Global dynamic analyzes of the discrete SIS models with application to daily reported cases

Abstract

Emerging infectious diseases, such as COVID-19, manifest in outbreaks of varying magnitudes. For large-scale epidemics, continuous models are often employed for forecasting, while discrete models are preferred for smaller outbreaks. We propose a discrete susceptible-infected-susceptible model that integrates interaction between parasitism and hosts, as well as saturation recovery mechanisms, and undertake a thorough theoretical and numerical exploration of this model. Theoretically, the model incorporating nonlinear recovery demonstrates complex behavior, including backward bifurcations and the coexistence of dual equilibria. And the sufficient conditions that guarantee the global asymptotic stability of a disease-free equilibrium have been obtained. Considering the challenges posed by saturation recovery in theoretical analysis, we then consider the case of linear recovery. Bifurcation analysis for of the linear recovery model displays a variety of bifurcations at the endemic equilibrium, such as transcritical, flip, and Neimark–Sacker bifurcations. Numerical simulations reveal complex dynamic behavior, including backward and fold bifurcations, periodic windows, period-doubling cascades, and multistability. Moreover, the proposed model could be used to fit the daily COVID-19 reported cases for various regions, not only revealing the significant advantages of discrete models in fitting, evaluating, and predicting small-scale epidemics, but also playing an important role in evaluating the effectiveness of prevention and control strategies. Furthermore, sensitivity analyses for the key parameters underscore their significant impact on the effective reproduction number during the initial months of an outbreak, advocating for better medical resource allocation and the enforcement of social distancing measures to curb disease transmission.

1 Introduction

Entering the new century, humanity has been recurrently challenged by the outbreak of major infectious diseases like SARS, H1N1, and COVID-19. These diseases have been characterized by rapid transmission, high mortality rates, and considerable impacts on global health infrastructure and economies. Mathematical modeling has become indispensable in the study of infectious diseases. Given the short life cycles of many pathogens, non-overlapping generations, and the small scale of each outbreak, discrete models are more practical. In such circumstances, there is an urgent need to develop suitable discrete models for analyzing interspecies interactions and disease propagation. Compared to their continuous counterparts [1–5], discrete models are capable of demonstrating a richer array of dynamical behaviors, which present further complexities for theoretical analysis. Notably, Allen [6] pointed out that in discrete susceptible-infected (SI) and susceptible-infected-susceptible (SIS) models, periodic doubling and chaotic behavior can occur. Xiang et al. [7] studied a model that incorporates Ricker-type recruitment and disease-induced mortality:

$$ \begin{aligned} S_{t+1}&=N_{t}e^{r-N_{t}}+\gamma _{1}e^{- \frac{\alpha I_{t}}{N_{t}}}S_{t} + \gamma _{2 }(1-\sigma )I_{t}, \\ I_{t+1}&=\gamma _{1} (1-e^{- \frac{\alpha I_{t}}{N_{t}}})S_{t} + \gamma _{2} \sigma I_{t}. \end{aligned} $$

(1.1)

In their model, $S_{t}$ and $I_{t}$ are defined as the susceptible and infectious populations in each generation t respectively [7]. Here, r denotes the intrinsic growth rate, $\gamma _{1}$ and $\gamma _{2}$ represent the survival probabilities of susceptible and infectious individuals respectively. Additionally, $1- \sigma$ is the recovery probability of infectious individuals. They demonstrate this model can undergo complex bifurcations such as transcritical and flip bifurcations, as well as Neimark–Sacker bifurcations. However, the biggest assumption of the above model is that the population from generation t to generation $t+1$ does not participate in the disease transmission process (i.e., the term $N_{t}e^{r-N_{t}}$), resulting in the superposition of the population numbers of two different generations. Notably, one of local asymptotic stability conditions for the disease-free equilibrium that may result in the intrinsic growth rate r greater than 2—a condition that contradicts the well-established stability criteria of the Ricker model [8].

In this paper, we consider a discrete SIS model with population dynamics, specifically focusing on characterizing nonlinear recovery mechanisms:

$$ \begin{aligned} S_{t+1}&=e^{r(1- \frac{S_{t}+I_{t}}{K})}e^{- \beta I_{t}}S_{t}+ \frac{1-\sigma}{1+\sigma _{1}I_{t}}I_{t}, \\ I_{t+1}&=(1-e^{- \beta I_{t}})S_{t}+\left (1- \frac{1-\sigma}{1+\sigma _{1}I_{t}}\right )I_{t}. \end{aligned} $$

(1.2)

Here, K is the carrying capacity, β is the transmission constant, and the term $e^{-\beta I_{t}}$ denotes the probability of susceptible individuals avoiding infection. The expression $\frac{1-\sigma}{1+\sigma _{1}I_{t}}$ represents the daily probability of an infected individual recovering, where $\sigma _{1}$ is the parameter governing the saturation effect of the recovery rate. This saturation phenomenon is influenced by factors such as the availability of healthcare resources, treatment effectiveness, and immune response. As the number of infections increases, the recovery rate gradually decreases.

In population dynamics, the interaction between parasitism and hosts is a critical factor influencing both population size and structure. Numerous studies [9–13] have explored this interactions using mathematical models, including differential equation and difference equation models. Among these, May’s work [14] is particularly enlightening, indicating that density-dependent host-parasite models can exhibit a range of bifurcations, including transcritical, flip, or Neimark–Sacker bifurcations. Additionally, the limitation of medical resources leads to the recovery rate appearing as a nonlinear function with a saturation effect. Non-linear recovery can facilitate backward bifurcations, a phenomenon where disease eradication is not assured even when the basic reproduction number $R_{0}$ is less than 1. The concept of backward bifurcation has been deeply explored in continuous models by Castillo–Chavez and Song [15], and extended to discrete models by Roumen Anguelov et al. [16]. Furthermore, Sulayman and Abdullah’s work [17] on a tuberculosis model with a saturated recovery function illustrates the potential for both backward and Hopf bifurcations, emphasizing the critical role of saturation parameters in driving these intricate dynamics. Tangjuan Li and Yanni Xiao [18] demonstrated various complex dynamical behaviors in their nonlinear recovery model, including backward bifurcation, Hopf bifurcation, saddle-node bifurcation, homoclinic bifurcation, and unstable limit cycle.

The main aim of our study is to examine the complex dynamics of the SIS model with nonlinear saturating recovery and to investigate the impact of saturation recovery on disease transmission. The organization of this paper is as follows: Sect. 2 provides a thorough theoretical analysis of the dynamics of model (1.2), demonstrating the existence of backward bifurcations and the possibility of coexisting dual positive equilibria. This section also compares these results with models that do not consider saturation recovery effects, offering a detailed exploration of the endemic equilibrium where various types of bifurcations, such as fold, transcritical [19], and Neimark–Sacker bifurcations may occur. Section 3 focuses on numerical simulations that expose complex dynamical behaviors. These simulations reveal a range of phenomena including backward bifurcations, fold bifurcations, period-doubling bifurcations, subcritical bifurcations, multiple attractors [20], Neimark–Sacker bifurcations and chaotic behaviors. In Sect. 4, we apply these models to actual epidemiological data to demonstrate their effectiveness. We further focus on the important relationship between parameter estimation values and the effectiveness of prevention and control strategies, in order to reveal the important role of non-pharmacological interventions in COVID-19 epidemic prevention and control. A sensitivity analysis follows, quantitatively evaluating how parameters influence the effective reproduction number of infections and the number of infections preceding epidemic outbreaks. Section 5 provides a discussion about this article.

2 Dynamical analysis of the system

2.1 The existence and stability of equilibria

Obviously, model (1.2) has a disease-free equilibrium $E_{0}=(K, 0)$. Employing the methodology outlined in the work of van den Driessche and Watmough [21], we obtain the basic reproduction number $R_{0}$ of model (1.2) is as follows,

$$ R_{0}=\frac{\beta K}{1-\sigma}. $$

The local stability of the disease-free equilibrium is as shown in the following theorem.

Theorem 2.1

The disease-free equilibrium $E_{0}$ is locally asymptotically stable if $R_{0}<1$ and $0 < r < 2$, and unstable if $R_{0}>1$. The bifurcation at $R_{0}=1$ is backward when $\sigma _{1} >\tilde{\sigma}_{1}$; the bifurcation at $R_{0}=1$ is forward when $\sigma _{1}<\tilde{\sigma}_{1}$, which means when $R_{0}=1$, $E_{0}$ is locally asymptotically stable if $\sigma _{1}<\tilde{\sigma}_{1}$ and unstable if $\sigma _{1} >\tilde{\sigma}_{1}$, where $\tilde{\sigma}_{1}=\frac{\beta ^{2}K+2\beta}{2(1-\sigma )}$.

Proof

The Jacobian matrix associated with the linearization of system (1.2) at $E_{0}$ is

$$ J(E_{0})=\left ( \textstyle\begin{array}{c@{\quad}c} -r+1&-\beta K-r+1-\sigma \\ 0& \beta K+\sigma \end{array}\displaystyle \right ). $$

The eigenvalues of the Jacobian matrix of the model (1.2) at the fixed point $E_{0}$ are $\lambda _{1}= 1-r$ and $\lambda _{2}=K\beta +\sigma$. These indicate that when $0 < r < 2$, then $|\lambda _{1}| < 1$, and when $R_{0}< 1$, then $|\lambda _{2}| < 1$. In summary of the aforementioned discussion, we can deduce that $E_{0}$ is locally asymptotically stable if $0 < r < 2$ and $R_{0}< 1$.

When $R_{0}=1$, 1 is a eigenvalue of $J(E_{0})$ and the other eigenvalue of $J(E_{0})$ has negative real part. Consequently, $E_{0}$ is a non-hyperbolic equilibrium, leading to inconclusive stability determination through linearization. However, one of the applications of the central manifold theorem is to determine the stability of equilibrium states. Below, we employ it to analyze the stability of $E_{0}$. Let $\hat{\sigma}=1-\beta K$, $\phi =\hat{\sigma}-\sigma$ be the bifurcation parameter. Introducing $x_{1}= S$, $x_{2}= I$, system (1.2) can be denoted as $X=(x_{1},x_{2})^{\prime }=(f_{1},f_{2})^{\prime }$. We denoted the left and right eigenvalues of $J(E_{0})$ as v and w respectively, where $v=(0, 1)^{\prime }$, $w=(-1, 1)^{\prime }$. Here, the left eigenvector v satisfies $vw = 1$. According to Theorem 2 in [16], the local dynamics near $R_{0}=1$ is determined by a and b:

$$ a=\sum ^{2}_{k,i,j=1}v_{k}w_{i}w_{j} \frac{\partial ^{2} f_{k}}{\partial x_{i}\partial x_{j}},\; b=\sum ^{2}_{k,i=1}v_{k}w_{i} \frac{\partial ^{2} f_{k}}{\partial x_{i}\partial \phi}. $$

The computation yields

$$ \frac{\partial ^{2} f_{2}}{\partial x_{1}\partial x_{2}}=\beta ,\; \frac{\partial ^{2} f_{2}}{\partial x_{2}\partial x_{1}}=\beta ,\; \frac{\partial ^{2} f_{2}}{\partial x_{2}^{2}}=-\beta ^{2}K+2(1- \sigma )\sigma _{1},\; \frac{\partial ^{2} f_{2}}{\partial x_{2}\partial \phi}=1. $$

The rest of the second derivatives of $f_{1}$ and $f_{2}$ are all zero. Hence,

$$ \begin{aligned} a&=w_{1}w_{2}\frac{\partial ^{2} f_{2}}{\partial x_{1}\partial x_{2}}+w_{2}w_{1} \frac{\partial ^{2} f_{2}}{\partial x_{2}\partial x_{1}}+w_{2}^{2} \frac{\partial ^{2} f_{2}}{\partial x_{2}^{2}} =-2\beta -\beta ^{2}K+2(1- \sigma ) \sigma _{1}, \\ b&=v_{2}w_{2}\frac{\partial ^{2} f_{2}}{\partial x_{2}\partial \phi}=1>0. \end{aligned} $$

Then, we have $a>0$ if and only if

$$ \sigma _{1}>\frac{\beta ^{2}K+2\beta}{2(1-\sigma )}\doteq \tilde{\sigma _{1}}. $$

(2.1)

So, we get system (1.2) undergoes backward bifurcation at $R_{0}=1$ when $\sigma _{1}>\tilde{\sigma _{1}}$; system (1.2) undergoes forward bifurcation at $R_{0}=1$ when $\sigma _{1}<\tilde{\sigma _{1}}$, that is to say when $R_{0}=1$, $E_{0}$ is locally asymptotically stable if $\sigma _{1}<\tilde{\sigma _{1}}$ and unstable if $\sigma _{1}>\tilde{\sigma _{1}}$. □

Below, we discuss the existence and local stability of endemic equilibrium.

Theorem 2.2

If $R_{0}>1$, system (1.2) has a unique endemic equilibrium, denoted by $E_{1}(S^{*},\;I^{*})$, which satisfies (2.2); If $R_{0}<1$, system (1.2) has no endemic equilibrium for $f_{min}>0$, but there exists at least two endemic equilibria for $f_{min}<0$.

Proof

The endemic equilibrium of system (1.2) satisfies:

$$ \begin{aligned} S=&e^{r(1- \frac{S+I}{K})}e^{- \beta I}S+ \frac{1-\sigma}{1+\sigma _{1}I}I, \\ I=&(1-e^{- \beta I})S+(1-\frac{1-\sigma}{1+\sigma _{1}I})I. \end{aligned} $$

(2.2)

Adding the above two equations and use the substitution $S=N-I$ yields $e^{-\beta I}(K-I)(e^{r(1-N/K)}-1)=0$, hence resulting in $N=K$. Then, substituting $N=K$ into the second equation of (2.2) yields

$$ \frac{1-\sigma}{1+\sigma _{1}I}I=(1-e^{- \beta I})(K-I). $$

(2.3)

Note $I = 0$ is a root of equation (2.3). Define $f_{1}(I)=\frac{1-\sigma}{1+\sigma _{1}I}I$, $f_{2}(I)=(1-e^{- \beta I})(K-I)$. Therefore, the existence of the root of equation (2.3) is equivalent to the the existence of intersection of $f_{1}(I)$ and $f_{2}(I)$. Algebraic calculation yields

$$\begin{aligned}& f'_{1}(I)=\frac{1-\sigma}{(1+\sigma _{1}I)^{2}}>0,\;f''_{1}(I)=- \frac{2(1-\sigma )\sigma _{1}}{(1+\sigma _{1}I)^{3}}< 0,\\& f'_{2}(I)=\beta e^{-\beta I}(K-I)-1+e^{-\beta I},\;f''_{2}(I)=-\beta ^{2} e^{-\beta I}(K-I)-2\beta e^{-\beta I}< 0. \end{aligned}$$

We observe that function $f_{1}(I)$ is a monotonically increasing concave function, while function $f_{2}(I)$ exhibits a concave shape characterized by an initial increase followed by a decrease. Consequently, when the slope of function $f_{1}(I)$ at the origin is less than that of function $f_{2}(I)$, denoted by $R_{0}>1$, there exists exactly one intersection point between functions $f_{1}(I)$ and $f_{2}(I)$. we denote the unique endemic equilibrium as $E_{1}(S^{*},I^{*})=(K-I^{*},I^{*})$.

When $R_{0}<1$, denote $f(I)=f_{1}(I)-f_{2}(I)$, we get $f(K)>0$ and $f'(I)>0$ for $I>K$, then $f(I)>0$ must hold true for $I>K$. Since $f_{1}(I)$ and $f_{2}(I)$ intersect at $I=0$ and the slope of $f_{1}(I)$ is smaller than that of $f_{2}(I)$, there must exist a $\tilde{I}\;\epsilon (0, K)$ such that $f(\tilde{I})>0$. Denote $f_{min}=min\{f(I)\}$, $f_{1}(I)$ and $f_{2}(I)$ have no intersection on $(0, K]$ when $f_{min}>0$. For $f_{min}<0$, there must be a $\hat{I}\;\epsilon (0, K)$ such that $f(\hat{I})<0$, so $f(0)f(\hat{I})<0$ and $f(\hat{I})f(k)<0$. Thus, according to the zero-point theorem, $f(I)$ has at least two positive equilibria. □

Next, we study the stability of the endemic equilibrium $E_{1}$.

Theorem 2.3

If $r< min\{r_{a},\;r_{b}\}$ and $R_{0}>1$, the unique endemic equilibrium of system (1.2) is locally asymptotically stable.

Proof

Define

$$ \begin{aligned} r_{a}&=\frac{2K}{e^{-\beta I^{*}}(K-I^{*})}, \\ r_{b}&=\frac{1}{(1-\sigma ) (K-I^{*})e^{-\beta I^{*}}}((-\beta \sigma _{1}^{2}{I^{*}}^{2}K(K-I^{*}) -2\beta \sigma _{1}I^{*}K(K-I^{*}) -\sigma _{1}^{2}{I^{*}}^{2}K \\ &-\beta K(K-I^{*}) -2\sigma _{1}I^{*}K-K)e^{-\beta I^{*}} +K((1+ \sigma _{1}I^{*})^{2}+(1-\sigma ))). \end{aligned} $$

The Jacobian matrix associated with the linearization of system (1.2) at $E_{1}$ is obtained as

$$ J(E_{1})=\left ( \textstyle\begin{array}{c@{\quad}c} -\frac{r}{K}e^{-\beta I^{*}}(K-I^{*})+e^{-\beta I^{*}}&(- \frac{r}{K}- \beta ) e^{-\beta I^{*}}(K-I^{*})+ \frac{1-\sigma}{(1+\sigma _{1}I^{*})^{2}} \\ 1-e^{-\beta I^{*}}& \beta e^{-\beta I^{*}}(K-I^{*})+1- \frac{1-\sigma}{(1+\sigma _{1}I^{*})^{2}} \end{array}\displaystyle \right ). $$

And the characteristic equation at the endemic equilibrium $E_{1}$ is expressed as $H(\lambda )=\lambda ^{2}-a_{1}\lambda +a_{2}$, where

$$ \begin{aligned} a_{1}&=(\beta -\frac{r}{K})e^{-\beta I^{*}}(K-I^{*})+e^{-\beta I^{*}}+1- \frac{1-\sigma}{(1+\sigma _{1}I^{*})^{2}}, \\ a_{2}&=(-\frac{r}{K}e^{-\beta I^{*}}(K-I^{*})+e^{-\beta I^{*}})( \beta e^{-\beta I^{*}}(K-I^{*})+1- \frac{(1-\sigma )}{(1+\sigma _{1}I^{*})^{2}})\\ &\quad -((- \frac{r}{K}-\beta ) e^{- \beta I^{*}}(K-I^{*}) \\ &\quad +\frac{1-\sigma}{(1+\sigma _{1}I^{*})^{2}}) (1-e^{-\beta I^{*}}). \end{aligned} $$

Further, we use the Jury criterion to determine the conditions required for local asymptotic stability of the endemic equilibrium $E_{1}$. Algebraic computation results in

$$\begin{aligned} \begin{aligned} &H(1)=-\frac{re^{-\beta I^{*}}(K-I^{*})}{K(1+\sigma _{1}I^{*})^{2}}((1+ \sigma _{1}I^{*})^{2}(\beta (K-I^{*})+1)e^{- \beta I^{*}}-(1+\sigma _{1}I^{*})^{2}-(1- \sigma )), \\ &H(-1)=- \frac{re^{- \beta I^{*}}(K-I^{*})-2K}{K(1+\sigma _{1}I^{*})^{2}}((1+\sigma _{1}I^{*})^{2}( \beta (K-I^{*})+1) e^{-\beta I^{*}}\\ &\quad +(1+\sigma _{1}I^{*})^{2}-(1- \sigma )), \\ &H(0)-1=-\frac{1}{K(1+\sigma _{1}I^{*})^{2}}(r(K-I^{*}) (\sigma _{1}I^{*}+1)^{2}( \beta (K-I^{*})+1)(e^{-\beta I^{*}})^{2}\\ &\quad + (-\beta \sigma _{1}^{2}{I^{*}}^{2}K(K-I^{*}) \\ & \quad -2\beta \sigma _{1}I^{*}K(K-I^{*}) -\sigma _{1}^{2}{I^{*}}^{2}K- \beta K(K-I^{*})-2\sigma _{1}I^{*}K-r(K-I^{*})(1-\sigma )-K)e^{- \beta I^{*}} \\ & \quad +K((1+\sigma _{1}I^{*})^{2}+(1-\sigma ))). \end{aligned} \end{aligned}$$

(2.4)

Since $-\frac{re^{-\beta I^{*}}(K-I^{*})}{K(1+\sigma _{1}I^{*})^{2}}<0$, $H(1)>0$ if and only if

$$ \begin{aligned} (1+\sigma _{1}I^{*})^{2}(\beta (K-I^{*})+1)e^{-\beta I^{*}}-(1+ \sigma _{1}I^{*})^{2}-(1-\sigma )< 0. \end{aligned} $$

(2.5)

(2.5) is equivalent to

$$ \begin{aligned} \beta (K-I^{*})e^{-\beta I^{*}}+e^{-\beta I^{*}}-1- \frac{1-\sigma}{(1+\sigma _{1}I^{*})^{2}}< 0. \end{aligned} $$

(2.6)

Equation (2.6) precisely represents the difference between the slopes of $f_{2}(I)$ and $f_{1}(I)$ at the endemic equilibrium point. It is easy to see that when $R_{0}>1$, the slope of $f_{1}(I)$ at the equilibrium point is always greater than that of $f_{2}(I)$, thus naturally validating (2.6).

Next, since $-\frac{1}{K(1+\sigma _{1}I^{*})^{2}}((1+\sigma _{1}I^{*})^{2} ( \beta (K-I^{*})+1)e^{-\beta I^{*}}+(1+\sigma _{1}I^{*})^{2}-(1- \sigma ))<0$, $H(-1)>0$ if and only if $r< r_{a}$. Finally, since $r(K-I^{*}) (\sigma _{1}I^{*}+1)^{2}(\beta (K-I^{*})+1)(e^{-\beta I^{*}})^{2}>0$, we have $H(0)-1<0$ if the inequality $(-\beta \sigma _{1}^{2}{I^{*}}^{2}K(K-I^{*})-2\beta \sigma _{1}I^{*}K(K-I^{*}) -\sigma _{1}^{2}{I^{*}}^{2}K-\beta K(K-I^{*})-2\sigma _{1}I^{*}K-r(K-I^{*})(1- \sigma )-K)e^{-\beta I^{*}} +K((1+\sigma _{1}I^{*})^{2}+(1-\sigma ))>0$ holds. This inequality is equivalent to $r< r_{b}$. Therefore, according to the Jury criterion, if $R_{0}>1$ and $r< min\{r_{a},\;r_{b}\}$ hold, the equilibrium is locally asymptotically stable. □

Following this, we analyze the global stability of the disease-free equilibrium. Define

$$ \hat{\sigma _{1}}= \frac{2(1-\sigma )}{2(1-\sigma )+\beta ^{2}K+2\beta}. $$

Theorem 2.4

If $\sigma _{1}<\hat{\sigma _{1}}$ and $R_{0}<\frac{r \hat{\sigma _{1}}}{e^{r-1}}$, the disease-free equilibrium of system (1.2) is globally asymptotically stable.

Proof

From the first equation of (1.2), we obtain $S_{t+1}=e^{r(1- \frac{S_{t}+I_{t}}{K})}e^{- \beta I_{t}}S_{t}\leq e^{r(1- \frac{S_{t}+I_{t}}{K})}S_{t}$ when $I_{t}=0$. Define $Y_{t}=e^{r(1- \frac{S_{t}+I_{t}}{K})}S_{t}$, then $Y_{t}\leq e^{r-1}\frac{K}{r}$ for $I\geq 0$. Consider a Lyapunov function: $V(S_{t}, I_{t})=I_{t}+1$. We have

$$ \begin{aligned} \vartriangle V(S_{t}, I_{t})&=V(S_{t+1}, I_{t+1})-V(S_{t}, I_{t}) \\ &=(1-e^{- \beta I_{t}})S_{t}-\frac{1-\sigma}{1+\sigma _{1}I_{t}}I_{t} \\ &\leq \beta I_{t}S_{t}-\frac{1-\sigma}{1+\sigma _{1}I_{t}}I_{t} \\ &\leq (\beta e^{r-1}\frac{K}{r} -\frac{1-\sigma}{1+\sigma _{1}I_{t}})I_{t} \\ &=(1-\sigma )(\frac{e^{r-1}}{r}R_{0}-\hat{\sigma _{1}})I_{t}. \end{aligned} $$

It follows from $R_{0}<\frac{r\hat{\sigma _{1}}}{e^{r-1}}$ that we have $\vartriangle V(S_{t}, I_{t})\leq 0$, and $\vartriangle V(S_{t}, I_{t})=0$ if and only if $I_{t}=0$. According to the comparison principle of difference equations [22] and LaSalle’s invariance principle [23], the disease-free equilibrium point is globally asymptotically stable. □

Due to the challenges that nonlinear recovery poses for theoretical analysis, we proceed to consider the scenario of $\sigma _{1} = 0$ in model (1.2).

When the recovery rate increases linearly, model (1.2) becomes

$$ \begin{aligned} S_{t+1}&=e^{r(1- \frac{S_{t}+I_{t}}{K})}e^{- \beta I_{t}}S_{t} + (1- \sigma )I_{t}, \\ I_{t+1}&=(1-e^{- \beta I_{t}})S_{t} + \sigma I_{t}. \end{aligned} $$

(2.7)

In the next step, we present the results on the existence and stability of fixed points of the discrete model (2.7). By computing the spectral radius of the next-generation matrix, it is easy to observe that the basic reproduction number and the disease-free equilibrium remain the same as those of model (1.2), and there is no change in the local asymptotic stability of the disease-free equilibrium. Further, we present the local stability of the endemic equilibrium $E_{2}(S_{*},\;I_{*})$, and denote $r_{0}=\frac{2Ke^{\beta I_{*}}}{ K-I_{*} }$.

Theorem 2.5

If $\sigma _{1} = 0$, $R_{0}>1$, $I_{*} < \frac{\beta K+1-Lambert\;W(0,\;2e^{\beta K +1})}{\beta}$, $\sigma <2-e^{-\beta I_{*}}(\beta (K-I_{*})+1)$ and $r< r_{0}$ hold, then the unique endemic equilibrium $E_{2}(S_{*},\;I_{*})$ of model (1.2) (or model (2.7)) is locally asymptotically stable.

Proof

The Jacobian matrix at the endemic equilibrium $E_{2}(S_{*},\;I_{*})$ is given by

$$ J(E_{2})=\left ( \textstyle\begin{array}{c@{\quad}c} (-\frac{rS_{*}}{K}+1)e^{r(1-\frac{S_{*}+I_{*}}{K})}e^{-\beta I_{*}} & (- \frac{rS^{*}}{K}-\beta S_{*})e^{r(1-\frac{S_{*}+I_{*}}{K})}e^{-\beta I_{*}}+(1- \sigma ) \\ 1-e^{-\beta I_{*}} & \beta e^{-\beta I_{*}}S_{*}+\sigma \end{array}\displaystyle \right ) $$

The two eigenvalues of the Jacobian matrix are $\lambda _{1}=1-\frac{r (K-I_{*})e^{-\beta I_{*}}}{K}$ and $\lambda _{2}=e^{-\beta I_{*}}(\beta (K-I_{*})+1)+\sigma -1$. Clearly, $\lambda _{1}<1$ and $\lambda _{2}>-1$ hold true. To ensure that both eigenvalues lie within the unit circle, it is necessary to satisfy $\lambda _{1}>-1$ and $\lambda _{2}<1$. Firstly, let $\lambda _{1}=1-\frac{r (K-I_{*})e^{-\beta I_{*}}}{K}>-1$, which holds if $r< r_{0}$. Secondly, let us now examine the requisite conditions to fulfill $\lambda _{2}<1$. It is easy to see $\lambda _{2}<1$ if $\beta e^{-\beta I_{*}}S_{*}+e^{-\beta I_{*}}+\sigma -2 <0$ (i.e., $\sigma <2-e^{-\beta I_{*}}(\beta (K-I_{*})+1)$). Given the values of σ range from 0 to 1, this criterion prompts a deeper analysis. Introducing the function $f_{4}(I)=2-e^{-\beta I}(\beta (K-I)+1)$, it becomes evident that $f_{4}(I)$ is consistently less than 1. The focus then shifts to identifying the conditions under which I causes $f_{4}(I)$ to be greater than 0. This involves determining the roots of I that satisfy $f_{4}(I)= 0$. Transforming the equation $2-e^{-\beta I}(\beta (K-I)+1)=0$, we arrive at $e^{-\beta I}(\beta (K-I)+1)=2$. Taking the logarithm of both sides leads to

$$ -\beta I+ln(\beta (K-I)+1)=ln2. $$

(2.8)

Then, we have $\beta (K-I)+1+ln(\beta (K-I)+1)=ln2+\beta K+1$. By applying the definition of the Lambert W function, we deduce that $\beta (K-I)+1=Lambert\;W(0, 2e^{\beta K+1})$. Thus, $I=\frac{\beta K+1-Lambert\;W(0,\;2e^{\beta K +1})}{\beta}$ when $f_{4}(I)= 0$. Observing equation (2.8), it is clear that that the left side of the equation is a decreasing function of I, indicating that $f_{4}(I)>0$ when $I< \frac{\beta K+1-Lambert\;W(0,\;2e^{\beta K +1})}{\beta}$. From this discussion, we conclude that $\lambda _{2}>-1$ if $R_{0}> 1$, $I_{*} <\frac{\beta K+1-Lambert\;W(0,\;2e^{\beta K +1})}{\beta}$ and $0<\sigma <2-e^{-\beta I_{*}}(\beta (K-I_{*})+1)$.

Therefore, $E_{2}(S_{*},\;I_{*})$ is locally asymptotically stable provided that $I_{*} < \frac{\beta K+1-Lambert\;W(0,\;2e^{\beta K +1})}{\beta}$, $0<\sigma <2-e^{-\beta I_{*}}(\beta (K-I_{*})+1)$ and $r< r_{0}$. □

To further analyze the endemic equilibrium, we provide the characteristic equation as $H(\lambda )=\lambda ^{2}-p(S_{*},\;I_{*})\lambda +q(S_{*},\;I_{*})=0$, and express the two roots of this equation as $\lambda _{1}$ and $\lambda _{2}$, where

$$ \begin{aligned} p(S_{*},\;I_{*})&={ \frac {\beta \,{{\mathrm{e}}^{-\beta I_{*}}}S_{*}K-r{{\mathrm{e}}^{-\beta I_{*}}}S_{*}+K{{\mathrm{e}}^{-\beta I_{*}}}+K\sigma}{K}}, \\ q(S_{*},\;I_{*})&={ \frac {{{\mathrm{e}}^{-\beta I_{*}}} \left ( -rS_{*}+K \right ) \left ( \beta \,{ {\mathrm{e}}^{-\beta I_{*}}}S_{*}+\sigma \right ) }{K}}\\ &\quad -{ \frac { \left ( -1+{{\mathrm{e}}^{ -\beta I_{*}}} \right ) \left ( {{\mathrm{e}}^{-\beta I_{*}}}S_{*}\beta \,K+r{{\mathrm{e}}^{-\beta I_{*} }}S_{*}+K\sigma -K \right ) }{K}}. \end{aligned} $$

Then, define

$$\begin{aligned} r_{1}&={ \frac {K \left ( \left ( K\beta -\beta \,I_{*}+1 \right ) {{\mathrm{e}}^{-\beta \,I_{*}}}+\sigma -2 \right ) }{{{\mathrm{e}}^{-\beta \,I_{*}}} \left ( K-I_{*} \right )\left ( \left ( K\beta -\beta \,I_{*}+1 \right ) {{\mathrm{e}}^{-\beta \,I_{*}}}+\sigma -1 \right ) }}, \\ r_{2}&={ \frac {K \left ( 1+ \left ( K-I_{*} \right ) \beta \right ) }{K-I_{*}}}+{ \frac {K\left ( \sigma -2 \right ) {{\mathrm{e}}^{\beta \,I_{*}}}}{K-I_{*}}}, \\ r_{3}&={ \frac {K \left ( 1+ \left ( K-I_{*} \right ) \beta \right ) }{K-I_{*}}}+{ \frac{K \left ( \sigma +2 \right ) {{\mathrm{e}}^{\beta \,I_{*}}}}{K-I_{*}}}, \\ r_{4}&={ \frac {K \left ( 1+ \left ( K-I_{*} \right ) \beta \right ) }{K-I_{*}}}+{ \frac {K \left ( \sigma -1 \right ) }{{{\mathrm{e}}^{-\beta \,I_{*}}} \left ( K-I_{*} \right ) }}, \\ r_{5}&={ \frac {K \left ( 1+ \left ( K-I_{*} \right ) \beta \right ) }{K-I_{*}}}+{ \frac {K \sigma }{{{\mathrm{e}}^{-\beta \,I_{*}}} \left ( K-I_{*} \right ) }}. \end{aligned}$$

Theorem 2.6

When $\sigma _{1}=0$, model (1.2) has the following results:

• (a) if $I_{*} <\frac{\beta K+1-Lambert\;W(0,\;2e^{\beta K +1})}{\beta}$, $0<\sigma <2-e^{-\beta I_{*}}(\beta (K-I_{*})+1)$ and $r< r_{0}$, then $|\lambda _{1,2}|<1$ and $E_{2}(S_{*},\;I_{*})$ is a stable node or focus;

• (b) if $I_{*} <\frac{\beta K+1-Lambert\;W(0,\;2e^{\beta K +1})}{\beta}$, $\sigma =2-e^{-\beta I_{*}}(\beta (K-I_{*})+1)$ and $r\neq r_{0}$, then $|\lambda _{1}|\neq 1$, $\lambda _{2}=1$ and $E_{2}(S_{*},\;I_{*})$ is non-hyperbolic;

• (c) if $r =r_{0}$, $I_{*} < \frac{\beta K+1-Lambert\;W(0,\;2e^{\beta K +1})}{\beta}$ and $\sigma \neq 2-e^{-\beta I_{*}}(\beta (K-I_{*})+1)$, then $\lambda _{1}=-1$, $|\lambda _{2}|\neq 1$ and $E_{2}(S_{*},\;I_{*})$ is non-hyperbolic;

• (d) if $r=r_{0}$, $I_{*} <\frac{\beta K+1-Lambert\;W(0,\;2e^{\beta K +1})}{\beta}$ and $\sigma =2-e^{-\beta I_{*}}(\beta (K-I_{*})+1)$, then $\lambda _{1}=-1$, $\lambda _{2}=1$, and $E_{2}(S_{*},\;I_{*})$ is non-hyperbolic;

• (e) if $r =r_{1}$ and $r_{2}< r< r_{3}$, then $\lambda _{1,2}$ are a pair of conjugate complex roots on the unit circle and $E_{2}(S_{*},\;I_{*})$ is non-hyperbolic.

Proof

For real eigenvalues, to ensure that their modulus is 1, they can only take the values of 1 or −1. According to the proof of Theorem 2.5, we know $\lambda _{1}<1$ and $\lambda _{2}>-1$, thus the possible scenarios to ensure that the modulus of the eigenvalues is 1 are narrowed down to $\lambda _{1}=-1$ and $\lambda _{2}=1$. Substituting $\lambda _{1}=-1$ yields $\frac{(\beta e^{-\beta I_{*}}S_{*}+e^{-\beta I_{*}}+\sigma )(-re^{-\beta I_{*}}S_{*}+2K)}{K}=-1$. Because of $\frac{\beta e^{-\beta I_{*}}S_{*}+e^{-\beta I_{*}}+\sigma}{K}>0$, there is

$$ \lambda _{1}=\left \{ \begin{aligned} =-1,\quad \quad &r=r_{0}, \\ \neq -1,\quad \quad &r\neq r_{0}. \end{aligned} \right . $$

Substituting $\lambda _{2}=1$ yields $1-\frac{r (K-I_{*})e^{-\beta I_{*}}}{K}=1$. So,

$$\begin{aligned} \lambda _{2}=\left \{ \begin{aligned} \neq 1,\quad \quad & I_{*} < \frac{\beta K+1-Lambert\;W(0,\;2e^{\beta K +1})}{\beta}\quad and \quad \sigma \neq 2-e^{-\beta I_{*}}(\beta (K-I_{*})+1), \\ =1,\quad \quad & I_{*} < \frac{\beta K+1-Lambert\;W(0,\;2e^{\beta K +1})}{\beta}\quad and \quad \sigma =2-e^{-\beta I_{*}}(\beta (K-I_{*})+1). \end{aligned} \right . \end{aligned}$$

From this, we can draw conclusions from (a) to (d). For (e), if $-2<-(\lambda _{1}+\lambda _{2})<2$ and $\lambda _{1}\lambda _{2}=1$ hold, the characteristic equation possesses a pair of conjugate complex roots. Computing $-2<-(\lambda _{1}+\lambda _{2})<2$ and $\lambda _{1}\lambda _{2}=1$ is respectively equivalent to $r_{2}< r< r_{3}$ and $r=r_{1}$. □

Remark 1

We investigate two specific cases of (e), namely $\lambda _{1,2}=\frac{-1\pm \sqrt{3}i}{2}$ and $\lambda _{1,2}=\pm \ i$. When $\lambda _{1,2}=\frac{-1\pm \sqrt{3}i}{2}$, then we obtain $-(\lambda _{1}+\lambda _{2})=-1$ and $\lambda _{1}\lambda _{2}=1$, that is $r =r_{1}=r_{4}$. When $\lambda _{1,2}=\pm \ i$, we obtain $-(\lambda _{1}+\lambda _{2})=0$ and $\lambda _{1}\lambda _{2}=1$, that is $r=r_{1}=r_{5}$.

2.2 Bifurcations

In this section, we study the existence of flip, transcritical and Neimark–Sacker bifurcations at the unique endemic equilibrium $E_{2}(S_{*},\;I_{*})$ when $R_{0 }> 1$ and $\sigma _{1}=0$, employing methodologies consistent with those outlined in reference [7].

We choose the intrinsic growth rate r as a bifurcation parameter to study the flip bifurcation of the endemic equilibrium $E_{2}(S_{*},\;I_{*})$ by using the center manifold theorem [24–27].

Theorem 2.7

If $R_{0} > 1$, $r = r_{0}$ and $\sigma _{1}=0$, then the model (1.2) or model (2.7) has a unique and non-hyperbolic endemic equilibrium $E_{2}(S_{*},\;I_{*})$. Moreover, if $I_{*} < \frac{\beta K+1-Lambert\;W(0,\;2e^{\beta K +1})}{\beta}$, $\sigma \neq 2-e^{-\beta I_{*}}(\beta (K-I_{*})+1)$, $c_{101}\neq 0$ and $c_{200}^{2}+\frac{c_{110}d_{200}}{p_{1}}\neq 0$, then a flip bifurcation occurs at $E_{2}$ as r passes through $r_{0}$.

Proof

Make $p_{1}$ be the value of p and $q_{1}$ be the value of q when $r=r_{0}$. Let

$$ \begin{aligned} p_{1} &=-\beta (K-I_{*}) e^{-\beta I_{*}}- e^{-\beta I_{*}}-\sigma +2, \\ q_{1} &=-\beta (K-I_{*}) e^{-\beta I_{*}} - e^{-\beta I_{*}}-\sigma +1. \end{aligned} $$

(2.9)

Then, the model (2.7) reduced on the center manifold is

$$ \begin{aligned} X\rightarrow {f }(X,\;Z)&=-X+c_{200}X^{2}+c_{101}XZ+ \frac{c_{110}d_{200}}{p_{1}}X^{3}+(\frac{c_{011}d_{200}}{p_{1}} \\ &\quad - \frac{c_{110}d_{101}}{2-p_{1}})X^{2}Z- \frac{c_{011}d_{101}}{2-p_{1}}XZ^{2}+\mathcal{O}(|X,\;Z)|^{4}), \end{aligned} $$

where

$$\begin{aligned} c_{200}&=-\frac{1}{a_{010}(p_{1}-2)(b_{010}+p_{1}-1)}(a_{020}(b_{010}+1)(b_{010}+p_{1}-1)^{2} \\ &\quad +a_{010}(a_{010}(a_{200}(b_{010}+1) \\ &\quad -b_{110}(b_{010}+p_{1}-1))+a_{110}(p_{1}+b_{010}^{2}+b_{010}p_{1}-1) -b_{020}(b_{010}+p_{1}-1)^{2})), \\ c_{020}&=-\frac{1}{a_{010}(p_{1}-2)(b_{010}+1)}(b_{010}+p_{1}-1)(a_{020}(b_{010}+1)^{2} \\ &\quad -a_{010}((b_{020}-a_{110})(b_{010}+1) \\ &\quad -a_{010}(a_{200}-b_{110})), \\ c_{110}&=-\frac{1}{a_{010}(p_{1}-2)(b_{010}+1)}(2a_{020}(b_{010}+1)^{2}(b_{010}+p_{1}-1) \\ &\quad +a_{010}(a_{010}(2a_{200}(b_{010}+1) \\ &\quad -b_{110}(2b_{010}+p_{1}))+(b_{010}+1)(a_{110}(2b_{010}+p_{1}) -2b_{020}(b_{010}+p_{1}-1)))), \\ c_{101}&=- \frac{a_{101}(b_{010}+1)(a_{010}+b_{010}+p_{1}-1)}{a_{010}(p_{1}-2)}, \\ c_{011}&=- \frac{a_{101}(a_{010}+b_{010}+1)(b_{010}+p_{1}-1)}{a_{010}(p_{1}-2)}, \\ d_{200}&=\frac{1}{a_{010}(p_{1}-2)(b_{010}+p_{1}-1)}(a_{020}(b_{010}+1)(b_{010}+p_{1}-1)^{2} +a_{010}(a_{010}(a_{200} \\ &\quad -b_{110})(b_{010}+1)+(a_{110}-b_{020})(b_{010}^{2}+b_{010}p_{1}+p_{1}-1))), \\ d_{020}&=\frac{1}{a_{010}(p_{1}-2)(b_{010}+1)}(a_{020}(b_{010}+1)^{2}(b_{010}+p_{1}-1) +a_{010}(-b_{020}(b_{010}+1)^{2} \\ &\quad +a_{010}(a_{200}(b_{010}+p_{1}-1)-b_{110}(b_{010}+1))+a_{110}(b_{010}^{2} +b_{010}p_{1}+p_{1}-1))), \\ d_{110}&=\frac{1}{a_{010}(p_{1}-2)(b_{010}+p_{1}-1)}(2a_{020}(b_{010}+1)(b_{010}+p_{1}-1)^{2} +a_{010}((b_{010}+p_{1}-1) \\ &\quad (-2b_{020}(b_{010}+1) +a_{110}(2b_{010}+p_{1}))+a_{010}(2a_{200}(b_{010} +p_{1}-1)\\ &\quad -b_{110}(2b_{010}+p_{1}))), \\ d_{101}&= \frac{a_{101}(b_{010}+1)(a_{010}+b_{010}+p_{1}-1)}{a_{010}(p_{1}-2)}, \\ d_{011}&= \frac{a_{101}(a_{010}+b_{010}+1)(b_{010}+p_{1}-1)}{a_{010}(p_{1}-2)}, \end{aligned}$$

and

$$ \begin{aligned} a_{010}&=-(\frac{2K\beta}{r_{0}}+\sigma +1),\;\;\; &a_{200}&= \frac{r_{0}}{K}-\frac{2}{K-I_{*}}, \\ a_{020}&=\frac{K\beta ^{2}}{r_{0}}+2\beta +\frac{r_{0}}{K},\;\;\; &a_{110}&=2 \frac{r_{0}}{K}+2\beta -\frac{2}{K-I_{*}}-\beta e^{-\beta I_{*}}, \\ a_{101}&=-\frac{2}{r_{0}},\;\;\; &b_{010}&=\frac{2K\beta}{r_{0}}+ \sigma , \\ b_{020}&=-\frac{K\beta ^{2}}{r_{0}},\;\;\; &b_{110}&=\beta e^{-\beta I_{*}}. \end{aligned} $$

The computation yields

$$ \begin{aligned} &\frac{\partial {f }}{\partial X}(0,\;0)=-1,\; \frac{\partial ^{2} {f }}{\partial Z\partial X}(0,\;0)=c_{101},\\ & -3( \frac{\partial ^{2} {f }}{\partial X^{2}}(0,\;0))^{2}-2 \frac{\partial ^{3} {f }}{\partial X^{3}}(0,\;0)= -12(c_{200}^{2}+ \frac{c_{110}d_{200}}{p_{1}}){.} \end{aligned} $$

Based on the conclusions drawn in references [26, 27], the model (2.7) undergoes a flip bifurcation if $c_{101}\neq 0$ and $c_{200}^{2}+\frac{c_{110}d_{200}}{p_{1}}\neq 0$. Hence, the corresponding results are obtained. □

We choose the recovery rate σ as a bifurcation parameter to study the transcritical bifurcation of the endemic equilibrium $E_{2}(S_{*},\;I_{*})$.

Theorem 2.8

If $R_{0} > 1$, $I_{*} <\frac{\beta K+1-Lambert\;W(0,\;2e^{\beta K +1})}{\beta}$, $\sigma = \sigma _{0}$ and $r\neq r_{0}$, then the model has a unique and non-hyperbolic endemic equilibrium $E_{2}(S_{*},\;I_{*})$, whose eigenvalues are $\lambda _{2}=1$ and $\lambda _{1}\neq \pm 1$. Moreover, if $g_{101}\neq 0$ and $g_{200}\neq 0$ also hold, then system (2.7) undergoes a transcritical bifurcation at $E_{2}$.

Proof

Make $p_{2}$ be the value of p and $q_{2}$ be the value of q when $I_{*} <\frac{\beta K+1-Lambert\;W(0,\;2e^{\beta K +1})}{\beta}$ and $\sigma =\sigma _{0}$. Through the calculation, we get the results below:

$$ \begin{aligned} p_{2}&=\frac {r e^{-\beta I_{*}} (K-I_{*})-2K}{K}, \\ q_{2}&=-\frac {r e^{-\beta I_{*}} (K-I_{*})-K}{K}, \\ \sigma _{0}&=2-e^{-\beta I_{*}}(\beta (K-I_{*})+1). \end{aligned} $$

(2.10)

At this moment, the Jacobian matrix exhibits two eigenvalues, $\lambda _{2}=1$ and $\lambda _{1}\neq \pm 1$, at the endemic equilibrium. The model (2.7) reduced to the center manifold is

$$ \begin{aligned} X\rightarrow \tilde{f }(X,\;\mu )&=X+g_{200}X^{2}+g_{101}X\mu + \frac{g_{110}h_{200}}{2+p_{2}}X^{3}+(\frac{g_{011}h_{200}}{2+p_{2}} \\ &\quad + \frac{g_{110}h_{101}}{2+p_{2}})X^{2}\mu + \frac{g_{011}h_{101}}{2+p_{2}}X\mu ^{2}+\mathcal{O}(|X,\;\mu )|^{3}), \end{aligned} $$

where

$$\begin{aligned} g_{200}&=-\frac{1}{e_{010}(p_{2}+2)(e_{100}-1)}(e_{020}(e_{100}-1)^{2}(e_{100}+p_{2}+1) \\ &\quad +e_{010}(e_{010}(e_{200}(e_{100}+p_{2}+1) \\ &\quad -f_{110}(e_{100}-1))+f_{020}(e_{100}-1)^{2}-e_{110}(e_{100}-1)(p_{2}+1 +e_{100}))), \\ g_{020}&=-\frac{e_{100}-1}{e_{010}(p_{2}+2)(e_{100}+p_{2}+1)}(e_{020}(e_{100}+p_{2}+1)^{2} \\ &\quad +e_{010}((e_{100}+p_{2}+1)(f_{020}-e_{110}) \\ &\quad +e_{010}(e_{200}-f_{110}))), \\ g_{110}&=-\frac{1}{e_{010}(p_{2}+2)(e_{100}+p_{2}+1)}(2e_{020}(e_{100}+p_{2}+1)^{2}(e_{100}-1) \\ &\quad +e_{010}(e_{010}(-f_{110}(2e_{100}+p_{2})+ \\ &\quad 2e_{200}(e_{100}+p_{2}+1))+(-e_{110}(2e_{100}+p_{2}) +2f_{020}(e_{100}-1))(e_{100}+p_{2}+1))), \\ g_{101}&=g_{011}=- \frac{(e_{100}-1)(e_{011}(e_{100}+p_{2}+1)+e_{010}f_{011})}{e_{010}(p_{2}+2)}, \\ h_{200}&=\frac{1}{e_{010}(p_{2}+2)(- 1+e_{100})}(e_{100}+p_{2}+1)(e_{020}(e_{100}-1)^{2} \\ &\quad +e_{010}(e_{010}(e_{200}-f_{110})+(f_{020}-e_{110}) \\ &\quad (e_{100}-1))), \\ h_{020}&=\frac{1}{e_{010}(p_{2}+2)(e_{100}+p_{2}+1)}(e_{020}(e_{100}+p_{2}+1)^{2}(e_{100}-1) \\ &\quad +e_{010}(f_{020}(e_{100}+p_{2}+1)^{2} \\ &\quad +e_{010}(e_{200}(e_{100}-1)-f_{110}(e_{100}+p_{2}+1)) -e_{110}(e_{100}-1)(e_{100}+p_{2}+1))), \\ h_{110}&=\frac{1}{e_{010}(p_{2}+2)(e_{100}-1)}(2e_{020}(e_{100}+p_{2}+1)(e_{100}-1)^{2} \\ &\quad +e_{010}(2f_{020}(e_{100}+p_{2}+1)(e_{100}-1) \\ &\quad -e_{110}(e_{100}-1) (2e_{100}+p_{2}) +e_{010}(2e_{200}(e_{100}-1) -f_{110}(2e_{100}+p_{2})))), \\ h_{101}&=h_{011}= \frac{(e_{100}+p_{2}+1)(e_{011}(e_{100}-1) +e_{010}f_{011})}{e_{010}(p_{2}+2)}, \end{aligned}$$

and

$$ \begin{aligned} e_{100}&=e^{-\beta I_{*}}-\frac{re^{-\beta I_{*}}(K-I_{*})}{K},\;\;\; &e_{010}&=- \frac{((K-I_{*})r-K)e^{-\beta I_{*}}}{K}-1, \\ e_{200}&=\frac{((K-I_{*})r-2K)r}{2K^{2}}e^{-\beta I_{*}},\;\;\; &e_{020}&= \frac{(K-I_{*})(\beta K+r)^{2}}{2K^{2}}e^{-\beta I_{*}}, \\ e_{110}&=-\frac{(Kr-I_{*}r-K)(K\beta +r)}{K^{2}}e^{-\beta I_{*}},\;\; \; &e_{011}&=-1, \\ f_{020}&=-\frac{\beta ^{2}(K-I_{*})}{2}e^{-\beta I_{*}},\;\;\; &f_{110}&= \beta e^{-\beta I_{*}}, \\ f_{011}&=1. \end{aligned} $$

We can deduce that

$$ \begin{aligned} \tilde{f }(0,\;0)=0,\;\frac{\partial \tilde{f }}{\partial X}(0,\;0)=1, \;\frac{\partial \tilde{f }}{\partial \mu}(0,\;0)=0,\; \frac{\partial ^{2} \tilde{f }}{\partial \mu \partial X}(0,\;0)=g_{101}, \;\frac{\partial ^{2} \tilde{f }}{\partial X^{2}}(0,\;0)=2g_{200}. \end{aligned} $$

Based on the conclusions drawn in references [26, 27], the model (2.7) undergoes a transcritical bifurcation if $g_{101}\neq 0$ and $g_{200}\neq 0$ and the corresponding results are obtained. □

Last, we study the Neimark–Scaker bifurcation at the endemic equilibrium $E_{2}(S_{*},\;I_{*})$.

Theorem 2.9

If $R_{0}>1$, $r = r_{1}$, $r_{3} < r < r_{4}$, $- \frac{e^{-\beta I_{*}}(K-I_{*})(e^{-\beta I_{*}}(K\beta -\beta I_{*}+1) +\sigma -1)}{K} \neq 0$, $r \neq r_{5}$, $r\neq r_{6}$, and $\hat{a} \neq 0$, then system (2.7) undergoes a Neimark–Sacker bifurcation at the endemic equilibrium $E_{2}$.

Proof

Make $p_{3}$ be the value of p and $q_{3}$ be the value of q when $r=r_{1}$. Through the calculation, we get the results below:

$$ \begin{aligned} p_{3}&=- \frac {((K\beta -\beta I_{*}+1)e^{-\beta I_{*}}+\sigma -1)^{2} +1}{(K\beta -\beta I_{*}+1)e^{-\beta I_{*}}+\sigma -1}, \\ q_{3}&=1. \end{aligned} $$

(2.11)

The transversality condition is $\frac{d|\lambda |}{dr}=- \frac{e^{-\beta I_{*}}(K-I_{*})((e^{-\beta I_{*}}(K\beta -\beta I_{*}+1) +\sigma -1)}{2K}$. It is calculated that if $\sigma \neq -(K\beta -\beta I_{*}+1)e^{-\beta I_{*}}+1$, then $\frac{d|\lambda (r)|}{dr} \neq 0$. Furthermore, $r_{1} \neq r_{4}$ and $r_{1} \neq r_{5}$ are required to satisfy $\lambda _{j} \neq 1 (j = 1,\;2,\;3,\;4)$. Then, the model (2.7) reduced on the center manifold is

$$ \begin{pmatrix} X(t+1) \\ Y(t+1) \end{pmatrix} = \begin{pmatrix} -\frac{p_{3}}{2}&\frac{\sqrt{4-p_{3}^{2}}}{2} \\ -\frac{\sqrt{4-p_{3}^{2}}}{2}&-\frac{p_{3}}{2} \end{pmatrix} \begin{pmatrix} X(t) \\ Y(t) \end{pmatrix} + \begin{pmatrix} f(X(t),\;Y(t),\;Z(t)) \\ g(X(t),\;Y(t),\;Z(t)) \end{pmatrix} , $$

(2.12)

where

$$\begin{aligned}& \begin{aligned} f(X(t),\;Y(t),\;Z(t))&=m_{20}X^{2}+m_{02}Y^{2}+m_{11}X(t)Y(t)+m_{30}X(t)^{3} +m_{03}Y(t)^{3} \\ &\quad +m_{21}X(t)^{2}Y(t)+m_{12}X(t)Y(t)^{2}+\mathcal{O}(|X(t),\;Y(t)|^{4}), \\ g(X(t)\;,Y(t),\;Z(t))&=n_{20}X^{2}+n_{02}Y^{2}+n_{11}X(t)Y(t)+n_{30}X(t)^{3}+n_{03}Y(t)^{3} \\ &\quad +n_{21}X(t)^{2}Y(t)+n_{12}X(t)Y(t)^{2} +\mathcal{O}(|X(t),\;Y(t)|^{4}), \end{aligned} \\& \begin{aligned} m_{20}&=\frac{1}{j_{01}}(j_{01}^{2}j_{20}+j_{01}j_{11}(k_{01}+ \frac{p_{3}}{2}) +\frac{j_{02}(p_{3}+2k_{01})^{2}}{4}), \\ m_{02}&=-\frac{1}{4j_{01}}(j_{02}(p_{3}+2)(p_{3}-2)), \\ m_{11}&=\frac{\sqrt{4-p_{3}^{2}}}{2j_{01}}(j_{02}(p_{3}+2k_{01})+j_{01}j_{11}), \\ m_{30}&=\frac{1}{j_{01}}(j_{01}^{3}j_{30}+j_{01}^{2} (j_{21}k_{01}+ \frac{1}{2}j_{21}p_{3})+j_{01}j_{12}(k_{01} +\frac{p_{3}}{2})^{2}+j_{03}(k_{01}+ \frac{p_{3}}{2})^{3}), \\ m_{03}&=\frac{j_{03}(4-p_{3})^{\frac{3}{2}}}{8j_{01}}, \\ m_{21}&=\frac{\sqrt{4-p_{3}^{2}}}{8j_{01}}(4j_{01}^{2}j_{21}+j_{01}j_{12} (4p_{3}+8k_{01})+3j_{03}(p_{3}+2k_{01})^{2}), \\ m_{12}&=-\frac{1}{8j_{01}}(p_{3}^{2}-4)(2j_{01}j_{12}+6j_{03}k_{01}+3j_{03}p_{3}), \\ n_{20}&=-\frac{1}{4j_{01}\sqrt{4-p_{3}^{2}}}(p_{3}+2k_{01})(4(j_{20}-k_{11})j_{01}^{2} \\ &\quad +2(p_{3}+2k_{01})(j_{11}-k_{02})j_{01}+j_{02}(p_{3}+2k_{01})^{2}), \\ n_{02}&=\frac{\sqrt{4-p_{3}^{2}}}{4j_{01}}(2j_{01}k_{02}-2j_{02}k_{01}-j_{02}p_{3}), \\ n_{11}&=-\frac{1}{2j_{01}}(p_{3}+2k_{01})(j_{01}j_{11}+j_{02}(2k_{01}+p_{3})) +k_{11}j_{01}+k_{02}p_{3}+2k_{02}k_{01}, \\ n_{30}&=-\frac{1}{8j_{01}\sqrt{4-p_{3}^{2}}}(p_{3}+2k_{01})(8j_{30}j_{01}^{3} +4(p_{3}+2k_{01})(j_{21}-k_{12})j_{01}^{2}, \\ &\quad +2(p_{3}+2k_{01})^{2}(j_{12}-k_{03})j_{01}+j_{03}(p_{3}+2k_{01})^{3}), \\ n_{03}&=-\frac{1}{8j_{01}}(p_{3}^{2}-4)(2j_{01}k_{03}-j_{03 }(2k_{01}+p_{3})), \\ n_{21}&=\frac{1}{8j_{01}}(-4j_{01}^{2}(p_{3}+2k_{01})(j_{21}-2k_{12}) -2j_{01}(p_{3}+2k_{01})^{2} \\ &\quad (2j_{12}-3k_{03})-3j_{03}(p_{3}+2k_{01})^{3}), \\ n_{12}&=\frac{\sqrt{4-p_{3}^{2}}}{8j_{01}}(4j_{01}^{2}k_{12}-2j_{01} (p_{3}+2k_{01})(j_{12}-3k_{03})-3j_{03}(p_{3}+2k_{01})^{2}){ ,} \end{aligned} \end{aligned}$$

(2.13)

and

$$\begin{aligned}& \begin{aligned} j_{01}&=-(\beta +\frac{r_{1}}{K})e^{-\beta I_{*}}(K-I_{*})-\sigma +1, \;\;\; &j_{20}&= \frac{r_{1}e^{-\beta I_{*}}(Kr_{1}-I_{*}r_{1}-2K)}{2K^{2}}, \\ j_{02}&=\frac{e^{-\beta I_{*}}(K\beta +r_{1})^{2}(K-I_{*})}{2K^{2}}, \; \;\; &j_{11}&= \frac{e^{-\beta I_{*}}(Kr_{1}-I_{*}r_{1}-K)(K\beta +r_{1})}{K^{2}}, \\ j_{30}&=- \frac{r^{2}_{1}e^{-\beta I_{*}}(Kr_{1}-I_{*}r_{1}-3K)}{6K^{3}}, \;\; \; &j_{03}&=- \frac{e^{-\beta I_{*}}(K\beta +r_{1})^{3}(K-I_{*})}{6K^{3}}, \end{aligned} \\& \begin{aligned} j_{21}&=- \frac{r_{1}e^{-\beta I_{*}}(Kr_{1}-I_{*}r_{1}-2K)(K\beta +r_{1})}{2K^{3}}, \;\;\; &j_{12}&=- \frac{e^{-\beta I_{*}}(Kr_{1}-I_{*}r_{1}-K)(K\beta +r_{1})^{2}}{2K^{3}}, \\ k_{01}&=e^{-\beta I_{*}}\beta (K-I_{*})+\sigma , \; \;\;&k_{02}&=- \frac{e^{-\beta I_{*}}\beta ^{2}(K-I_{*})}{2}, \\ k_{11}&=e^{-\beta I_{*}}\beta , \;\;\;&k_{03}&= \frac{e^{-\beta I_{*}}\beta ^{3}(K-I_{*})}{6}, \\ k_{12}&=-\frac{e^{-\beta I_{*}}\beta ^{2}}{2}. \end{aligned} \end{aligned}$$

A complex calculation gives

$$ \begin{aligned} \hat{a}=&\frac{1}{16}(-\frac{1}{2+p_{3}}((p_{3}+2)((p_{3}+1)p_{3}-3)(-m_{02}^{2}+m_{02}n_{11}+(n_{02} + n_{20})(m_{11}+n_{02}-n_{20}) \\ &+m_{20}(m_{20}+n_{11})) +\sqrt{4-p_{3}^{2}}((p_{3} + 3)p_{3} + 1)(-m_{02}(m_{11}+ 2n_{02}) \\ &- m_{11}m_{20}+2 m_{20}n_{20} \\ & +n_{11}(n_{02}+ n_{20})))+8( -\frac{1}{4}(m_{02} + m_{20})^{2}- \frac{1}{4}(n_{02} + n_{20})^{2})+\sqrt{4-p_{3}^{2}}(-3 m_{03} -m_{21} \\ &+ n_{12} + 3n_{30})-(m_{02}-m_{20}+n_{11})^{2}-(n_{02}-m_{11 } - n_{20})^{2}\\ &-p_{3}(m_{12} + 3(m_{30} + n_{03}) + n_{21})). \end{aligned} $$

If the quantity â is not zero, the corresponding results are obtained. □

3 Numerical simulation

This section will employ numerical simulation to validate the correctness of theoretical derivations and explore the complex dynamical behavior of model (1.2) and model (2.7).

3.1 Numerical bifurcation analyzes of model (1.2)

For convenience, parameter η is defined as $\eta =1-\sigma$ such that as it increases, the recovery rate decreases. A backward bifurcation is shown in Fig. 1, which illustrates the fold bifurcation at $R_{0}=0.808651$, the flip bifurcation at $R_{0}=0.882538$ and the backward bifurcations at $R_{0}=1$. When $R_{0} < 0.808651$ ($\eta > 0.8904$), system (1.2) has a unique disease-free equilibrium. When $0.808651 < R_{0} < 1$ ($0.72 < \eta < 0.8904$), two endemic equilibria coexist with disease-free equilibrium. As parameter η decreases, system (1.2) has an endemic equilibrium.

Figure 1

Backward bifurcation diagram of model (1.2), where $K=8000$, $\beta =0.00009$, $\sigma _{1}=0.0004$, $r=5.2$

In order to gain a better understanding of the flip bifurcation shown by the model (1.2), we analyze a bifurcation plot with parameter r as the bifurcation parameter in the Fig. 2 (a), where r ranges from 6.5 to 15.5. The model reveals complex dynamical behavior, including period-doubling bifurcations that evolve into chaotic dynamics as r increases from 6.5 to 10.8. Notably, within the interval $10.8< r<11.1$, there is a critical transition characterized by a sudden halt in chaotic behavior and the appearance of periodic solutions. This dramatic shift is indicative of a chaos crisis. As r further grows, the model transitions back from chaotic to periodic behavior through a series of period-halving bifurcations, ultimately stabilizing into single-period solutions. Throughout the process of varying r, period-doubling cascades and windows of complex periods are also observed.

Figure 2

(a): Flip bifurcation diagram of model (1.2) for $K=8000$, $\sigma = 0.1841$, $\beta = 0.000008$, $\sigma _{1}=0.009$. (b): Localized magnification of (a) for $13.9< r<14.6$

In order to study the impact of parameter changes on the stability and periodicity of the system, Fig. 3 presents a two-dimensional bifurcation diagram with r and σ as the parameters. The diagram is color-coded to distinguish between different dynamical states: blue for 1-period point, fresh green for 2-period point, and aqua for chaotic states with periods exceeding 17. Observing from Fig. 3, it is noted that as r decreases and σ increases, the system tends to stabilize at the fixed point.

Figure 3

Two dimensional bifurcation diagram of model (1.2) for $K=8000$, $\beta = 0.000008$, $\sigma _{1}=0.009$

Further analysis is conducted with r maintained as the primary bifurcation parameter, while keeping other parameters unchanged, as shown in Fig. 4 (a). Under this arrangement, model (1.2) demonstrates diverse dynamics: a stable period-2 solution exists within the range of $9< r<10.3$. Moreover, multiple attractors emerge and vanish as r increases, particularly between $9.725< r<9.915$ and $10.175< r<10.21$, suggesting a diverse range of attractor structures. To probe these phenomena in greater depth, we set r at 9.632 and experiment with different initial conditions. The time series of the last 50 steps for the infected population, after 5000 iterations, are plotted to examine the long-term outcomes. With an initial condition of $(4000,4000)$, the model stabilizes into a 10-cycle solution. Conversely, starting with $(5000,3000)$, a stable 2-cycle solution is observed (refer to Fig. 4 (b) and Fig. 4 (c)). Subsequently, setting $r = 10.184$, different initial values are chosen and the resulting time series for the last 50 steps post 5000 iterations are again plotted. Here, the initial condition $(4000,4000)$ leads to a stable 2-cycle solution, while $(5000,3000)$ results in a stable 10-cycle solution (refer to Fig. 4 (d) and Fig. 4 (e)).

Figure 4

(a): Flip bifurcation diagram of model (1.2) in the $(r, I)$-plane for $K=8000$, $\sigma = 0.6$, $\beta = 0.00004$, $\sigma _{1}=0.0002$. (b)–(c): Time series of infected individuals, where $r = 9.632$ with initial values $(4000,4000)$ and $(5000,3000)$. (d)–(e): Time series of infected individuals, where $r = 10.184$ with initial values $(4000,4000)$ and $(5000,3000)$

The sensitivity of the system to initial values is a key factor in determining its final state, as slight variations in starting points can lead to diverse outcomes by converging to different attractors. To thoroughly explore this aspect, we have constructed basins of attraction [28] for two periodic solution attractors across varying parameter settings, as depicted in Fig. 5. The initial values in Fig. 5 are $(4000,4000)$ and $(5000,3000)$ respectively. The distinct color variations in the diagram represent the coexistence of different periodic attractors: trajectories that start in the blue regions tend to converge towards the 2-cycle attractors, whereas those beginning in the orange regions typically evolve towards the 10-cycle attractors.

Figure 5

Basins of attraction of two coexisting attractors of system (1.2)

3.2 Numerical bifurcation analyzes of system (2.7)

In this section, we present the bifurcation diagrams and maximum Lyapunov exponents (MLEs), and show the complex dynamical behaviours by using numerical simulations. We initially set the parameters as follows: $K=3.5$, $\sigma = 0.6$, $\beta = 0.3$, and varied r. Subsequent calculations yielded $R_{0} = 2.625> 1$, and specific values for r as $r_{0} = 6.93769$, $r_{1 }= -3.62385$ and $r_{2} = 10.56150$. Further, we have $c_{101} = - 0.11481$, $c_{200}^{2} + \frac{c_{110}d_{200}}{p_{1}}=39.22021$, and the endemic equilibrium $E_{2}(S_{*},\;I_{*})=$ (1.7207, 1.7793). Analysis reveals that r exactly equals $r_{0}$, but does not equal $r_{1}$ or $r_{2}$, with $c_{110}\neq 0$ and $c_{200}^{2}+\frac{c_{110}d_{200}}{p_{1}}\neq 0$. According to Theorem 2.7, these conditions indicate a flip bifurcation at $E_{2}$ when r crosses $r_{0}$. The bifurcation diagram, mapped in the (r, S)-plane, is depicted in Fig. 6 (a). Accompanying this, the maximum Lyapunov exponent [27] corresponding to Fig. 6 (a) is given in Fig. 6 (b). Figures 6 (c)–(d) provide local amplification plots of the bifurcation diagram for ranges $6.5 < r < 11.12$ and $11< r < 11.6$, respectively. In fact, if $r < r_{0} = 6.93769$, there is a locally stable endemic equilibrium and loses its stability at the flip bifurcation parameter value $r =9.36$. Subsequently, periodic orbits of period 2, 4, 6, and 8 emerge successively. When r becomes larger, following the disappearance of periodic orbits, a chaotic set emerges.

Figure 6

(a): Flip bifurcation diagram of model (2.7) for $K=3.5$, $\sigma = 0.6$, $\beta = 0.3$. (b): Maximum Lyapunov exponents corresponding to (a). (c): The local amplification for (a) for $6.5 < r < 11.12$. (d): The local amplification for (a) for $11< r < 11.6$

To analyze the periodicity of solutions of model (2.7) for various parameter values, we revisit the parameters as in Fig. 6 with $r \in (2, 5)$ and $\sigma \in (0.5, 1)$. The corresponding two-dimensional bifurcation diagram is depicted in Fig. 7. The different period attractors are indicated in different colors. For example, if the parameters lie on the red area, model (2.7) will be ultimately stabilized at a period-1 point, and the blue at a period-2 point. When the parameter resides within the white region, the system stabilizes at a periodic point greater than 20 cycles. In such instances, we infer that, in most cases, the system stabilizes towards a chaotic attractor. On the other hand, when the parameter takes values within the black region, the system is deemed unstable.

Figure 7

Two dimensional bifurcation diagram of model (2.7) in the (r, σ)-parameter plane

Then, we consider a special case of system (2.7) by choosing parameters $K = 0.68$, $\sigma = 0.64$ and $\beta =0.15$. Figures 8 (a)–(b) show that the three equilibria of the system simultaneously undergo a Neimark–Sacker bifurcation. When $r < 6.1395$, a locally stable endemic equilibrium exists, which loses its stability at $r = 6.1395$; subsequently, an invariant circle emerges as the parameter r exceeds 6.1395. Moreover, the maximum Lyapunov exponent corresponding to Fig. 8 (a) is given in Fig. 8 (c). The local amplification of Fig. 8 (a) for $0.001< S < 0.6$ is shown in Fig. 8 (d).

Figure 8

(a) and (b): Neimark–Sacker bifurcation diagram of model (2.7) for $K=3.5$, $\sigma = 0.6$, $\beta = 0.3$. (c): Maximum Lyapunov exponents plot corresponding to the N-S bifurcation in (a). (d): The local amplification for (a) for $0.001 < S < 0.6$

4 Validate the model with actual data

As mentioned in the Introduction, the discrete difference model is suitable for characterizing small-scale quantitative changes, and we note that since the outbreak of COVID-19 in China, more than 100 waves of small-scale outbreaks have occurred under strong prevention and control strategies [29]. Therefore, in this section, we will employ the proposed model (see (1.2) and (2.7)) and select some small-scale clustered epidemic data sets with aims at achieve model validation and parameter estimation.

For data fitting and model validation, we collected daily reported COVID-19 case data from various outbreaks across four provinces in China: Jiangsu (June 28 to July 19, 2022) [30], Shaanxi (December 9, 2021, to January 20, 2022) [31], Sichuan (March 30 to April 14, 2022) [32], and Shandong (July 24 to August 7, 2022) [33]. Employing the least square estimation method, we fitted the data from those provinces to estimate relevant parameters and evaluate the model’s efficacy, as shown in Table 1 and Table 2, reveal a goodness-of-fit due to R-square exceeding 0.8 for all data sets, indicating strong agreement between the model predictions and actual data. This conclusion is supported by the fitting curve graphs depicted in Fig. 9 and Fig. 10, where minimal discrepancies between the fitted and actual values underscore the model’s reliability. Specifically, the subplots in the first column depict the modeled trends of COVID-19 case counts over time in the mentioned provinces, with the horizontal and vertical axes representing time (in days) and daily reported cases, respectively. In the corresponding second column, residual plots associated with these fitted curves are presented, providing insights into the accuracy of the fitting outcomes.

Figure 9

Data fitting daily reported COVID-19 cases, with corresponding residual distributions of model (1.2)

Figure 10

Data fitting daily reported COVID-19 cases, with corresponding residual distributions of model (2.7)

Table 1 Parameter estimations for four different small-scale outbreaks of model (1.2)

Table 2 Parameter estimations for four different small-scale outbreaks of model (2.7)

Regarding the parameter estimation of intrinsic growth rate and carrying capacity, we have drawn the following important conclusions on infectious disease prevention and control closely related to the effectiveness of non-pharmacological intervention (NPI) measures: 1) Compared to the total population in the outbreak area, the estimated carrying capacity is very small, i.e., the population size involved in each small-scale epidemic is very small; 2) As the NPI strategy continues to strengthen, the scale of the population size involved will become smaller and smaller. In fact, based on the estimated negative intrinsic growth rate, it fully demonstrates that prevention and control measures, including close contact tracing, are very effective in controlling the epidemic by rapidly reducing the size of susceptible populations.

However, the analysis of the residual distribution graphs reveals minor details about model performance across the provinces. In model (1.2) and model (2.7), the residuals in Shaanxi province demonstrated an approximate uniform distribution around zero, implying a robust model fit and stability. Conversely, in the remaining provinces, the residuals exhibited patterns suggestive of systematic deviations. These deviations could be attributed to factors such as limited sample sizes or inherent non-random elements within the model’s application.

The study reveals that the model incorporating saturation recovery demonstrates significantly improved fitting accuracy in Shaanxi and Shandong provinces compared to counterparts. Nevertheless, this enhanced fitting precision is not observed in the remaining provinces. Such variance may be attributed to divergences in the epidemic progression stages across distinct regions during the study period. Specifically, some regions may have already passed the peak of the epidemic, thereby reducing the noticeable impact of saturation recovery.

4.1 Sensitivity analysis

Global sensitivity analysis plays a crucial role in identifying key factors that have a significant impact on the transmission dynamics of infectious diseases. This helps in developing more effective strategies to control the spread of such diseases. The Partial Rank Correlation Coefficient (PRCC) [34–36], a robust statistical tool, is widely utilized in sensitivity analysis to assess the impact of various parameters. It achieves this by measuring the partial correlations between input parameters and model outputs, effectively isolating the influence of individual parameters by eliminating the linear effects of others [35]. This methodology is particularly adept at handling non-linear relationships. PRCC is typically implemented alongside Latin Hypercube Sampling (LHS), primarily because LHS significantly reduces the number of simulations necessary for analysis. In our study, LHS and PRCC were used to assess the effective reproduction number and the infection count, as described in Equation (1.2). The formula for the effective reproduction number, $R(t)$, is derived via the next-generation matrix method [21]:

$$ R(t)=\frac{\beta K S(t)}{(1-\sigma ) (S(t)+I(t))} $$

Figure 11 (a) presents the impact of various parameters on $R(t)$ throughout a specified time period. In the early stages of an outbreak, the transmission constant β, the parameter σ, and the environmental carrying capacity K are strongly positively correlated with $R(t)$. This highlights the potential effectiveness of interventions such as disrupting transmission channels, promoting mask usage, ensuring timely medical interventions, and bolstering immunity to decrease $R(t)$ and thus manage the spread of the disease. As the outbreak evolves, while K continues to exhibit a robust positive correlation with $R(t)$, the correlation with σ weakens, indicating changes in disease transmission and recovery dynamics. The parameters, including the recovery rate adjustment parameter $\sigma _{1}$, intrinsic growth rate r, and transmission constant β, display a notable negative correlation with the effective reproduction number $R(t)$. This suggests that strategic allocation of medical resources and population control measures can significantly slow down disease progression. Figure 11 (b) illustrates the influence of various factors on the number of infected individuals $I(t)$ over time, illustrating that measures such as reducing birth rates and transmission rates, combined with increasing the environmental carrying capacity K, can effectively reduce infection levels.

Figure 11

PRCC sensitivity analysis plots for $R(t)$ and $I(t)$

To further analyze the impact of these parameters on the effective reproduction number and the infection count, contour maps were generated for day 50. Figure 12 (a) reveals that a decrease in the transmission constant β coupled with an increase in the recovery rate adjustment constant $\sigma _{1}$ raises the effective reproduction number $R(t)$, potentially leading to a higher number of infections. Conversely, Fig. 12 (b) shows that lower values of both β and $\sigma _{1}$ tend to amplify the scale of the infection.

Figure 12

Contour plots of $R(t)$ and $I(t)$ with respect to β and $\sigma _{1}$

5 Discussion

During the outbreak of the COVID-19 epidemic, various continuous infectious disease dynamics models were used to study the development and evolution of the epidemic, and even to evaluate the effectiveness of prevention and control strategies. In fact, under China’s strong prevention and control strategy, most of the over 100 waves of outbreaks have been very small in scale, and consequently the discrete infectious disease models with population dynamics could be employed to depict the small-scale outbreaks.

In this study, we developed a simplified discrete SIS model that integrates Ricker-type recruitment and embodies principles of host-parasite interaction, offering a framework for studying the transmission mechanisms of infectious diseases within host populations and their interactions with pathogens. We also enhance the model’s accuracy in describing disease transmission dynamics by introducing saturation recovery [37], thereby providing more effective tools for preventing and controlling infectious diseases.

When the basic reproduction number, $R_{0}$, is less than 1, the model displays a variety of complex dynamical behavior, including backward bifurcations, fold bifurcations, and flip bifurcations. The model also exhibits phenomena such as period-doubling cascades, chaos crisis, and multiple attractors, as illustrated in the bifurcation diagrams. Notably, the model considering saturated recovery exhibits dynamic behavior of period doubling leading to chaos, followed by period doubling leading to chaos again, and subsequently period halving to fixed points, and period halving to fixed points again. This behavior is not observed in a model that does not consider saturated recovery, underscoring the impact of saturation recovery on the model’s dynamics. Furthermore, we establish the conditions for the global asymptotic stability of the disease-free equilibrium by constructing a Lyapunov function. Given the complexity introduced by nonlinear recovery functions, we also explore a situation with linear recovery. Using the method of linearized approximation, we analyze the local asymptotic stability of equilibrium. Bifurcation analysis uncovers various bifurcations, including transcritical, flip, and Neimark–Sacker bifurcations, with the latter occurring simultaneously at three equilibria. Numerical simulations reveal the complex dynamics of the non-saturated model, showing complex periodic windows, periodic solutions, and chaotic behaviors.

To quantitatively validate our model, we apply the least squares method to fit daily new COVID-19 case data, enabling the estimation of previously unknown parameters. Notably, the saturation recovery model occasionally outperforms the non-saturation model when applied to data from the same province, emphasizing the importance of considering different stages of epidemic progression. A subsequent sensitivity analysis identifies key factors that significantly influence both the effective reproduction number and the number of infected individuals. Integrating these findings with numerical analysis demonstrates a strong negative correlation between modulating the recovery rate parameters and the effective reproduction number.

Despite not considering disease-induced mortality, our model extended previous works by incorporating nuanced aspects that increase its accuracy and versatility [7]. Primarily, our model aligns with the stability conditions of the well-established Ricker model by limiting that the intrinsic growth rate, r, should lie between 0 and 2 to ensure the local asymptotic stability of disease-free equilibria. Notably, when r is set to zero, our model simplifies to a discrete SIS model with Ricker-type recruitment, thus showcasing its adaptability and broad applicative potential. Furthermore, our approach includes data fitting, which validates the practical applicability of our model through excellent goodness-of-fit metrics. However, our model simplifies several key elements present in real-world disease transmission, such as stochasticity, acquired immunity, and disease mortality rates. These factors are crucial for a comprehensive understanding of disease dynamics and spread. Future research should focus on integrating these elements into the model to improve its accuracy and applicability in real-world scenarios.