Mazurek, J.: 1,000,000 cases of COVID19 outside of China: the date predicted by a simple heuristic. Glob. Epidemiol. 2, 100023 (2020). https://doi.org/10.1016/j.gloepi.2020.100023
Holloway, J.L.: Smoothing and filtering of time series and space fields. In: Landsberg, H.E., Van Mieghem, J. (eds.) Advances in Geophysics, vol. 4, pp. 351–389. Academic Press, New York (1958). https://doi.org/10.1016/S00652687(08)604872. https://www.sciencedirect.com/science/article/pii/S0065268708604872
Takefuji, Y.: Fourier analysis using the number of COVID19 daily deaths in the US. Epidemiol. Infect. 149, 64 (2021). https://doi.org/10.1017/S0950268821000522
Kuzmenko, O.V., Smiianov, V.A., Rudenko, L.A., Kashcha, M.O., Vasilyeva, T.A., Kolomiiets, S.V., Antoniuk, N.A.: Impact of vaccination on the COVID19 pandemic: bibliometric analysis and cross country forecasting by Fourier series. Wiad. Lek. 74(10 pt 1), 2359–2367 (2021)
Kashcha, M., Palienko, M., Marchenko, R.: Forecast of COVID19 progress considering the seasonal fluctuations. Health Econ. Manag. Rev. 2, 71–82 (2021)
Wang, X., Washington, D., Weber, G.F.: Complex systems analysis informs on the spread of COVID19. Epidemiol. Methods 10(s1), 20210019 (2021). https://doi.org/10.1515/em20210019
Eubank, R.L.: A Kalman Filter Primer. Chapman & Hall/CRC, Boca Raton (2006)
Bisgaard, S., Kulahci, M.: Time Series Analysis and Forecasting by Example, 1st edn. Wiley, Hoboken (2011)
Box, G.E.P., Jenkins, G.M., Reinsel, G.C., Ljung, G.M.: TIME SERIES ANALYSIS Forecasting and Control, 5th edn. Wiley, Hoboken, New Jersey (2016)
Alzahrani, S.I., Aljamaan, I.A., AlFakih, E.A.: Forecasting the spread of the COVID19 pandemic in Saudi Arabia using ARIMA prediction model under current public health interventions. J. Infect. Publ. Health 13(7), 914–919 (2020). https://doi.org/10.1016/j.jiph.2020.06.001
Papastefanopoulos, V., Linardatos, P., Kotsiantis, S.: COVID19: a comparison of time series methods to forecast percentage of active cases per population. Appl. Sci. 10(11) (2020). https://doi.org/10.3390/app10113880
Claris, S., Peter, N.: ARIMA model in predicting of COVID19 epidemic for the southern Africa region. Afr. J. Infect. Dis. 17(1), 1–9 (2023). https://doi.org/10.21010/Ajidv17i1.1
Shahid, F., Zameer, A., Muneeb, M.: Predictions for Covid19 with deep learning models of LSTM, GRU and BiLSTM. Chaos Solitons Fractals 140, 110212 (2020). https://doi.org/10.1016/j.chaos.2020.110212
Chaurasia, V., Pal, S.: COVID19 pandemic: ARIMA and regression modelbased worldwide death cases predictions. SN Comput. Sci. 1(5), 288 (2020). https://doi.org/10.1007/s42979020002986
Mills, T.C.: Applied Time Series Analysis, 1st edn. Academic Press, New York (2019)
Ahmar, A.S., del Val, E.B.: SutteARIMA: shortterm forecasting method, a case: COVID19 and stock market in Spain. Sci. Total Environ. 729, 138883 (2020). https://doi.org/10.1016/j.scitotenv.2020.138883
Malki, Z., Atlam, E.S., Ewis, A., Dagnew, G., Alzighaibi, A.R., ELmarhomy, G., Elhosseini, M.A., Hassanien, A.E., Gad, I.: ARIMA models for predicting the end of COVID19 pandemic and the risk of second rebound. Neural Comput. Appl. 33(7), 2929–2948 (2021). https://doi.org/10.1007/s00521020054340
Chakraborty, T., Ghosh, I.: Realtime forecasts and risk assessment of novel coronavirus (COVID19) cases: a datadriven analysis. Chaos Solitons Fractals 135, 109850 (2020). https://doi.org/10.1016/j.chaos.2020.109850
Chintalapudi, N., Battineni, G., Amenta, F.: COVID19 virus outbreak forecasting of registered and recovered cases after sixty day lockdown in Italy: a data driven model approach. J. Microbiol. Immunol. Infect. 53(3), 396–403 (2020). https://doi.org/10.1016/j.jmii.2020.04.004
Liao, Z., Lan, P., Liao, Z., Zhang, Y., Liu, S.: TWSIR: timewindow based sir for COVID19 forecasts. Sci. Rep. 10(1), 22454 (2020). https://doi.org/10.1038/s41598020800078
Maleki, M., Mahmoudi, M.R., Heydari, M.H., Pho, K.H.: Modeling and forecasting the spread and death rate of coronavirus (COVID19) in the world using time series models. Chaos Solitons Fractals 140, 110151 (2020). https://doi.org/10.1016/j.chaos.2020.110151
Cheng, C., Jiang, W.M., Fan, B., Cheng, Y.C., Hsu, Y.T., Wu, H.Y., Chang, H.H., Tsou, H.H.: Realtime forecasting of COVID19 spread according to protective behavior and vaccination: autoregressive integrated moving average models. BMC Public Health 23(1), 1500 (2023). https://doi.org/10.1186/s12889023164198
Rahimi, I., Chen, F., Gandomi, A.H.: A review on COVID19 forecasting models. Neural Comput. Appl. (2021). https://doi.org/10.1007/s00521020056268
Wang, L., Zhang, Y., Wang, D., Tong, X., Liu, T., Zhang, S., Huang, J., Zhang, L., Chen, L., Fan, H., Clarke, M.: Artificial Intelligence for COVID19: A Systematic Review. Frontiers in Medicine, vol. 8 (2021). https://doi.org/10.3389/fmed.2021.704256
Benvenuto, D., Giovanetti, M., Vassallo, L., Angeletti, S., Ciccozzi, M.: Application of the ARIMA model on the COVID2019 epidemic dataset. Data Brief 29, 105340 (2020). https://doi.org/10.1016/j.dib.2020.105340
Rath, S., Tripathy, A., Tripathy, A.R.: Prediction of new active cases of coronavirus disease (COVID19) pandemic using multiple linear regression model. Diabetes Metab. Syndr. Clin. Res. Rev. 14(5), 1467–1474 (2020). https://doi.org/10.1016/j.dsx.2020.07.045
Tan, C.V., Singh, S., Lai, C.H., Zamri, A.S.S.M., Dass, S.C., Aris, T.B., Ibrahim, H.M., Gill, B.S.: Forecasting COVID19 case trends using SARIMA models during the third wave of COVID19 in Malaysia. Int. J. Environ. Res. Public Health 19(3) (2022). https://doi.org/10.3390/ijerph19031504
Amisha, M.P., Pathania, M., Rathaur, V.K.: Overview of artificial intelligence in medicine. J. Family Med. Prim. Care 8(7), 2328–2331 (2019). https://doi.org/10.4103/jfmpc.jfmpc_440
Khemani, D.: A First Course in Artificial Intelligence, 1st edn. McGraw Hill, New Delhi (2013). http://repo.darmajaya.ac.id/3793/1/A
Warwick, K.: Artificial Intelligence: The Basics. Routledge, New York (2011)
Markiewicz, T., Zheng, J.: Getting Started with Artificial Intelligence, 2nd edn. O’Reilly, Sebastopol, California (2020)
Flasiński, M.: Introduction to Artificial Intelligence, 1st edn. Springer, Cham, Switzerland (2016). https://doi.org/10.1007/9783319400228
Khan, M., Mehran, M.T., Haq, Z.U., Ullah, Z., Naqvi, S.R., Ihsan, M., Abbass, H.: Applications of artificial intelligence in COVID19 pandemic: a comprehensive review. Expert Syst. Appl. 185, 115695 (2021). https://doi.org/10.1016/j.eswa.2021.115695
Wieczorek, M., Siłka, J., Woźniak, M.: Neural network powered COVID19 spread forecasting model. Chaos Solitons Fractals 140, 110203 (2020). https://doi.org/10.1016/j.chaos.2020.110203
Hssayeni, M.D., Chala, A., Dev, R., Xu, L., Shaw, J., Furht, B., Ghoraani, B.: The forecast of COVID19 spread risk at the county level. J. Big Data 8(1), 99 (2021). https://doi.org/10.1186/s40537021004911
Haouari, M., Mhiri, M.: A particle swarm optimization approach for predicting the number of COVID19 deaths. Sci. Rep. 11(1), 16587 (2021). https://doi.org/10.1038/s41598021960575
Salgotra, R., Gandomi, M., Gandomi, A.H.: Time series analysis and forecast of the COVID19 pandemic in India using genetic programming. Chaos Solitons Fractals 138, 109945 (2020). https://doi.org/10.1016/j.chaos.2020.109945
Chimmula, V.K.R., Zhang, L.: Time series forecasting of COVID19 transmission in Canada using LSTM networks. Chaos Solitons Fractals 135, 109864 (2020). https://doi.org/10.1016/j.chaos.2020.109864
Zhao, C., Tepekule, B., Criscuolo, N.G., Wendel Garcia, P.D., Hilty, M.P., RISC19ICU Investigators for Switzerland, Fumeaux, T., Van Boeckel, T.: icumonitoring.ch: a platform for shortterm forecasting of intensive care unit occupancy during the COVID19 epidemic in Switzerland. Swiss Med. Wkly. 150, 20277 (2020). https://doi.org/10.4414/smw.2020.20277
Stanke, Z., Spouge, J.L.: Estimating agestratified transmission and reproduction numbers during the early exponential phase of an epidemic: a case study with COVID19 data. Epidemics 44, 100714 (2023). https://doi.org/10.1016/j.epidem.2023.100714
Sarker, I.H.: Machine learning: algorithms, realworld applications and research directions. SN Comput. Sci. 2(3), 160 (2021). https://doi.org/10.1007/s4297902100592x
Vega, R., Flores, L., Greiner, R.: SIMLR: machine learning inside the SIR model for COVID19 forecasting. Forecasting 4(1), 72–94 (2022). https://doi.org/10.3390/forecast4010005
Watson, G.L., Xiong, D., Zhang, L., Zoller, J.A., Shamshoian, J., Sundin, P., Bufford, T., Rimoin, A.W., Suchard, M.A., Ramirez, C.M.: Pandemic velocity: forecasting COVID19 in the US with a machine learning & Bayesian time series compartmental model. PLoS Comput. Biol. 17(3), 1–20 (2021). https://doi.org/10.1371/journal.pcbi.1008837
Delli Compagni, R., Cheng, Z., Russo, S., Van Boeckel, T.P.: A hybrid neural networkSEIR model for forecasting intensive care occupancy in Switzerland during COVID19 epidemics. PLoS ONE 17(3), 1–15 (2022). https://doi.org/10.1371/journal.pone.0263789
Botmart, T., Sabir, Z., Javeed, S., Sandoval Núñez, R.A., Wajaree weera, Ali, M.R., Sadat, R.: Artificial neural networkbased heuristic to solve COVID19 model including government strategies and individual responses. Inform. Med. Unlocked 32, 101028 (2022). https://doi.org/10.1016/j.imu.2022.101028
Alsayed, A., Sadir, H., Kamil, R., Sari, H.: Prediction of epidemic peak and infected cases for COVID19 disease in Malaysia. Int. J. Environ. Res. Public Health 17(11) (2020). https://doi.org/10.3390/ijerph17114076. 2020
Liao, Z., Lan, P., Fan, X., Kelly, B., Innes, A., Liao, Z.: SIRVDDL: a COVID19 deep learning prediction model based on timedependent SIRVD. Comput. Biol. Med. 138, 104868 (2021). https://doi.org/10.1016/j.compbiomed.2021.104868
Zheng, N., Du, S., Wang, J., Zhang, H., Cui, W., Kang, Z., Yang, T., Lou, B., Chi, Y., Long, H., Ma, M., Yuan, Q., Zhang, S., Zhang, D., Ye, F., Xin, J.: Predicting COVID19 in China using hybrid AI model. IEEE Trans. Cybern. 50(7), 2891–2904 (2020). https://doi.org/10.1109/TCYB.2020.2990162
GhafouriFard, S., MohammadRahimi, H., Motie, P., Minabi, M.A.S., Taheri, M., Nateghinia, S.: Application of machine learning in the prediction of COVID19 daily new cases: a scoping review. Heliyon 7(10), 08143 (2021). https://doi.org/10.1016/j.heliyon.2021.e08143
Adebayo Ifeoluwapo, R., Supriyanto, E., Taheri, S.: COVID19 death risk assessment in Iran using Artificial Neural Network. J. Phys. Conf. Ser. 1964(6), 062117 (2021). https://doi.org/10.1088/17426596/1964/6/062117
Shanbehzadeh, M., Nopour, R., KazemiArpanahi, H.: Developing an artificial neural network for detecting COVID19 disease. Int. J. Health Promot. Educ. 11(2), 387 (2022). https://doi.org/10.4103/jehp.jehp_387
Abdulaal, A., Patel, A., Charani, E., Denny, S., Mughal, N., Moore, L.: Prognostic modeling of COVID19 using Artificial Intelligence in the United Kingdom: model development and validation. J. Med. Internet Res. 22(8), 20259 (2020). https://doi.org/10.2196/20259
Yesilkanat, C.M.: Spatiotemporal estimation of the daily cases of COVID19 in worldwide using random forest machine learning algorithm. Chaos Solitons Fractals 140, 110210 (2020). https://doi.org/10.1016/j.chaos.2020.110210
Moulaei, K., Shanbehzadeh, M., MohammadiTaghiabad, Z., KazemiArpanahi, H.: Comparing machine learning algorithms for predicting COVID19 mortality. BMC Med. Inform. Decis. Mak. 22(1), 2 (2022). https://doi.org/10.1186/s12911021017420
Assaf, D., Gutman, Y., Neuman, Y., Segal, G., Amit, S., GefenHalevi, S., Shilo, N., Epstein, A., MorCohen, R., Biber, A., Rahav, G., Levy, I., Tirosh, A.: Utilization of machinelearning models to accurately predict the risk for critical COVID19. Intern. Emerg. Medicine 15(8), 1435–1443 (2020). https://doi.org/10.1007/s11739020024750
da Silva, R.G., Ribeiro, M.H.D.M., Mariani, V.C., dos Santos Coelho, L.: Forecasting Brazilian and American COVID19 cases based on artificial intelligence coupled with climatic exogenous variables. Chaos Solitons Fractals 139, 110027 (2020). https://doi.org/10.1016/j.chaos.2020.110027
Moslehi, S., Rabiei, N., Soltanian, A.R., Mamani, M.: Application of machine learning models based on decision trees in classifying the factors affecting mortality of COVID19 patients in Hamadan, Iran. BMC Med. Inform. Decis. Mak. 22(1), 192 (2022). https://doi.org/10.1186/s1291102201939x
Giotta, M., Trerotoli, P., Palmieri, V.O., Passerini, F., Portincasa, P., Dargenio, I., Mokhtari, J., Montagna, M.T., De Vito, D.: Application of a Decision Tree model to predict the outcome of nonintensive inpatients hospitalized for COVID19. Int. J. Environ. Res. Public Health 19(20) (2022). https://doi.org/10.3390/ijerph192013016
Yao, H., Zhang, N., Zhang, R., Duan, M., Xie, T., Pan, J., Peng, E., Huang, J., Zhang, Y., Xu, X., Xu, H., Zhou, F., Wang, G.: Severity detection for the Coronavirus Disease 2019 (COVID19) patients using a Machine Learning model based on the blood and urine tests. Front. Cell Dev. Biol. 8, 683 (2020). https://doi.org/10.3389/fcell.2020.00683
Fu, L., Li, Y., Cheng, A., Pang, P., Shu, Z.: A novel Machine Learningderived Radiomic Signature of the whole lung differentiates stable from progressive COVID19 infection: a retrospective cohort study. J. Thorac. Imaging 35(6) (2020)
Sun, L., Song, F., Shi, N., Liu, F., Li, S., Li, P., Zhang, W., Jiang, X., Zhang, Y., Sun, L., Chen, X., Shi, Y.: Combination of four clinical indicators predicts the severe/critical symptom of patients infected Covid19. J. Clin. Virol. 128, 104431 (2020). https://doi.org/10.1016/j.jcv.2020.104431
Islam, M.Z., Islam, M.M., Asraf, A.: A combined deep CNNLSTM network for the detection of novel coronavirus (COVID19) using Xray images. Inform. Med. Unlocked 20, 100412 (2020). https://doi.org/10.1016/j.imu.2020.100412
Arora, P., Kumar, H., Panigrahi, B.K.: Prediction and analysis of COVID19 positive cases using deep learning models: a descriptive case study of India. Chaos Solitons Fractals 139, 110017 (2020). https://doi.org/10.1016/j.chaos.2020.110017
Niazkar, H.R., Niazkar, M.: Application of artificial neural networks to predict the COVID19 outbreak. Glob. Health Res. Policy 5(1), 50 (2020). https://doi.org/10.1186/s4125602000175y
Mollalo, A., Rivera, K.M., Vahedi, B.: Artificial neural network modeling of novel coronavirus (COVID19) incidence rates across the continental United States. Int. J. Environ. Res. Public Health 17(12) (2020). https://doi.org/10.3390/ijerph17124204
Ribeiro, M.H.D.M., da Silva, R.G., Mariani, V.C., dos Santos Coelho, L.: Shortterm forecasting COVID19 cumulative confirmed cases: perspectives for Brazil. Chaos Solitons Fractals 135, 109853 (2020). https://doi.org/10.1016/j.chaos.2020.109853
Commandeur, J.F.F., Koopman, S.J.: An Introduction to StateSpace Time Series Analysis, Practical Econometrics, 1st edn. Oxford University Press, Oxford, UK (2007)
Watson, O., Alhaffer, M., Mehchy, Z., Whittaker, C., et al.: Report 31: estimating the burden of COVID19 in Damascus, Syria: an analysis of novel data sources to infer mortality underascertainment (2020). https://doi.org/10.25561/82443
Proverbio, D., Kemp, F., Magni, S., Ogorzaly, L., Cauchie, H.M., Gonçalves, J., Skupind, A., Aalto, A.: Modelbased assessment of COVID19 epidemic dynamics by wastewater analysis. Sci. Total Environ. 827, 154235 (2022). https://doi.org/10.1016/j.scitotenv.2022.154235
Woolf, S.H., Chapman, D.A., Sabo, R.T., Weinberger, D.M., Hill, L., Taylor, D.D.H.: Excess deaths from COVID19 and other causes, March–July 2020. JAMA 324(15), 1562–1564 (2020). https://doi.org/10.1001/jama.2020.19545
Achilleos, S., Quattrocchi, A., Gabel, J., Heraclides, A., Kolokotroni, O., Constantinou, C., Pagola Ugarte, M., Nicolaou, N., RodriguezLlanes, J.M., Bennett, C.M., Bogatyreva, E., Schernhammer, E., Zimmermann, C., Costa, A.J.L., Lobato, J.C.P., Fernandes, N.M., SemedoAguiar, A.P., Jaramillo Ramirez, G.I., Martin Garzon, O.D., Mortensen, L.H., Critchley, J.A., Goldsmith, L.P., Denissov, G., Rüütel, K., Le Meur, N., Kandelaki, L., Tsiklauri, S., O’Donnell, J., Oza, A., Kaufman, Z., Zucker, I., Ambrosio, G., Stracci, F., Hagen, T.P., Erzen, I., Klepac, P., Arcos González, P., Camporro, Á.F., Burström, B., Pidmurniak, N., Verstiuk, O., Huang, Q., Mehta, N.K., Polemitis, A., Charalambous, A., Demetriou, C.A.: Excess allcause mortality and COVID19related mortality: a temporal analysis in 22 countries, from January until August 2020. Int. J. Epidemiol. 51(1), 35–53 (2020). https://doi.org/10.1093/ije/dyab123
Lee, D., Heo, K., Seo, Y.: COVID19 in South Korea: lessons for developing countries. World Dev. 135, 105057 (2020). https://doi.org/10.1016/j.worlddev.2020.105057
Lindner, A.K., Sarma, N., Rust, L.M., Hellmund, T., KrasovskiNikiforovs, S., Wintel, M., Klaes, S.M., Hoerig, M., Monert, S., Schwarzer, R., Edelmann, A., Martinez, G.E., Mockenhaupt, F.P., Kurth, T., Seybold, J.: Monitoring for COVID19 by universal testing in a homeless shelter in Germany: a prospective feasibility cohort study. BMC Infect. Dis. 21(1), 1241 (2021). https://doi.org/10.1186/s12879021069454
O’Shea, T., Mbuagbaw, L., Mokashi, V., Bulir, D., Gilchrist, J., Smieja, N., Chong, S., Marttala, S., Vera, V., Cvetkovic, A., Smieja, M.: Comparison of four COVID19 screening strategies to facilitate early case identification within the homeless shelter population: a structured summary of a study protocol for a randomised controlled trial. Trials 21(1), 941 (2020). https://doi.org/10.1186/s13063020048902
Silva, P.J.S., Pereira, T., Sagastizábal, C., Nonato, L., Cordova, M.M., Struchiner, C.J.: Smart testing and critical care bed sharing for COVID19 control. PLoS ONE 16(10), 1–17 (2021). https://doi.org/10.1371/journal.pone.0257235
Arons, M.M., Hatfield, K.M., Reddy, S.C., Kimball, A., James, A., Jacobs, J.R., Taylor, J., Spicer, K., Bardossy, A.C., Oakley, L.P., Tanwar, S., Dyal, J.W., Harney, J., Chisty, Z., Bell, J.M., Methner, M., Paul, P., Carlson, C.M., McLaughlin, H.P., Thornburg, N., Tong, S., Tamin, A., Tao, Y., Uehara, A., Harcourt, J., Clark, S., BrostromSmith, C., Page, L.C., Kay, M., Lewis, J., Montgomery, P., Stone, N.D., Clark, T.A., Honein, M.A., Duchin, J.S., Jernigan, J.A.: Presymptomatic SARSCoV2 infections and transmission in a skilled nursing facility. N. Engl. J. Med. 382(22), 2081–2090 (2020). https://doi.org/10.1056/NEJMoa2008457
Nishiura, H., Kobayashi, T., Yang, Y., Hayashi, K., Miyama, T., Kinoshita, R., Linton, N.M., Jung, S.m., Yuan, B., Suzuki, A., Akhmetzhanov, A.R.: The rate of underascertainment of Novel Coronavirus (2019nCoV) infection: estimation using Japanese passengers data on evacuation flights. J. Clin. Med. 9(2) (2020). https://doi.org/10.3390/jcm9020419
Mizumoto, K., Kagaya, K., Zarebski, A., Chowell, G.: Estimating the asymptomatic proportion of coronavirus disease 2019 (COVID19) cases on board the diamond princess cruise ship. Euro Surveill. 25(10) (2020). https://doi.org/10.2807/15607917.ES.2020.25.10.2000180. 2020
Ing, A.J., Cocks, C., Green, J.P.: COVID19: in the footsteps of Ernest Shackleton. Thorax 75(8), 693–694 (2020). https://doi.org/10.1136/thoraxjnl2020215091
Yoo, K.J., Kwon, S., Choi, Y., Bishai, D.M.: Systematic assessment of South Korea’s capabilities to control COVID19. Health Policy 125(5), 568–576 (2021). https://doi.org/10.1016/j.healthpol.2021.02.011
Majeed, A., Seo, Y., Heo, K., Lee, D.: Can the UK emulate the South Korean approach to COVID19? BMJ 369 (2020). https://doi.org/10.1136/bmj.m2084. https://www.bmj.com/content/369/bmj.m2084.full.pdf
ArroyoMarioli, F., Bullano, F., Kucinskas, S., RondónMoreno, C.: Tracking R of COVID19: a new realtime estimation using the Kalman filter. PLoS ONE 16(1), 1–16 (2021). https://doi.org/10.1371/journal.pone.0244474
Song, J., Xie, H., Gao, B., Zhong, Y., Gu, C., Choi, K.S.: Maximum likelihoodbased extended Kalman filter for COVID19 prediction. Chaos Solitons Fractals 146, 110922 (2021). https://doi.org/10.1016/j.chaos.2021.110922
Koyama, S., Horie, T., Shinomoto, S.: Estimating the timevarying reproduction number of COVID19 with a statespace method. PLoS Comput. Biol. 17(1), 1–18 (2021). https://doi.org/10.1371/journal.pcbi.1008679
Aslam, M.: Using the Kalman filter with ARIMA for the COVID19 pandemic dataset of Pakistan. Data Brief 31, 105854 (2020). https://doi.org/10.1016/j.dib.2020.105854
Assimakis, N., Ktena, A., Manasis, C., Mele, E., Kunicina, N., Zabasta, A., Juhna, T.: Using the time varying Kalman filter for prediction of COVID19 cases in Latvia and Greece. In: 2020 IEEE 61th International Scientific Conference on Power and Electrical Engineering of Riga Technical University (RTUCON), pp. 1–7 (2020). https://doi.org/10.1109/RTUCON51174.2020.9316598
Singh, K.K., Kumar, S., Dixit, P., Bajpai, M.K.: Kalman filter based short term prediction model for COVID19 spread. Appl. Intell. 51(5), 2714–2726 (2021). https://doi.org/10.1007/s10489020019481
Flaxman, S., Mishra, S., Gandy, A., Unwin, H.J.T., Mellan, T.A., Coupland, H., Whittaker, C., Zhu, H., Berah, T., Eaton, J.W., Monod, M., PerezGuzman, P.N., Schmit, N., Cilloni, L., Ainslie, K.E.C., Baguelin, M., Boonyasiri, A., Boyd, O., Cattarino, L., Cooper, L.V., Cucunubá, Z., CuomoDannenburg, G., Dighe, A., Djaafara, B., Dorigatti, I., van Elsland, S.L., FitzJohn, R.G., Gaythorpe, K.A.M., Geidelberg, L., Grassly, N.C., Green, W.D., Hallett, T., Hamlet, A., Hinsley, W., Jeffrey, B., Knock, E., Laydon, D.J., NedjatiGilani, G., Nouvellet, P., Parag, K.V., Siveroni, I., Thompson, H.A., Verity, R., Volz, E., Walters, C.E., Wang, H., Wang, Y., Watson, O.J., Winskill, P., Xi, X., Walker, P.G.T., Ghani, A.C., Donnelly, C.A., Riley, S., Vollmer, M.A.C., Ferguson, N.M., Okell, L.C., Bhatt, S., Team, I.C.C..R.: Estimating the effects of nonpharmaceutical interventions on COVID19 in Europe. Nature 584(7820), 257–261 (2020). https://doi.org/10.1038/s4158602024057
Ritchie, H., Mathieu, E., RodésGuirao, L., Appel, C., Giattino, C., OrtizOspina, E., Hasell, J., Macdonald, B., Beltekian, D., Roser, M.: Coronavirus Pandemic (COVID19). https://ourworldindata.org/coronavirus
Mathieu, E., Ritchie, H., OrtizOspina, E., Roser, M., Hasell, J., Appel, C., Giattino, C., RodésGuirao, L.: A global database of COVID19 vaccinations. Nat. Hum. Behav. 5(7), 947–953 (2021). https://doi.org/10.1038/s41562021011228
Hu, H., Nigmatulina, K., Eckhoff, P.: The scaling of contact rates with population density for the infectious disease models. Math. Biosci. 244(2), 125–134 (2013). https://doi.org/10.1016/j.mbs.2013.04.013
Read, J.M., Bridgen, J.R.E., Cummings, D.A.T., Ho, A., Jewell, C.P.: Novel coronavirus 2019nCoV (COVID19): early estimation of epidemiological parameters and epidemic size estimates. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 376(1829), 20200265 (2021). https://doi.org/10.1098/rstb.2020.0265
Yap, F.F., Yong, M.: Implementation of a realtime, datadriven online epidemic calculator for tracking the spread of COVID19 in Singapore and other countries. Infect. Dis. Model. 6, 1159–1172 (2021). https://doi.org/10.1016/j.idm.2021.10.002
Rawat, S., Deb, S.: A spatiotemporal statistical model to analyze COVID19 spread in the USA. J. Appl. Stat. 50(11–12), 2310–2329 (2023). https://doi.org/10.1080/02664763.2021.1970122
Bergquist, S., Otten, T., Sarich, N.: COVID19 pandemic in the United States. Health Policy Technol. 9(4), 623–638 (2020). https://doi.org/10.1016/j.hlpt.2020.08.007
Unwin, H.J.T., Mishra, S., Bradley, V.C., Gandy, A., Mellan, T.A., Coupland, H., IshHorowicz, J., Vollmer, M.A.C., Whittaker, C., Filippi, S.L., Xi, X., Monod, M., Ratmann, O., Hutchinson, M., Valka, F., Zhu, H., Hawryluk, I., Milton, P., Ainslie, K.E.C., Baguelin, M., Boonyasiri, A., Brazeau, N.F., Cattarino, L., Cucunuba, Z., CuomoDannenburg, G., Dorigatti, I., Eales, O.D., Eaton, J.W., van Elsland, S.L., FitzJohn, R.G., Gaythorpe, K.A.M., Green, W., Hinsley, W., Jeffrey, B., Knock, E., Laydon, D.J., Lees, J., NedjatiGilani, G., Nouvellet, P., Okell, L., Parag, K.V., Siveroni, I., Thompson, H.A., Walker, P., Walters, C.E., Watson, O.J., Whittles, L.K., Ghani, A.C., Ferguson, N.M., Riley, S., Donnelly, C.A., Bhatt, S., Flaxman, S.: Statelevel tracking of COVID19 in the United States. Nat. Commun. 11(1), 6189 (2020). https://doi.org/10.1038/s41467020196526
Billah, M.A., Miah, M.M., Khan, M.N.: Reproductive number of coronavirus: a systematic review and metaanalysis based on global level evidence. PLoS ONE 15(11), 1–17 (2020). https://doi.org/10.1371/journal.pone.0242128
Lewes, F.M.M.: A note on the origin of the net reproduction ratio. Popul. Stud. 38(2), 321–324 (1984). Accessed 20230813
Levin, B.R., Bull, J.J., Stewart, F.M.: The intrinsic rate of increase of HIV/AIDS: epidemiological and evolutionary implications. Math. Biosci. 132(1), 69–96 (1996). https://doi.org/10.1016/00255564(95)000534
Li, Q., Guan, X., Wu, P., Wang, X., Zhou, L., Tong, Y., Ren, R., Leung, K.S.M., Lau, E.H.Y., Wong, J.Y., Xing, X., Xiang, N., Wu, Y., Li, C., Chen, Q., Li, D., Liu, T., Zhao, J., Liu, M., Tu, W., Chen, C., Jin, L., Yang, R., Wang, Q., Zhou, S., Wang, R., Liu, H., Luo, Y., Liu, Y., Shao, G., Li, H., Tao, Z., Yang, Y., Deng, Z., Liu, B., Ma, Z., Zhang, Y., Shi, G., Lam, T.T.Y., Wu, J.T., Gao, G.F., Cowling, B.J., Yang, B., Leung, G.M., Feng, Z.: Early transmission dynamics in Wuhan, China, of novel coronavirus–infected pneumonia. N. Engl. J. Med. 382(13), 1199–1207 (2020). https://doi.org/10.1056/NEJMoa2001316