Stability and bifurcations for a 3D Filippov SEIS model with limited medical resources

Abstract

In this paper, a three-dimensional (3D) Filippov SEIS epidemic model is proposed to characterize the impact of limited medical resources on disease transmission with discontinuous treatment functions. Qualitative analysis of non-smooth dynamical behaviors are performed on two subsystems and sliding modes. Criteria on the stability of various kinds of feasible equilibria and bifurcations, e.g., saddle-node bifurcation, transcritical bifurcation, and boundary equilibrium bifurcation, are established. The theoretical results are illustrated by numerical simulation, from which we find there could exist bistable phenomena, e.g., endemic and pseudo-equilibria, endemic equilibria of the two subsystems, or endemic and disease-free equilibria, even the basic reproduction numbers of two subsystems are less than 1. The disease spread is dependent both on the limited medical resources and latent compartment, which are more beneficial to effective disease control than planar Filippov and smooth models.

1 Introduction

The Filippov system is a discontinuous dynamical system composed of two or more different smooth subsystems connected by various threshold boundaries [1], which has extensive applications in many fields [2, 3], e.g., forest fires [4], pest control [5], infectious disease precaution [6, 7], feedback circuit [8], etc. Recently, many important work have been done on Filippov models [3, 9], mainly focused on stability of equilibria (or periodic solutions) [10], bifurcations [5], sliding dynamics [11], etc.

Tang and Liang et al. [12] presented a planar Filippov population model based on larval density. They showed that threshold strategy control can be used to prevent outbreaks of multiple pests or stabilize pest density at the desired level. Zhou and Xiao et al. [13] proposed a planar Filippov model for the West Nile virus. They found that intermittent mosquito control and bird protection are more effective on curbing virus spread.

Medical resources (e.g., medicines [14], paramedics [15], hospital beds [16], quarantines [17], etc.) play a crucial role in the control of diseases spread, e.g., in 2001, vaccination avoided 61 percent of measles deaths, 78 percent of whooping cough deaths, 98 percent of polio deaths, etc. [18]. Specially, sufficient hospital beds and the construction of health facilities can effectively prevent disease transmission and reduce the risk of infection [19], e.g., in 2019, during the COVID-19 in Wuhan, the establishment of Huoshen Mountain, Leishen Mountain and Fangcang Shelter Hospital avoided 22786 infections and saved 6524 lives [20]. Many scholars [21–23] studied the impact of limited medical resources on diseases spread. Wang et al. [23] established the following planar Filippov SI model with linear treatment function:

$$ \textstyle\begin{array}{l} \left \{ { \textstyle\begin{array}{l} {\dot{S}(t) = \Lambda - \mu S - \beta SI,} \\ {\dot{I}(t) = \beta SI - {\mu I - {v_{1}} } I - \gamma I,} \end{array}\displaystyle \quad \quad \quad {\mathrm{{}}} I < {I_{c}},} \right . \\ \left \{ { \textstyle\begin{array}{l} {\dot{S}(t) = \Lambda - \mu S - \beta SI,} \\ {\dot{I}(t) = \beta SI - {\mu I - {v_{1}} } I - \gamma {I_{c}},} \end{array}\displaystyle \quad \quad \quad {\mathrm{{}}}I > {I_{c}},} \right . \end{array} $$

(1)

where $S(t)$ and $I(t)$ represent the susceptible and infected individuals at time t, respectively; Λ is the recruitment rate of susceptible individuals; μ represents the natural mortality rate; β is the transmission rate; γ represents the treatment rate of infected individuals, and $v_{1}$ represents the mortality rate due to illness. The authors assumed that when the number of infected individuals is less than the threshold $I_{c}$, the treatment rate is directly proportional to the number of infected individuals, otherwise it is the maximum value $\gamma I_{c}$. They found that when medical resources are insufficient, backward bifurcation would occur.

Furthermore, if medical resources are insufficient, infected individuals will not receive timely treatment, which will further seriously affect the treatment efficiency, and slow down the speed of recovery of infected individuals [24]. Therefore, many researchers [25–28] considered the effect of delayed treatment on epidemic models. Liu et al. [28] established the following planar Filippov SI model with saturated treatment function:

$$ \textstyle\begin{array}{l} \left \{ { \textstyle\begin{array}{l} {\dot{S}(t) = \Lambda - \mu S - \beta SI,} \\ {\dot{I}(t) = \beta SI - {\mu _{1}}I - {v_{1}}I,} \end{array}\displaystyle } \right .\quad \quad \quad \quad \quad{\mathrm{{ }}}I < {I_{c}}, \\ \left \{ { \textstyle\begin{array}{l} {\dot{S}(t) = \Lambda - \mu S - \beta SI,} \\ {\dot{I}(t) = \beta SI - {\mu _{1}}I - {v_{1}}I - \frac{{\gamma I}}{{1 + \alpha I}},} \end{array}\displaystyle } \right .\quad \quad{\mathrm{{ }}}I > {I_{c}}, \end{array} $$

(2)

when the number of infected individuals exceeds the threshold $I_{c}$, the saturated treatment function ${\frac{{\gamma I}}{{1 + \alpha I}}}$ is implemented, γ is the treatment rate, and α is the degree of impact of delayed treatment on infected individuals. They found that saturation therapy functions can generate more complex dynamic behaviors, such as saddle-node bifurcation, backward bifurcation, etc.

Recently, the Filippov systems have been successfully extended to 3D models [29], which have the following advantages comparing with planar Filippov systems: i) provide a more detailed description of the real world [30], e.g., earthquakes are a complex phenomenon involving the fracture, movement, and stress release of underground rocks. The three-dimensional Filippov system could be used to describe the spatially development process of earthquake dynamics [31]; In terms of pest control, 3D Filippov system could characterize the complex interactions between pest quantity, environment, and control strategies, etc. [29]; ii) exhibit richer dynamic behaviors, e.g., pseudo-equilibra, periodic orbits, and chaotic phenomena, which can further dynamic explore the full properties of models [32]. Tang et al. [33] established a 3D Filippov crops-pests-natural enemies model and found that there are multiple pseudo-equilibria in the sliding region, with rich bifurcations appearing. Tiago de Carvalho et al. [34] considered a 3D Filippov HIV infection model and found that intermittent treatment can effectively control viral infections and reduce the side effect of drugs.

Meanwhile, compared with planar Filippov epidemic models, 3D Filippov epidemic models also face the following challenges [35]: i) The threshold strategies change the switching manifold from a line to a plane; ii) Sliding mode dynamics is governed by a set of two-dimensional differential equations. Complex dynamic phenomena, such as saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation, etc., may occur in the sliding mode domain within the switching surface.

Moreover, as well known, many infectious diseases have latent period ranging from a few days to several months [36], e.g., HIV [37], Hepatitis-B [38], bacterial and amoebic dysenter [39], etc. The length of latent period can affect the speed and scope of disease transmission, as well as control strategies [40]. A large number of smooth ODEs have incorporated the latent compartment into the epidemic model [19, 41]. The classic SEIS epidemic model with latent compartment is as follows

$$ \textstyle\begin{cases} \dot{S}\left (t\right )=\Lambda -\mu S-\beta SI+\gamma I, \\ \dot{E}\left (t\right )=\beta SI-\mu E-\sigma E, \\ \dot{I}\left (t\right )=\sigma E-\mu _{1}I-\gamma I, \end{cases} $$

(3)

where $E(t)$ represents latent individual at time t, and $1/\sigma$ is the latent period. It has been found that introducing latent compartment can better predict the speed and scope of disease transmission [41]. To our knowledge, the three-dimensional Filippov epidemic model with latent compartment has not been fully considered. Moreover, since the constraints in medical resources, the mortality rate due to illness would rise [20]. Therefore, in order to better model disease transmission, introducing a latent compartment in the planar Filippov epidemic model is more consistent with the real process of disease.

Based on the above motivation, we establish the following three-dimensional Filippov SEIS epidemic model with limited medical resources depending on the infected individuals.

$$ \textstyle\begin{array}{l} \left \{ { \textstyle\begin{array}{l} {\dot{S}(t) = \Lambda - \beta SI - \mu S + \gamma I,} \\ {\dot{E}(t) = \beta SI - \mu E - \sigma E,} \\ {\dot{I}(t) = \sigma E - \gamma I - {\mu _{1}}I,} \end{array}\displaystyle } \right . \quad \quad \quad \quad \quad I < I_{c}, \\ \left \{ { \textstyle\begin{array}{l} {\dot{S}(t) = \Lambda - \beta SI - \mu S + \frac{{\gamma I}}{{1 + \alpha I}},} \\ {\dot{E}(t) = \beta SI - \mu E - \sigma E,} \\ {\dot{I}(t) = \sigma E - \frac{{\gamma I}}{{1 + \alpha I}} - {\mu _{2}}I,} \end{array}\displaystyle } \right . \quad \quad \quad \quad I > I_{c}, \end{array} $$

(4)

where $\mu _{1}$, $\mu _{2}$ are the death rates that include both natural death and death due to disease if $I< I_{c}$ or $I>I_{c}$, respectively. Our primary objective is to investigate how the thresholds switch of therapy functions could lead to dynamic variation, particularly the emergence of multiple steady states, multiple bifurcations, ect. Additionally, we aim to describe how these complicated phenomena in the model (4) impact the allocation of medical resources.

The rest of the paper is organized as follows. In Sect. 2, the basic definitions and preliminary theories of the Filippov system are presented. In Sect. 3, the basic reproduction numbers ($R_{01}$, $R_{02}$) are defined and the stability of the equilibria of the two subsystems is obtained. The sliding mode dynamics is studied in Sect. 4. In Sect. 5, the dynamical behaviors of the system (4) is discussed with $R_{01}$, $R_{02}$. A brief conclusion is given in the last section.

2 Basic definition and preliminary theories of Filippov system

First, we introduce some properties and definitions of Filippov systems [1, 30]. Let $A \subset \mathbb{R}^{3}$ be an open set and $\Sigma = \left \{ {\left ( {S,E,I} \right ) \in A| H\left ( Z \right ) = 0} \right \}$ and $H\left ( Z \right ) = I - I_{c}$ with vector $Z = {\left ( {S,E,I} \right )^{T}}$. The discontinuous switching surface Σ is the boundary between the regions ${S_{1}} = \left \{ {\left ( {S,E,I} \right ) \in A|H\left ( Z \right ) < 0} \right \}$ and ${S_{2}} = \left \{ {\left ( {S,E,I} \right ) \in A|H\left ( Z \right ) > 0} \right \}$. Obviously $R_{+} ^{3} = {S_{1}} \cup \Sigma \cup {S_{2}}$. For convenience, we define the Filippov systems in regions $S_{1}$ and $S_{2}$ as subsystems $S_{1}$ and $S_{2}$, respectively. We define ${\chi ^{r}}$ the space of $C^{r}$-vector fields with $C^{r}$-topology on A, where $r \ge 1$. A segmented smooth vector field on an open set $A \subset \mathbb{R}^{3}$ can be written as

$$ F\left ( Z \right ) = \left \{ { \textstyle\begin{array}{l} {{F_{{S_{1}}}} = {{\left ( {\Lambda - \beta SI - \mu S + \gamma I, \beta SI - \mu E - \sigma E,\sigma E - \gamma I - {\mu _{1}}I} \right )}^{T},}} \\ {{F_{{S_{2}}}} = {{\left ( {\Lambda - \beta SI - \mu S + \frac{{\gamma I}}{{1 + \alpha I}},\beta SI - \mu E - \sigma E,\sigma E - \frac{{\gamma I}}{{1 + \alpha I}} - {\mu _{2}}I} \right )}^{T},}} \end{array}\displaystyle } \right . $$

system (4) can be written as the following Filippov system

$$ \dot{Z}\left ( t \right ) = \left \{ { \textstyle\begin{array}{l} {{F_{{S_{1}}}}\left ( Z \right ),Z \in {S_{1}}}, \\ {{F_{{S_{2}}}}\left ( Z \right ),Z \in {S_{2}}}. \end{array}\displaystyle } \right . $$

(5)

The type of contact between the smooth vector field ${F_{{S_{i}}}} \in {\chi ^{r}}$ and the switching surface Σ is determined by the following directional Lie derivatives

$$ {L_{{F_{{S_{i}}}}}}H = \left \langle {\nabla H,{F_{{S_{i}}}}} \right \rangle \left ( {i = 1,2} \right ), $$

where ∇H is the gradient of the smooth function H, $\left \langle { \cdot , \cdot } \right \rangle$ is the standard vector product. The higher order Lie derivative is $L_{{F_{{S_{i}}}}}^{m}H = \left \langle {L_{{F_{{S_{i}}}}}^{m - 1}H,{F_{{S_{i}}}}} \right \rangle ,m \ge 2$.

If there exists a point $Z_{*} \in \Sigma$ satisfying ${L_{{F_{{S_{i}}}}}}H\left ( {Z_{*}} \right ) = 0$ and $L_{{F_{{S_{i}}}}}^{2}H\left ( {Z_{*}} \right ) \ne 0$, then $Z_{*}$ is a folding point [32]. If $L_{{F_{{S_{i}}}}}^{2}H\left ( {Z_{*}} \right ) > 0$, then $Z_{*}$ is a visible folding point, and if $L_{{F_{{S_{i}}}}}^{2}H\left ( {Z_{*}} \right ) < 0$, then $Z_{*}$ is an invisible folding point [32]. If ${L_{{F_{{S_{i}}}}}}H\left ( {Z_{*}} \right ) = L_{{F_{{S_{i}}}}}^{2}H \left ( {Z_{*}} \right ) = 0$, $L_{{F_{{S_{i}}}}}^{3}H\left ( {Z_{*}} \right ) \ne 0$, then $Z_{*}$ is a cusp point [32].

The trajectory of the solution of the Filippov system on the discontinuity surface Σ can be classified into three scenarios [1]. The first is that the trajectory crosses Σ. The second is that the trajectory arrives at Σ from both sides and slides along Σ. The third is that the trajectory leaves Σ from both sides. These scenarios can be mathematically described, i.e., if the trajectory is in the regions ${S_{1}}$ and ${S_{2}}$, the local trajectory is defined in terms of the vector fields ${F_{{S_{1}}}}$ and ${F_{{S_{2}}}}$. The following classification of Σ is applicable:

• Crossing region (Fig. 1(a)):

Figure 1

Crossing, attractive sliding and repulsive sliding regions, respectively

$\Sigma _{c}^{+} = \left \{ {Z \in \Sigma |{L_{{F_{{S_{1}}}}}}H > 0,{L_{{F_{{S_{2}}}}}}H > 0} \right \}$ and $\Sigma _{c}^{-} = \left \{ {Z \in \Sigma |{L_{{F_{{S_{1}}}}}}H < 0,{L_{{F_{{S_{2}}}}}}H < 0} \right \}$.

• Attractive sliding region (Fig. 1(b)):

$$ \Sigma s = \left \{ {Z \in \Sigma |{L_{{F_{{S_{1}}}}}}H > 0,{L_{{F_{{S_{2}}}}}}H < 0} \right \}. $$

• Repulsive sliding region (Fig. 1(c)):

$$ \Sigma e = \left \{ {Z \in \Sigma |{L_{{F_{{S_{1}}}}}}H < 0,{L_{{F_{{S_{2}}}}}}H > 0} \right \}. $$

The definitions of all other types of equilibria for the Filippov system [30] are as follows:

Definition 1

If $Z_{*} \in {S_{1}}$ and ${F_{{S_{1}}}}\left ( {Z_{*}} \right ) = 0$ or $Z_{*} \in {S_{2}}$ and ${F_{{S_{2}}}}\left ( {Z_{*}} \right ) = 0$, then $Z_{*}$ is called a real equilibrium. If $Z_{*} \in {S_{1}}$ and ${F_{{S_{2}}}}\left ( {Z_{*}} \right ) = 0$ or $Z_{*} \in {S_{2}}$ and ${F_{{S_{1}}}}\left ( {Z_{*}} \right ) = 0$, then $Z_{*}$ is called a virtual equilibrium.

Remark 1

Both real and virtual equilibria are called regular equilibria.

Definition 2

If ${F_{{S_{1}}}}\left ( {Z_{*}} \right ) = 0$, $H\left ( {Z_{*}} \right ) = 0$ or ${F_{{S_{2}}}}\left ( {Z_{*}} \right ) = 0$, $H\left ( {Z_{*}} \right ) = 0$, then $Z_{*}$ is called a boundary equilibrium.

Definition 3

If $Z_{*} \in \Sigma$, ${L_{{F_{{S_{i}}}}}}H\left ( {Z_{*}} \right ) = 0,\left ( {i = 1,2} \right )$, then $Z_{*}$ is called a tangency point.

Definition 4

If a point $Z_{*}$ satisfies

$$ {F_{{\Sigma _{S}}}}\left ( {{Z_{*} }} \right ) = \lambda {F_{{S_{1}}}} \left ( {{Z_{*} }} \right ) + \left ( {1 - \lambda } \right ){F_{{S_{2}}}} \left ( {{Z_{*} }} \right ), $$

where $\lambda = \frac{{\left \langle {\nabla H,{F_{{S_{2}}}}} \right \rangle }}{{\left \langle {\nabla H,{F_{{S_{2}}}} - {F_{{S_{1}}}}} \right \rangle }} \in \left ( {0,1} \right )$, then $Z_{*}$ is called a pseudo-equilibrium.

3 Dynamic behaviour of subsystems

3.1 Dynamic behaviour of subsystem $S_{1}$

According to the method of next generation matrices [42, 43], we calculate the basic reproduction number of subsystem $S_{1}$ as ${R_{01}} = \frac{{\beta {\Lambda }{\sigma }}}{{\mu \left ( {\mu + {\sigma }} \right )\left ( {{\mu _{1}} + \gamma } \right )}}$. Meanwhile, there are two equilibria in the subsystem $S_{1}$. A disease-free equilibrium $P_{{{{S1}}}}^{0}\left ( {\frac{{{\Lambda }}}{\mu },0,0} \right )$ and an endemic equilibrium $P_{{{{S1}}}}^{1}\left ( {S_{{{S1}}}^{1},E_{{{{S1}}}}^{1},I_{{{{S1}}}}^{1}} \right )$, where

$$ {S_{{{S1}}}^{1}} = \frac{{\left ( {{\mu _{1}} + \gamma } \right )\left ( {\mu + {\sigma }} \right )}}{{\beta {\sigma }}}, \quad I_{{{{S1}}}}^{1} = \frac{{ - \mu \left ( {\gamma + {\mu _{1}}} \right )\left ( {\mu + \sigma } \right ) + \Lambda \beta \sigma }}{{\left ( {\gamma + {\mu _{1}}} \right )\left ( {\mu + \sigma } \right )\beta - \sigma \gamma \beta }}, \quad {E_{{{S1}}}^{1}} = \frac{{\left ( {{\mu _{1}} + \gamma } \right ){I_{s_{1}}^{1}}}}{{{\sigma }}}. $$

The dynamics of subsystem $S_{1}$ has been analyzed in [41], therefore we obtain the following results:

Lemma 1

If ${R_{01}} < 1$, the disease-free equilibrium $P_{{{{S1}}}}^{0}$ of subsystem $S_{1}$ is globally asymptotically stable; if ${R_{01}} > 1$, the endemic equilibrium $P_{{{{S1}}}}^{1}$ of subsystem $S_{1}$ is globally asymptotically stable.

3.2 Dynamic behaviour of subsystem $S_{2}$

According to the method of next generation matrices [42, 43], the basic reproduction number of subsystem $S_{2}$ can be given as ${R_{02}} = \frac{{{\Lambda }{\sigma }\beta }}{{\mu \left ( {\mu + {\sigma }} \right )\left ( {{\mu _{2}} + \gamma } \right )}}$. It is obvious that there is always a disease-free equilibrium $P_{{{{S2}}}}^{0}\left ( {\frac{{{\Lambda }}}{\mu },0,0} \right )$ in the subsystem $S_{2}$.

The endemic equilibria of subsystem $S_{2}$ are solutions of

$$ \left \{ { \textstyle\begin{array}{l} {\Lambda - \beta SI - \mu S + \frac{{\gamma I}}{{1 + \alpha I}} = 0,} \\ {\beta SI - \mu E - \sigma E = 0,} \\ {\sigma E - \frac{{\gamma I}}{{1 + \alpha I}} - {\mu _{2}}I = 0,} \end{array}\displaystyle } \right .{\mathrm{{ }}} $$

(6)

which yields

$$ A{I^{2}} + BI + C = 0, $$

(7)

where

$$\begin{aligned}& A = - \alpha \beta {\mu _{2}}\left ( {\mu + \sigma } \right ) < 0,\\& B = - {\mu _{2}}\left ( {\mu + \sigma } \right )\left ( {\beta + \alpha \mu } \right ) + \left ( {\Lambda \alpha \sigma - \gamma \mu } \right )\beta ,\\& C = - \left ( {\mu + \sigma } \right )\left ( {{\mu _{2}}{\mkern 1mu} + \gamma } \right )\mu + \sigma \beta {\mkern 1mu} \Lambda . \end{aligned}$$

Therefore, we obtain two possible positive roots

$$ I_{{{S2}}}^{1} = \frac{{ - B + \sqrt {\Delta}}}{{2A}}, \quad \quad \quad I_{{{S2}}}^{2} = \frac{{ - B - \sqrt {\Delta}}}{{2A}}, $$

where

$$ \Delta = {B^{2}} - 4AC. $$

(8)

Accordingly, subsystem $S_{2}$ has two possible endemic equilibria $P_{{{{S2}}}}^{i}\left ( {S_{{{S2}}}^{i},E_{{{{S2}}}}^{i},I_{{{{S2}}}}^{i}} \right ), (i=1,2)$, where

$$ S_{{{S2}}}^{i} = \frac{{\left ( {\gamma + {\mu _{2}}\left ( {1 + \alpha {\mkern 1mu} I_{{{S2}}}^{i}} \right )} \right )\left ( {\mu + \sigma } \right )}}{{\left ( {\alpha {\mkern 1mu} I_{{{S2}}}^{i} + 1} \right )\sigma \beta }}, \quad \quad \quad E_{{{S2}}}^{i} = \frac{{I_{{{S2}}}^{i}\left ( {\gamma + {\mu _{2}}\left ( {1 + \alpha {\mkern 1mu} I_{{{S2}}}^{i}} \right )} \right )}}{{\left ( {\alpha {\mkern 1mu} I_{{{S2}}}^{i} + 1} \right )\sigma }}. $$

If ${R_{02}}<1$, $\Delta \ge 0$ and $B<0$, the Eq. (7) has no positive roots. Hence, the subsystem $S_{2}$ has no endemic equilibrium; If ${R_{02}}>1$, the Eq. (7) has one positive root (see Fig. 2(a)), the subsystem $S_{2}$ has a unique endemic equilibrium $P_{{{S2}}}^{2}\left ( {S_{{{S2}}}^{2},E_{{{S2}}}^{2},I_{{{S2}}}^{2}} \right )$; If ${R_{02}}<1$, $\Delta =0$ and $B>0$, the subsystem $S_{2}$ has a unique endemic equilibrium $P_{{{S2}}}^{*}$ (i.e., $P_{{{S2}}}^{1}$ = $P_{{{S2}}}^{2}$); if ${R_{02}}<1$, $\Delta > 0$ and $B> 0$, the Eq. (7) has two positive roots (see Fig. 2(b)). Therefore, the subsystem $S_{2}$ has two endemic equilibria $P_{{{S2}}}^{i}\left ( {S_{{{S2}}}^{i},E_{{{S2}}}^{i},I_{{{S2}}}^{i}} \right ), \left ( {i = 1,2} \right )$.

Figure 2

The existence of positive solution for Eq. (7). We fix all other parameters as follows: $\Lambda =10$, $\mu =0.32$, $\gamma =0.65$, $\sigma =1/7$, $\mu _{2}=0.6$, $\alpha =4$ (a) $\beta =0.15$ (b) $\beta =0.11$

Theorem 1

If ${R_{02}} < 1$, the disease-free equilibrium $P_{{{{S2}}}}^{0}$ of subsystem $S_{2}$ is locally asymptotically stable; if ${R_{02}} > 1$, the endemic equilibrium $P_{{{{S2}}}}^{2}$ of subsystem $S_{2}$ is locally asymptotically stable; if ${R_{02}}<1$, $\Delta > 0$ and $B> 0$, which exists two endemic equilibria $P_{{{{S2}}}}^{i}$ (i=1,2), the equilibrium $P_{{{{S2}}}}^{1}$ is an unstable saddle point and the equilibrium $P_{{{{S2}}}}^{2}$ is locally asymptotically stable.

Proof

The Jacobian matrix for subsystem $S_{2}$ is:

$$ {J_{S2}^{i}}{|_{P_{{{{S2}}}}^{i}\left ( {S,E,I} \right )}} = \left [ { \textstyle\begin{array}{c@{\quad}c@{\quad}c} { - \beta I - \mu }&0&{ - \beta S + \frac{\gamma }{{{{\left ( {1 + \alpha I} \right )}^{2}}}}} \\ {\beta I}&{ - \mu - \sigma }&{\beta S} \\ 0&\sigma &{ - \frac{\gamma }{{{{\left ( {1 + \alpha I} \right )}^{2}}}} - {\mu _{2}}} \end{array}\displaystyle } \right ]{|_{P_{{S2}}^{i}\left ( {S,E,I} \right )}}, $$

(9)

where $i=0,1, 2$. If ${R_{02}} < 1$, it is easy to obtain $tr\left ( {J_{{S2}}^{0}} \right ) < 0$, $\det \left ( {J_{{S2}}^{0}} \right ) > 0$, then the disease-free equilibrium $P_{{{{S2}}}}^{0}$ is locally asymptotically stable. The subsystem $S_{2}$ satisfies the characteristic equation at the equilibrium $P_{S2}^{i} (i=1,2)$ as follows:

$$ {\lambda ^{3}} + {B_{1}}{\lambda ^{2}} + {B_{2}}\lambda + {B_{3}} = 0, $$

(10)

where

according to the Routh–Hurwitz criterion [44], when ${B_{1}}\left ( {P_{{S2}}^{2}} \right ) \cdot {B_{2}}\left ( {P_{{S2}}^{2}} \right ) > {B_{3}}\left ( {P_{{S2}}^{2}} \right )$ and ${B_{3}}\left ( {P_{{S2}}^{2}} \right )>0$, the equilibrium $P_{{{{S2}}}}^{2}$ is locally asymptotically stable; it is easy to obtain ${B_{3}}\left ( {P_{{S2}}^{1}} \right ) < 0$, the equilibrium $P_{{{{S2}}}}^{1}$ is an unstable saddle point. □

Based on the above discussion, we give the theorem that subsystem $S_{2}$ undergoes a saddle-node bifurcation at $P_{{{S2}}}^{*}$.

Theorem 2

Subsystem $S_{2}$ has a saddle-node bifurcation at $P_{{{S2}}}^{*}(S_{S2}^{*},E_{S2}^{*},I_{S2}^{*})$, with respect to the bifurcation parameter $\gamma =\gamma ^{*}$ if

(i) ${B_{1}}(P_{{{S2}}}^{*}), {B_{2}}(P_{{{S2}}}^{*}) \ne 0, {B_{3}}(P_{{{S2}}}^{*}) =0$;

(ii) $2\alpha \gamma \mu \left ( {\mu + {\mu _{2}}} \right )\left ( {\mu + \sigma } \right )\left ( {\beta I + \mu } \right ) \ne \beta {\mu _{2}}{ \left ( {1 + \alpha I} \right )^{3}}\left ( {{\mu _{2}}\left ( {\mu + \sigma } \right )\left ( {\beta I + \mu } \right ) - \beta \mu \sigma I} \right )$;

(iii) $\Delta = 0$.

Proof

For the characteristic Eq. (10), if ${B_{1}}(P_{{{S2}}}^{*}), {B_{2}}(P_{{{S2}}}^{*})\ne 0$, i.e.

$$ \left \{ { \textstyle\begin{array}{l} { \frac{{{\alpha ^{2}}\beta {I^{3}} + \left ( {\left ( {2\mu + \sigma + {\mu _{2}}} \right )\alpha + 2\beta } \right )\alpha {I^{2}} + \left ( {\left ( {4\mu + 2\sigma + 2{\mu _{2}}} \right )\alpha + \beta } \right )I + 2\mu + \gamma + \sigma + {\mu _{2}}}}{{{{\left ( {1 + \alpha I} \right )}^{2}}}} \ne 0,} \\ \textstyle\begin{array}{l} \beta {\alpha ^{2}}\left ( {\mu + \sigma \! +\! {\mu _{2}}} \right ){I^{3}} + \bigl( {\alpha ^{2}}\left ( \mu \left ( {\mu + \sigma } \right ) + {\mu _{2}}\left ( {2\mu + \sigma } \right ) \right ) \\ + 2\alpha \beta \left ( {\mu \! +\! {\mu _{2}} + \sigma } \right ) - {\alpha ^{2}} \beta \sigma S \bigr){I^{2}} + 2\mu \left ( {\gamma + {\mu _{2}}} \right ) \\ +\left ({2\alpha \left ( {\mu \left ( {\mu + \sigma + 2{\mu _{2}}} \right ) + \sigma {\mu _{2}}} \right ) + \beta \left ( {\gamma + \mu + \sigma + {\mu _{2}}} \right ) - 2\alpha \beta \sigma S} \right )I\\ + \sigma \left ( {\gamma + \mu + {\mu _{2}}} \right ) - \beta \sigma S \ne 0. \end{array}\displaystyle \end{array}\displaystyle } \right . $$

Combining with Eq. (8), we can obtain the equation satisfied with $\gamma =\gamma ^{*}$ as follows:

$$ \left \{ { \textstyle\begin{array}{l} {{\left ( { - {\mu _{2}}\left ( {\mu + \sigma } \right )\left ( { \beta + \alpha \mu } \right ) + \left ( {\Lambda \alpha \sigma - \gamma \mu } \right )\beta } \right )}^{2}} \\ + 4\alpha \beta {\mu _{2}} \left ( {\mu + \sigma } \right )\left ( {\sigma \beta {\mkern 1mu} \Lambda - \left ( {\mu + \sigma } \right )\left ( {{\mu _{2}}{\mkern 1mu} + \gamma } \right )\mu } \right ) = 0, \\ \textstyle\begin{array}{l} {\mu _{2}}\beta {\alpha ^{2}}\left ( {\mu + \sigma } \right ){I^{3}} + \left ( {{\mu _{2}}\alpha \left ( {\mu + \sigma } \right )\left ( { \alpha \mu + 2\beta } \right ) - {\alpha ^{2}}\beta \mu \sigma S} \right ){I^{2}} \\ + \left ( {2\alpha \mu {\mu _{2}}\left ( {\mu + \sigma } \right ) - 2\alpha \beta \mu \sigma S} \right )I \\ + \beta \left ( {\gamma \mu + {\mu _{2}}\left ( {\mu + \sigma } \right )} \right )I + \mu \left ( {\mu + \sigma } \right )\left ( { \gamma + {\mu _{2}}} \right ) - \beta \mu \sigma S= 0. \end{array}\displaystyle \end{array}\displaystyle } \right . $$

According to Sotomayor’s theorem [45, 46], let V and W denote the eigenvectors corresponding to the matrices ${J_{P_{{S2}}^{*}}}$ and $J_{P_{{S2}}^{*}}^{T}$ at zero eigenvalue, respectively.

$$ V = \left ( { \textstyle\begin{array}{c} {{V_{1}}} \\ {{V_{2}}} \\ {{V_{3}}} \end{array}\displaystyle } \right ) = \left ( { \textstyle\begin{array}{c} { \frac{{\gamma - \beta S{{\left ( {1 + \alpha I} \right )}^{2}}}}{{{{\left ( {1 + \alpha I} \right )}^{2}}\left ( {\beta I + \mu } \right )}}} \\ { \frac{{\gamma + {\mu _{2}}{{\left ( {1 + \alpha I} \right )}^{2}}}}{{{{\left ( {1 + \alpha I} \right )}^{2}}\sigma }}} \\ 1 \end{array}\displaystyle } \right ), \quad \quad \quad W = \left ( { \textstyle\begin{array}{c} {{W_{1}}} \\ {{W_{2}}} \\ {{W_{3}}} \end{array}\displaystyle } \right ) = \left ( { \textstyle\begin{array}{c} {\sigma \beta I} \\ {\left ( {\beta I + \mu } \right )\sigma } \\ {\left ( {\mu + \sigma } \right )\left ( {\beta I + \mu } \right )} \end{array}\displaystyle } \right ). $$

According to the above calculation, we can obtain

$$ {F_{\gamma }}\left ( {P_{{S2}}^{*};\gamma ^{*}} \right ) = \left ( { \textstyle\begin{array}{c} {\frac{1}{{1 + \alpha I}}} \\ 0 \\ { - \frac{1}{{1 + \alpha I}}} \end{array}\displaystyle } \right ), \quad \quad {D^{2}}F\left ( {P_{{S2}}^{*};\gamma ^{*}} \right )\left ( {V,V} \right ) = \left ( { \textstyle\begin{array}{c} { \frac{{ - 2\beta \gamma + 2{\beta ^{2}}S{{\left ( {1 + \alpha I} \right )}^{2}} - 2\alpha \gamma \left ( {\beta I + \mu } \right )}}{{{{\left ( {1 + \alpha I} \right )}^{2}}\left ( {\beta I + \mu } \right )}}} \\ { \frac{{2\beta \gamma - 2{\beta ^{2}}S{{\left ( {1 + \alpha I} \right )}^{2}}}}{{{{\left ( {1 + \alpha I} \right )}^{2}}\left ( {\beta I + \mu } \right )}}} \\ {\frac{{2\alpha \gamma }}{{{{\left ( {1 + \alpha I} \right )}^{2}}}}} \end{array}\displaystyle } \right ). $$

Therefore, we have

$$\begin{aligned} &{W^{T}}{F_{\gamma }}\left ( {P_{{S2}}^{*};\gamma ^{*}} \right ) = \frac{{ - \mu \left ( {\beta I + \mu + \sigma } \right )}}{{\left ( {1 + \alpha I} \right )}} \ne 0,\\ &{W^{T}}\left [ {{D^{2}}F\left ( {P_{{S2}}^{*};\gamma ^{*}} \right ) \left ( {V,V} \right )} \right ] \\ &\quad= \frac{{2\alpha \gamma \mu \left ( {\beta I + \mu + \sigma } \right )\left ( {\beta I + \mu } \right ) - 2\beta {\mu ^{2}}\gamma - 2\beta \mu {\mu _{2}}\left ( {\mu + \sigma } \right )\left ( {1 + \alpha I} \right )}}{{{{\left ( {1 + \alpha I} \right )}^{2}}\left ( {\beta I + \mu } \right )}}. \end{aligned}$$

If ${W^{T}}\left [ {{D^{2}}F\left ( {P_{{S2}}^{*};\gamma ^{*}} \right ) \left ( {V,V} \right )} \right ] \ne 0$, i.e., $2\alpha \gamma \mu \left ( {\beta I + \mu + \sigma } \right )\left ( { \beta I + \mu } \right ) \ne 2\beta {\mu ^{2}}\gamma + 2\beta \mu { \mu _{2}}\left ( {\mu + \sigma } \right )\left ( {1 + \alpha I} \right )$, according to Sotomayor’s theorem, the subsystem $S_{2}$ undergoes a saddle-node bifurcation at $P_{S2}^{*}$ when $\gamma = \gamma ^{*}$. □

The saddle-node (LP) bifurcation mentioned in Theorem 2 is shown in Fig. 3(a)-(c). By solving the Eq. (8) and using the software Matcont, we can find that when $\gamma = 4.0405702$, subsystem $S_{2}$ exhibits a saddle-node bifurcation at the equilibrium $P_{{S2}}^{*} = \left ( {11.512452,5.5010698,1.7918869} \right )$, where the LP point is marked by Matcont. When the treatment rate γ is less than the critical value $\gamma ^{*}$, there are two endemic equilibria, and one endemic equilibrium when $\gamma =\gamma ^{*}$. When γ increases further, there is no endemic equilibrium, which means that the disease disappears.

Figure 3

(a)–(c) and (d)–(f) represent the bifurcation diagram of subsystem $S_{2}$ with respect to the bifurcation parameter γ at $P_{S2}^{*}$ and $P_{S2}^{0}$, respectively. The blue lines represent stable equilibria, while the red and magenta lines represent unstable equilibria

Theorem 3

Subsystem $S_{2}$ has a transcritical bifurcation at $P_{{{S2}}}^{0}$ with respect to the bifurcation parameter $\gamma = \tilde{\gamma}= \frac{{\beta \Lambda \sigma }}{{\mu \left ( {\mu + \sigma } \right )}} - {\mu _{2}}$ if $2\beta {\mu ^{2}} \ne {\sigma ^{2}}\left ( {\gamma \mu - \beta \Lambda } \right )$.

Proof

According to the characteristic equation at $P_{{{S2}}}^{0}$, we can easily know that one characteristic root is $\lambda _{1}=-\mu$. The other characteristic roots satisfy the characteristic equation as follows:

$$ {\lambda ^{2}} - Tr\left ( {{J_{S_{2}}^{0}}} \right )\lambda + Det \left ( {{J_{S_{2}}^{0}}} \right ) = 0, $$

(11)

where

$$ Tr\left ( {{J_{S_{2}}^{0}}} \right ) = - \left ( {{\mu _{2}} + \gamma + \mu + {\sigma }} \right ) \neq 0, \quad \quad \quad \quad \quad Det \left ( {{J_{S_{2}}^{0}}} \right )= \frac{{\mu \left ( {\mu + {\sigma }} \right )\left ( {{\mu _{2}} + \gamma } \right ) - \beta {\Lambda }{\sigma }}}{\mu }. $$

If $\mu \left ( {\mu + \sigma } \right )\left ( {{\mu _{2}} + \gamma } \right ) = \beta \Lambda \sigma$, the Eq. (11) has a unique zero eigenvalue. According to Sotomayor’s theorem [45, 46], let V and W denote the eigenvectors corresponding to the matrices ${J_{P_{{S2}}^{0}}}$ and $J_{P_{{S2}}^{0}}^{T}$ at zero eigenvalue, respectively.

$$ V = \left ( { \textstyle\begin{array}{c} {{V_{1}}} \\ {{V_{2}}} \\ {{V_{3}}} \end{array}\displaystyle } \right ) = \left ( { \textstyle\begin{array}{c} { \frac{{\left ( {\gamma \mu - \beta \Lambda } \right )\sigma }}{{{\mu ^{2}}}}} \\ {\gamma + {\mu _{2}}} \\ \sigma \end{array}\displaystyle } \right ), \quad \quad \quad W = \left ( { \textstyle\begin{array}{c} {{W_{1}}} \\ {{W_{2}}} \\ {{W_{3}}} \end{array}\displaystyle } \right ) = \left ( { \textstyle\begin{array}{c} 0 \\ \sigma \\ {\mu + \sigma } \end{array}\displaystyle } \right ). $$

Similarly, we have

$$\begin{aligned} &{F_{\gamma }}\left ( {P_{{S2}}^{0};\tilde{\gamma}} \right ) = \left ( { \textstyle\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\displaystyle } \right ),\quad D{F_{\gamma }}\left ( {P_{{S2}}^{0};\tilde{\gamma}} \right )V = \left ( { \textstyle\begin{array}{c} \sigma \\ 0 \\ { - \sigma } \end{array}\displaystyle } \right ),\\ & {D^{2}}F\left ( {P_{{S2}}^{0};\tilde{\gamma}} \right )\left ( {V,V} \right ) = \left ( { \textstyle\begin{array}{c} { \frac{{ - 2\beta \left ( {\gamma \mu - \beta \Lambda } \right ){\sigma ^{2}}}}{{\mu {}^{2}}}} \\ { \frac{{2\beta \left ( {\gamma \mu - \beta \Lambda } \right ){\sigma ^{2}}}}{{\mu {}^{2}}}} \\ 0 \end{array}\displaystyle } \right ). \end{aligned}$$

Thus, we can get

$$\begin{aligned}& {W^{T}}{F_{\gamma }}\left ( {P_{{S2}}^{0};\tilde{\gamma}} \right ) = 0,\\& {W^{T}}D{F_{\gamma }}\left ( {P_{{S2}}^{0};\tilde{\gamma}} \right )V = - \sigma \left ( {\mu + \sigma } \right ) \ne 0,\\& {W^{T}}\left [ {{D^{2}}F\left ( {P_{{S2}}^{0};\tilde{\gamma}} \right ) \left ( {V,V} \right )} \right ] = 2\beta \sigma - \frac{{{\sigma ^{3}}\left ( {\gamma \mu - \beta \Lambda } \right )}}{{{\mu ^{2}}}}. \end{aligned}$$

If ${W^{T}}\left [ {{D^{2}}F\left ( {P_{{S2}}^{0};\tilde{\gamma}} \right )\left ( {V,V} \right )} \right ] \ne 0$, ie., $2\beta {\mu ^{2}} \ne {\sigma ^{2}}\left ( {\gamma \mu - \beta \Lambda } \right )$, according to Sotomayor’s theorem, the parameter γ undergoes the critical threshold $\gamma = \tilde{\gamma}$, the subsystem $S_{2}$ undergoes a transcritical bifurcation at $P_{{S_{2}}}^{0}$.

The transcritical bifurcations (BP) mentioned in Theorem 3 are shown in Fig. 3(d)-(f). We give the bifurcation diagram when γ is chosen as the bifurcation parameter around $P_{S2}^{0}$ and $P_{S2}^{2}$, respectively. According to Theorem 3, at $P_{S2}^{0}$, the condition of transcritical bifurcation is satisfied. When $\gamma =\tilde{\gamma}=1.9$, a transcritical bifurcation occurs around $P_{S2}^{0}$. The stability of $P_{S2}^{0}$ changes from unstable to stable, while $P_{S2}^{2}$ changes from stable to unstable, where the BP point is detected by Matcont. In other words, transcritical bifurcation occurs between $P_{S2}^{0}$ and $P_{S2}^{2}$. From a biological perspective, the treatment rate γ plays a crucial role in controlling disease spread. When γ is greater than the critical value γ̃, $P_{S2}^{2}$ is unstable, implying that the disease ultimately disappears. □

4 Sliding mode dynamics

4.1 Existence of sliding mode

In this subsection, according to the definition of sliding region on Σ, we have

$$ \begin{aligned} \sigma \left ( {S,E,I} \right ) &= \left \langle {\nabla H\left ( {S,E,I} \right ),{F_{{S_{1}}}}\left ( {S,E,I} \right )} \right \rangle \left \langle {\nabla H\left ( {S,E,I} \right ),{F_{{S_{2}}}}\left ( {S,E,I} \right )} \right \rangle \\ &= \left ( {{\sigma }E - \gamma {I_{c}} - {\mu _{1}}{I_{c}}} \right ) \left ( {{\sigma }E - \frac{{\gamma {I_{c}}}}{{1 + \alpha {I_{c}}}} - { \mu _{2}}{I_{c}}} \right ), \end{aligned} $$

we can easily obtain that $\sigma \left ( {S,E,I} \right ) < 0$ is equivalent to the following equation:

$$ \left ( {{\sigma }E - \gamma {I_{c}} - {\mu _{1}}{I_{c}}} \right ) \left ( {{\sigma }E - \frac{{\gamma {I_{c}}}}{{1 + \alpha {I_{c}}}} - { \mu _{2}}{I_{c}}} \right ) < 0, $$

i.e., $E > \frac{{\left ( {\gamma + {\mu _{1}}} \right ){I_{c}}}}{{{\sigma}}}$ and $E < \frac{{\left ( {\gamma + {\mu _{2}}\left ( {1 + \alpha {I_{c}}} \right )} \right ){I_{c}}}}{{\left ( {1 + \alpha {I_{c}}} \right ){\sigma }}}$. Due to the uncertainty about the size relationship between $\frac{{\left ( {\gamma + {\mu _{1}}} \right ){I_{c}}}}{{{\sigma}}}$ and $\frac{{\left ( {\gamma + {\mu _{2}}\left ( {1 + \alpha {I_{c}}} \right )} \right ){I_{c}}}}{{\left ( {1 + \alpha {I_{c}}} \right ){\sigma }}}$, we will discuss it in different situations. For convenience, we define

$$ {E_{\min }} = \frac{{\left ( {\gamma + {\mu _{1}}} \right ){I_{c}}}}{{{\sigma}}}, $$

and

$$ {E_{\max }} = \frac{{\left ( {\gamma + {\mu _{2}}\left ( {1 + \alpha {I_{c}}} \right )} \right ){I_{c}}}}{{\left ( {1 + \alpha {I_{c}}} \right ){\sigma }}}. $$

(i) If $\left ( {1 + \alpha {I_{c}}} \right )\left ( {\gamma + {\mu _{1}}} \right ) < {\mu _{2}}\left ( {1 + \alpha {I_{c}}} \right ) + \gamma$, there is an attractive sliding region in the system (4), i.e.,

$$ \Sigma s \stackrel{\textstyle .}{=} \left \{ {\left ( {S,E,I} \right )|S \ge 0,{E_{\min }} < E < {E_{\max }},I = {I_{c}}} \right \}. $$

(ii) If $\left ( {1 + \alpha {I_{c}}} \right )\left ( {\gamma + {\mu _{1}}} \right ) > {\mu _{2}}\left ( {1 + \alpha {I_{c}}} \right ) + \gamma$, there is a repulsive sliding region in the system (4), i.e.,

$$ \Sigma e \stackrel{\textstyle .}{=} \left \{ {\left ( {S,E,I} \right )|S \ge 0,{E_{\max }} < E < {E_{\min }},I = {I_{c}}} \right \}. $$

The possible sliding region cases are shown in Fig. 4. In Fig. 4, the green part represents the attractive sliding region, while the cyan part represents the repulsive sliding region.

Figure 4

Phase diagram of the sliding region of the Filippov system (4) in the $(S, E, I_{c})$ plane, where the red and blue dashed lines indicate the tangential lines $T_{S1}$ and $T_{S2}$ of the subsystems $S_{1}$ and $S_{2}$, respectively. The blue and red solid dots indicate the cusp singularities $Z^{*}_{S1}$ and $Z^{*}_{S2}$ of the subsystems $S_{1}$ and $S_{2}$, respectively

4.2 Tangential singularities

According to Definition 3, we know that the tangent singularities ${T_{{Si}}}\left ( {{S_{i}},{E_{i}},{I_{i}}} \right ),i = 1,2$ of the Filippov system (4) satisfy the following equation:

$$ \left \{ { \textstyle\begin{array}{l} {H\left ( {S,E,I} \right ) = 0,} \\ {\left \langle {\nabla H\left ( {S,E,I} \right ),{F_{{S1}}}\left ( {S,E,I} \right )} \right \rangle = 0,} \end{array}\displaystyle } \right . $$

and

$$ \left \{ { \textstyle\begin{array}{l} {H\left ( {S,E,I} \right ) = 0,} \\ {\left \langle {\nabla H\left ( {S,E,I} \right ),{F_{{S2}}}\left ( {S,E,I} \right )} \right \rangle = 0.} \end{array}\displaystyle } \right . $$

The tangential sets of ${{F_{{S1}}}\left ( {S,E,I} \right )}$ and ${F_{S2}\left ( {S,E,I} \right )}$ are given, respectively, by the straight lines:

$$\begin{aligned}& {T_{{S1}}} \stackrel{\textstyle .}{=} \left \{ {\left ( {S,E,I} \right ) \in \Sigma |S \ge 0,E = {E_{\min }} > 0,I = {I_{c}}} \right \},\\& {T_{{S2}}} \stackrel{\textstyle .}{=} \left \{ {\left ( {S,E,I} \right ) \in \Sigma |S \ge 0,E = {E_{\max }} > 0,I = {I_{c}}} \right \}. \end{aligned}$$

Furthermore, the classification of the tangential singularities $T_{S1}$ and $T_{S2}$ is as follows:

(i) For all $Z_{{S1}}^{*} \left ( {S_{{S1}}^{*} ,E_{{S1}}^{*} ,{I_{c}}} \right ) \in {T_{{S1}}}$, we obtain that all points in ${T_{{S1}}}$ are visible fold singularities for $S < S_{{S1}}^{*}$ and invisible fold singularities for $S > S_{{S1}}^{*}$.

In light of ${L_{{F_{{S1}}}}}H\left ( {Z_{{S1}}^{*} } \right ) = L_{{F_{{S1}}}}^{2}H \left ( {Z_{{S1}}^{*} } \right ) = 0$ and $L_{{F_{{S1}}}}^{3}H\left ( {Z_{{S1}}^{*} } \right ) \ne 0$ (i.e., $Z_{{S1}}^{*} \ne P_{{{{S1}}}}^{1}$), one obtains that the point $Z_{{S1}}^{*} \left ( {S_{{S1}}^{*} ,E_{{S1}}^{*} ,{I_{c}}} \right )$ with

$$ S_{{S1}}^{*} = \frac{{\left ( {\mu + {\sigma}} \right )\left ( {\gamma + {\mu _{1}}} \right )}}{{{\sigma }\beta }}, \quad \quad \quad E_{{S1}}^{*} = \frac{{\left ( {\gamma + {\mu _{1}}} \right ){I_{c}}}}{{{\sigma}}}, $$

is a cusp singularity, i.e., ${F_{{S1}}}$ deviates from Σ by a trajectory through the point $Z_{{S1}}^{*}$. The point $Z_{{S1}}^{*}$ divides ${T_{{S1}}}$ into two fold singularity branches.

(ii) Similarly, for all $Z_{{S2}}^{*} \left ( {S_{{S_{2}}}^{*} ,E_{{S2}}^{*} ,{I_{c}}} \right ) \in {T_{{S2}}}$, we obtain that all points in ${T_{{S2}}}$ are visible fold singularities for $S >S_{{S_{2}}}^{*}$ and invisible fold singularities for $S < S_{{S_{2}}}^{*}$.

In light of ${L_{{F_{{S2}}}}}H\left ( {Z_{{S2}}^{*}} \right ) = L_{{F_{{S2}}}}^{2}H \left ( {Z_{{S2}}^{*}} \right ) = 0$ and $L_{{F_{{S2}}}}^{3}H\left ( {Z_{{S2}}^{*}} \right ) \ne 0$ (i.e., $Z_{{S2}}^{*} \ne P_{{{{S2}}}}^{2}$), one obtains that the point $Z_{{S2}}^{*} \left ( {S_{{S2}}^{*} ,E_{{S2}}^{*} ,{I_{c}}} \right )$ with

$$ S_{{S2}}^{*} = \frac{{\left ( {\gamma + {\mu _{2}}\left ( {1 + \alpha {\mkern 1mu} {I_{c}}} \right )} \right )\left ( {\mu + {\sigma}} \right ) }}{{\left ( {\alpha {\mkern 1mu} {I_{c}} + 1} \right ){\sigma}{\mkern 1mu} \beta }}, \quad \quad \quad E_{{S2}}^{*} = \frac{{{\mkern 1mu} {I_{c}}\left ( {\gamma + {\mu _{2}}\left ( {1 + \alpha {\mkern 1mu} {\mkern 1mu} {I_{c}}} \right )} \right )}}{{\left ( {\alpha {\mkern 1mu} {\mkern 1mu} {I_{c}} + 1} \right ){\sigma}}}, $$

is a cusp singularity, i.e., ${F_{{S2}}}$ deviates from Σ by a trajectory through the point $Z_{{S2}}^{*}$. The point $Z_{{S2}}^{*}$ divides ${T_{{S2}}}$ into two fold singularity branches. The cusp singularities $Z_{{S1}}^{*}$ and $Z_{{S2}}^{*}$, as well as the tangential lines $T_{S1}$ and $T_{S2}$, are shown in Fig. 4.

4.3 Regular equilibrium, boundary equilibrium, pseudo-equilibrium

According to Sect. 2, we know that there are four types of equilibria in the Filippov system (4), that is, virtual equilibrium, real equilibrium, boundary equilibrium, and pseudo-equilibrium. Next, we analyze the four types of equilibria in detail as follows:

(i) For the subsystem $S_{1}$, there are two equilibria

$$ P_{{{{S1}}}}^{0}\left ( {\frac{{{\Lambda }}}{\mu },0,0} \right ), \quad \quad \quad P_{{{{S1}}}}^{1}\left ( {S_{{{S1}}}^{1},E_{{{{S1}}}}^{1},I_{{{{S1}}}}^{1}} \right ), $$

where $P_{{{{S1}}}}^{0}$ is always real equilibrium, denoted $P_{{{{S1}}}}^{0 + }$; if $I_{{{{S1}}}}^{1} < {I_{c}}$, $P_{{{{S1}}}}^{1}$ is real equilibrium, denoted $P_{{{{S1}}}}^{1 + }$; if $I_{{{{S1}}}}^{1} > {I_{c}}$, $P_{{{{S1}}}}^{1}$ is virtual equilibrium, denoted $P_{{{{S1}}}}^{1 - }$; if $I_{{{{S1}}}}^{1} = {I_{c}}$, $P_{{{{S1}}}}^{1}$ is boundary equilibrium, denoted $P_{{{{S1}}}}^{1 B }$.

(ii) Similarly, for the subsystem $S_{2}$, there are at most three equilibria

$$ P_{{{{S2}}}}^{0}\left ( {\frac{{{\Lambda}}}{\mu },0,0} \right ), \quad \quad \quad P_{{{{S1}}}}^{i}\left ( {S_{{{S1}}}^{i},E_{{{{S1}}}}^{i},I_{{{{S1}}}}^{i}} \right ), \quad i=1,2, $$

where $P_{{{{S2}}}}^{0}$ is always virtual equilibrium, denoted $P_{{{{S2}}}}^{0 - }$; if $I_{{{{S2}}}}^{i} < {I_{c}}$, $P_{{{{S2}}}}^{i}$ is virtual equilibrium, denoted $P_{{{{S2}}}}^{i - }$; if $I_{{{{S2}}}}^{i} > {I_{c}}$, $P_{{{{S2}}}}^{i}$ is real equilibrium, denoted $P_{{{{S2}}}}^{i + }$; if $I_{{{{S2}}}}^{i} = {I_{c}}$, $P_{{{{S2}}}}^{i}$ is boundary equilibrium, denoted $P_{{{{S2}}}}^{i B }$.

According to the Filippov convex method [1], the pseudo-equilibrium for the Filippov system (4) satisfies the following sliding vector field:

$$ {\mathord{ \stackrel{{\lower 3pt\hbox{$\scriptscriptstyle \frown $}} }{F}} _{s}} = \frac{1}{{\dot{I}|{ _{{F_{{S_{2}}}}}} - \dot{I}{|_{{F_{{S_{1}}}}}}}} \left [ { \textstyle\begin{array}{c} {\dot{I}|{ _{{F_{{S_{2}}}}}} \cdot \dot{S}{|_{{F_{{S_{1}}}}}} - \dot{I}|{ _{{F_{{S_{1}}}}}} \cdot \dot{S}{|_{{F_{{S_{2}}}}}}} \\ {\dot{I}|{ _{{F_{{S_{2}}}}}} \cdot \dot{E}{|_{{F_{{S_{1}}}}}} - \dot{I}|{ _{{F_{{S_{1}}}}}} \cdot \dot{E}{|_{{F_{{S_{2}}}}}}} \\ 0 \end{array}\displaystyle } \right ], $$

(12)

where

$$\begin{aligned}& \dot{I}|{ _{{F_{{S_{2}}}}}} - \dot{I}{|_{{F_{{S_{1}}}}}} = \gamma I - \frac{{\gamma I}}{{1 + \alpha I}} - {\mu _{2}}I + {\mu _{1}}I,\\& \textstyle\begin{array}{l} \dot{I}|{ _{{F_{{S_{2}}}}}} \cdot \dot{S}{|_{{F_{{S_{1}}}}}} - \dot{I}|{ _{{F_{{S_{1}}}}}} \cdot \dot{S}{|_{{F_{{S_{2}}}}}} = \sigma E\left ( {\gamma I - \frac{{\gamma I}}{{1 + \alpha I}}} \right ) - \left ( { \frac{{\gamma I}}{{1 + \alpha I}} + {\mu _{2}}I - \gamma I - {\mu _{1}}I} \right ) \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad{\mathrm{{ }}}\left ( {\Lambda - \beta SI - \mu S} \right ) - { \mu _{2}}\gamma {I^{2}} + \frac{{{\mu _{1}}\gamma {I^{2}}}}{{1 + \alpha I}}, \end{array}\displaystyle \\& \dot{I}|{ _{{F_{{S_{2}}}}}} \cdot \dot{E}{|_{{F_{{S_{1}}}}}} - \dot{I}|{ _{{F_{{S_{1}}}}}} \cdot \dot{E}{|_{{F_{{S_{2}}}}}} = \left ( {\gamma I - \frac{{\gamma I}}{{1 + \alpha I}} + {\mu _{1}}I - {\mu _{2}}I} \right ) \cdot \left ( {\beta SI - \mu E - \sigma E} \right ). \end{aligned}$$

According to Definition 4, we can obtain the pseudo-equilibrium $P_{S} (S_{P}^{*},E_{P}^{*}, I_{c})$, where

$$\begin{aligned} &S_{p}^{*} = \frac{{\left ( {\left ( {{\mu _{2}}\gamma (1 + \alpha {I_{c}}) - {\mu _{1}}\gamma } \right ){I_{c}} - \Lambda \left ( {\left ( {\gamma + {\mu _{1}} - {\mu _{2}}} \right )(1 + \alpha {I_{c}}) - \gamma } \right )} \right )(\mu + \sigma )}}{{ - (\beta {I_{c}} + \mu )(\mu + \sigma )\left ( {\left ( {\gamma + {\mu _{1}} - {\mu _{2}}} \right )(1 + \alpha {I_{c}}) - \gamma } \right ) + \alpha \sigma \beta \gamma I_{c}^{2}}}, \\ & E_{P}^{*} = \frac{{S_{P}^{*} \beta {I_{c}}}}{{\left ( {\mu + \sigma } \right )}}. \end{aligned}$$

We normalize the sliding mode dynamics Eq. (12) and obtain

$$ {F_{s}}\left ( {S,E} \right ) = \left [ { \textstyle\begin{array}{c} {\dot{I}|{ _{{F_{{S_{2}}}}}} \cdot \dot{S}{|_{{F_{{S_{1}}}}}} - \dot{I}|{ _{{F_{{S_{1}}}}}} \cdot \dot{S}{|_{{F_{{S_{2}}}}}}} \\ {\dot{I}|{ _{{F_{{S_{2}}}}}} \cdot \dot{E}{|_{{F_{{S_{1}}}}}} - \dot{I}|{ _{{F_{{S_{1}}}}}} \cdot \dot{E}{|_{{F_{{S_{2}}}}}}} \end{array}\displaystyle } \right ]. $$

The Jacobian matrix at the pseudo-equilibrium $P_{S}$ is

$$ J{|_{P_{S}}} = \left [ { \textstyle\begin{array}{c@{\quad}c} { - \left ( {\beta I + \mu } \right )\left ( {\gamma I - \frac{{\gamma I}}{{1 + \alpha I}} + {\mu _{1}}I - {\mu _{2}}I} \right )}&{\sigma \left ( {\gamma I - \frac{{\gamma I}}{{1 + \alpha I}}} \right )} \\ {\beta I\left ( {\gamma I - \frac{{\gamma I}}{{1 + \alpha I}} + {\mu _{1}}I - {\mu _{2}}I} \right )}&{ - \left ( {\mu + \sigma } \right )\left ( { \gamma I - \frac{{\gamma I}}{{1 + \alpha I}} + {\mu _{1}}I - {\mu _{2}}I} \right )} \end{array}\displaystyle } \right ]. $$

The characteristic equation is given by

$$ {\lambda ^{2}} - Tr(J{_{{P_{S}}}})\lambda + Det(J{_{{P_{S}}}}) = 0, $$

where

$$\begin{aligned}& Tr(J{_{{P_{S}}}})=-\left ( {\gamma I - \frac{{\gamma I}}{{1 + \alpha I}} + {\mu _{1}}I - {\mu _{2}}I} \right )\left ( {\beta I + 2\mu + \sigma } \right ), \\& \begin{aligned} Det(J{_{{P_{S}}}})&= (\beta I + \mu )\left ( {\mu + \sigma } \right ){ \left ( {\gamma I - \frac{{\gamma I}}{{1 + \alpha I}} + {\mu _{1}}I - { \mu _{2}}I} \right )^{2}} \\ &-\left ( {\gamma I - \frac{{\gamma I}}{{1 + \alpha I}}} \right ) \left ( {\gamma I - \frac{{\gamma I}}{{1 + \alpha I}} + {\mu _{1}}I - { \mu _{2}}I} \right ). \end{aligned} \end{aligned}$$

If $Det\left ( {J{|_{P_{S}}}} \right ) > 0$, $Tr\left ( {J{|_{P_{S}}}} \right ) < 0$, the pseudo-equilibrium $P_{S}$ is a locally asymptotically stable focus or node. If $Det\left ( {J{|_{P_{S}}}} \right ) < 0$, the pseudo-equilibrium $P_{S}$ is an unstable saddle.

Remark 2

When the pseudo-equilibrium $P_{S}$ is in the sliding region, it is a real equilibrium, denoted ${P_{S}^{+}}$; otherwise, it is a virtual equilibrium, denoted ${P_{S}^{-}}$.

4.4 Boundary equilibrium bifurcation

To demonstrate the boundary equilibrium bifurcation of system (4), we select $I_{c}$ as the bifurcation parameter and fix on the other parameters. From Fig. 5, we can see that the system (4) undergoes two boundary equilibrium bifurcations. When $I_{c} = 0.3$, there is a stable real equilibrium $P_{{S2}}^{2 + }$, and the pseudo-equilibrium ${P_{S}}$ is on Σc, which is a virtual equilibrium, denoted as ${P_{S}^{-}}$. When $I_{c} = 0.6490$, $P_{S}^{+}$, $P_{{S2}}^{2 + }$ and $Z_{{S2}}^{*}$ collide with together, denoted $E_{{S_{2}}}^{B}$. This results in a boundary equilibrium bifurcation in the system (4), as shown in Fig. 5(b). When $I_{c} = 1$, there exists a stable admissible pseudo-equilibrium ${P_{S}^{+}}$, as in Fig. 5(c). When $I_{c} = 1.3586$, ${P_{S}^{+}}$, $P_{{S1}}^{1 + }$ and $Z_{{S1}}^{*}$ collide with together, denoted as $E_{{S_{1}}}^{B}$. At this moment, the system (4) generates a boundary equilibrium bifurcation, as in Fig. 5(d). When $I_{c} = 1.5$, pseudo-equilibrium ${P_{S}}$ disappears and $P_{{S1}}^{1}$ becomes real equilibrium, as shown in Fig. 5(e).

Figure 5

Boundary bifurcation of Filippov system (4) when the threshold $I_{c}$ is selected as (a) $I_{c}=0.3$ (b) $I_{c}=0.6490$ (c) $I_{c}=1$ (d)$I_{c}=1.3586$ (e) $I_{c}=1.5$. We fix all other parameters as follows: $\Lambda =2, \beta =0.31, \mu =0.3, \gamma =0.31, \sigma =0.52, \alpha =0.5, \mu _{1}=0.35, \mu _{2}=0.61$

5 Dynamics of the system (4)

In this section, we mainly discuss the whole dynamical behaviors of system (4). The disease-free equilibrium $P_{{S1}}^{0}$ is always a real equilibrium for the subsystem $S_{1}$, denoted $P_{{S1}}^{0+}$. For subsystem $S_{2}$, the disease-free equilibrium $P_{{S2}}^{0}$ is always a virtual equilibrium, denoted $P_{{S2}}^{0-}$. Note that for selecting different threshold $I_{c}$, the system (4) can exhibit various dynamic behaviors. The existence of endemic equilibria depends on the threshold $R_{0i}(i = 1, 2)$ with respect to 1. Therefore, we consider the following three cases:

Case (A): $1 > {R_{01}} > {R_{02}}$

In this case, there is no endemic equilibrium for subsystem $S_{1}$. If $\Delta \ge 0$, $B > 0$, the subsystem $S_{2}$ has at most two endemic equilibria $P_{{S2}}^{1}$ and $P_{{S2}}^{2}$. According to Theorem 1, $P_{{S2}}^{2}$ is a stable node, while $P_{{S2}}^{1}$ is an unstable saddle. For the system (4), the disease-free equilibrium $P_{{S1}}^{0}$ is always stable. Next, we discuss the size relationship between $I_{{S2}}^{2}$, $I_{{S2}}^{1}$ and ${I_{c}}$ ($I_{{S2}}^{2} > I_{{S2}}^{1}$).

Subcase (A1): If ${I_{c}} < I_{{S2}}^{1}$, the endemic equilibria $P_{{S2}}^{1}$, $P_{{S2}}^{2}$ are real equilibria, denoted $P_{{S2}}^{1+}$, $P_{{S2}}^{2+}$, respectively. We find that the solution trajectory of the system (4) eventually tends to the endemic equilibrium $P_{{S2}}^{2+}$ or the disease-free equilibrium $P_{S1}^{0+}$. Therefore, both $P_{{S2}}^{2+}$ and $P_{S1}^{0+}$ are bistable in Fig. 6(a).

Figure 6

(a)–(c): Phase diagram of the Filippov system (4) for subcase (A1–A3) when the threshold $I_{c}$ is selected as (a) $I_{c}=0.1$ (b) $I_{c}=1$ (c) $I_{c}=2$. We fix all other parameters as follows: $\Lambda =10, \beta =0.11, \mu =0.32, \gamma =0.86, \sigma =1/7, \alpha =4, \mu _{1}=0.5, \mu _{2}=0.6$. (d)–(e): Phase diagram of the Filippov system (4) for subcase (B1–B2) when the threshold $I_{c}$ is selected as (d) $I_{c}=0.2$ (e) $I_{c}=3$. We fix all other parameters as follows: $\Lambda =5, \beta =0.31, \mu =0.35, \gamma =0.53, \sigma =1/7, \alpha =0.9, \mu _{1}=0.36, \mu _{2}=0.76$

Subcase (A2): When $I_{{S2}}^{2} > {I_{c}} > I_{{S2}}^{1}$, the endemic equilibrium $P_{{S2}}^{1}$ is virtual equilibrium, denoted $P_{{S2}}^{1-}$, and $P_{{S2}}^{2}$ is a real equilibrium, denoted $P_{{S2}}^{2+}$. We find that the solution trajectories of system (4) starting from region $S_{1}$ tend directly the disease-free equilibrium $P_{S1}^{0+}$, or cross the switching surface Σ and finally converge to the endemic equilibrium $P_{{S2}}^{2+}$; the solution trajectories of system (4) starting from region $S_{2}$ tend directly to the endemic equilibrium $P_{{S2}}^{2+}$ or cross the switching surface Σ and finally tend the disease-free equilibrium $P_{S1}^{0+}$. Therefore, both $P_{{S2}}^{2+}$ and $P_{S1}^{0+}$ are bistable, as shown in Fig. 6(b).

Subcase (A3): When ${I_{c}} > I_{{S2}}^{2}$, the endemic equilibria $P_{{S2}}^{1}$, $P_{{S2}}^{2}$ are virtual equilibria, denoted as $P_{{S2}}^{1-}$, $P_{{S2}}^{2-}$, respectively. The solution trajectories from different regions eventually converge to the disease-free equilibrium $P_{S1}^{0+}$ in Fig. 6(c).

Case (B): ${R_{01}} > 1 > {R_{02}}$

In this case, subsystem $S_{1}$ has one endemic equilibrium $P_{{S1}}^{1}$, while subsystem $S_{2}$ has at most two endemic equilibria $P_{{S2}}^{1}$ and $P_{{S2}}^{2}$. The disease-free equilibrium $P_{{S1}}^{0}$ is unstable for subsystem $S_{1}$. Next, we discuss the size relationships with respect to $I_{{S2}}^{2}$, $I_{{S2}}^{1}$, $I_{{S1}}^{1}$ and ${I_{c}}$.

Subsystem $S_{1}$ has an endemic equilibrium $P_{{S1}}^{1}$ if $\Delta \ge 0$, $B < 0$, and there is no endemic equilibrium for subsystem $S_{2}$.

Subcase (B1): When ${I_{c}} > I_{{S1}}^{1}$, the endemic equilibrium $P_{{S1}}^{1}$ is virtual equilibrium, denoted as $P_{{S1}}^{1-}$; at this point, the system (4) exists a pseudo-equilibrium and is asymptotically stable, denoted as $P_{S}^{+}$. The solution trajectory of the system (4) eventually tends to the pseudo-equilibrium $P_{S}^{+}$, as shown in Fig. 6(d);

Subcase (B2): When ${I_{c}} < I_{{S1}}^{1}$, the endemic equilibrium $P_{{S1}}^{1}$ is real equilibrium, denoted as $P_{{S1}}^{1+}$, the solution trajectories from different regions eventually converge to the endemic equilibrium $P_{{S1}}^{1+}$, as shown in Fig. 6(e);

If $\Delta \ge 0$, $B > 0$, there is one endemic equilibrium for subsystem $S_{1}$ and two endemic equilibria for subsystem $S_{2}$.

Subcase (B3): When ${I_{c}} > I_{{S2}}^{2} > I_{{S1}}^{1} > I_{{S2}}^{1}$, the endemic equilibrium $P_{{S1}}^{1}$ is real equilibrium, denoted $P_{{S1}}^{1 + }$; the endemic equilibria $P_{{S2}}^{1}$, $P_{{S2}}^{2}$ are both virtual equilibria, denoted $P_{{S2}}^{1 - }$, $P_{{S2}}^{2 - }$, respectively. The solution trajectories of the system (4) eventually tend to the endemic equilibrium $P_{{S1}}^{1 + }$, as shown in Fig. 7(a);

Figure 7

(a)–(c): Phase diagram of the Filippov system (4) for subcase (B3-B5) when the threshold $I_{c}$ is selected as (a) $I_{c}=2$ (b) $I_{c}=0.65$ (c) $I_{c}=0.06$. We fix all other parameters as follows: $\Lambda =11, \beta =0.19, \mu =0.38, \gamma =0.86, \sigma =1/6, \alpha =2, \mu _{1}=0.54, \mu _{2}=0.89$. (d)–(f): Phase diagram of the Filippov system (4) for subcase (C1–C3) when the threshold $I_{c}$ is selected as (d) $I_{c}=2$ (e) $I_{c}=4$ (f) $I_{c}=6$. We fix all other parameters as follows: $\Lambda =10, \beta =0.31, \mu =0.25, \gamma =0.61, \sigma =1/7, \alpha =2, \mu _{1}=0.35, \mu _{2}=0.6$

Subcase (B4): When $I_{{S2}}^{2} > {I_{c}} > I_{{S1}}^{1} > I_{{S2}}^{1}$, the endemic equilibria $P_{{S1}}^{1}$, $P_{{S2}}^{2}$ are both real equilibria, denoted as $P_{{S1}}^{1 + }$, $P_{{S2}}^{2 + }$, respectively, and endemic equilibrium $P_{{S2}}^{1}$ is virtual equilibrium, denoted as $P_{{S2}}^{1 - }$. The solution trajectories from different regions eventually converge to the equilibrium $P_{{S1}}^{1 + }$ or $P_{{S2}}^{2 + }$. Therefore, both $P_{{S1}}^{1 + }$ and $P_{{S2}}^{2 + }$ are bistable, as shown in Fig. 7(b);

Subcase (B5): When $I_{{S2}}^{2} > I_{{S1}}^{1} > I_{{S2}}^{1} > {I_{c}}$, the endemic equilibria $P_{{S2}}^{1}$, $P_{{S2}}^{2}$ are both real equilibria, denoted as $P_{{S2}}^{1 + }$, $P_{{S2}}^{2 + }$, respectively. The endemic equilibrium $P_{{S1}}^{1}$ is virtual equilibrium, denoted as $P_{{S1}}^{1 - }$. At this point, the system exists a pseudo-equilibrium point $P_{S}$ and is asymptotically stable, denoted as $P_{S}^{+}$. We can find that the solution trajectories from different regions eventually converge to the equilibrium $P_{{S2}}^{2 + }$ or $P_{S}^{+}$. Therefore, both $P_{{S2}}^{2 + }$ and $P_{S}^{+}$ are bistable, as shown in Fig. 7(c).

Case (C): ${R_{01}} > {R_{02}} > 1$

In this case, there is an endemic equilibrium $P_{{S1}}^{1}$ for subsystems $S_{1}$ and $P_{{S2}}^{2}$ for subsystem $S_{2}$ when $\Delta > 0$. Additionally, the endemic equilibrium $P_{{S1}}^{0}$ is unstable for subsystem $S_{1}$. Next, we will analyze the size relationship between $I_{{S2}}^{2}$, $I_{{S1}}^{1}$ and ${I_{c}}$.

Subcase (C1): When $I_{{S2}}^{2} > I_{{S1}}^{1} > {I_{c}}$, the endemic equilibrium $P_{{S1}}^{1}$ is virtual equilibrium, denoted as $P_{{S1}}^{1-}$, and endemic equilibrium $P_{{S2}}^{2}$ is real equilibrium, denoted as $P_{{S2}}^{2+}$. We find that solution trajectories starting from different regions eventually tend to the endemic equilibrium $P_{{S2}}^{2+}$, as shown in Fig. 7(d).

Subcase (C2): When $I_{{S2}}^{2} > {I_{c}} > I_{{S1}}^{1}$, the endemic equilibria $P_{{S1}}^{1}$, $P_{{S2}}^{2}$ are both real equilibria, denoted as $P_{{S1}}^{1+}$, $P_{{S2}}^{2+}$, respectively. At this point, the solution trajectories the system (4) converge partly to the endemic equilibrium $P_{{S2}}^{2+}$ and partly to the endemic equilibrium $P_{{S1}}^{1+}$. The system is bistable, as shown in Fig. 7(e).

Subcase (C3): When ${I_{c}} > I_{{S2}}^{2} > I_{{S1}}^{1}$, the endemic equilibrium $P_{{S1}}^{1}$ is real equilibrium, denoted $P_{{S1}}^{1+}$, and $P_{{S2}}^{2}$ is virtual equilibrium, denoted $P_{{S2}}^{2-}$. The solution trajectories from different regions eventually converge to the endemic equilibrium $P_{{S1}}^{1+}$, as shown in Fig. 7(f).

Subcase (C4): When $I_{{S1}}^{1} > I_{{S2}}^{2} > {I_{c}}$, the endemic equilibrium $P_{{S1}}^{1}$ is virtual equilibrium, denoted as $P_{{S1}}^{1-}$, and endemic equilibrium $P_{{S2}}^{2}$ is real equilibrium, denoted as $P_{{S2}}^{2+}$. The solution trajectories of the system (4) eventually tend to the endemic equilibrium $P_{{S2}}^{2+}$, as shown in Fig. 8(a).

Figure 8

Phase diagram of the Filippov system (4) for subcase (C4–C6) when the threshold $I_{c}$ is selected as (a) $I_{c}=1$ (b) $I_{c}=3$ (c) $I_{c}=5$. We fix all other parameters as follows: $\Lambda =10, \beta =0.31, \mu =0.25, \gamma =0.61, \sigma =1/7, \alpha =0.2, \mu _{1}=0.35, \mu _{2}=0.75$

Subcase (C5): When $I_{{S1}}^{1} > {I_{c}} > I_{{S2}}^{2}$, the endemic equilibria $P_{{S1}}^{1}$, $P_{{S2}}^{2}$ are both virtual equilibria, denoted as $P_{{S1}}^{1-}$, $P_{{S2}}^{2-}$, respectively. At this point, the pseudo-equilibrium $P_{S}$ exists and is asymptotically stable, denoted as $P_{S}^{+}$. The solution trajectories of system (4) starting from different regions, eventually converge to pseudo-equilibrium $P_{S}^{+}$, as shown in Fig. 8(b).

Subcase (C6): When ${I_{c}} > I_{{S1}}^{1} > I_{{S2}}^{2}$, the endemic equilibrium $P_{{S1}}^{1}$ is real equilibrium, denoted $P_{{S1}}^{1+}$, and $P_{{S2}}^{2}$ is virtual equilibrium, denoted $P_{{S2}}^{2-}$. We find that the solution trajectories of system (4) starting from different regions eventually tend to the equilibrium $P_{{S1}}^{1+}$, as shown in Fig. 8(c).

In summary, the solution trajectories of the system (4), for different initial values, eventually converge to i) either the endemic or disease-free equilibria; ii) either the endemic or pseudo-equilibria; iii) the endemic equilibria of the two subsystems. In other words, the system (4) can exhibit bistability of endemic and disease-free equilibria, bistability of endemic and pseudo-equilibria, and bistability of the endemic equilibria of the two subsystems.

Next, we analyze the impact of medical resources and latent compartment on dynamical behaviors of system (4). According to Fig. 9(a)–(b), we found that when the latent period is not considered(i.e.,$1/\sigma =1/100$ day), the number of infected individuals will be overestimated. In Fig. 9(c)–(e), $P_{S}^{0}$, $P_{S}^{+}$ represent the disease-free equilibrium and pseudo-equilibrium, respectively. $P_{S}^{1}$, $P_{S1}^{-}$, $P_{S2}^{+}$ represent the endemic equilibria. We fix all parameters as follows: $\Lambda =10, \beta =0.31, \mu =0.25, \gamma =0.61, \sigma =1/7, \alpha =0.2, \mu _{1}=0.35, \mu _{2}=0.75, I_{c}=3$. By comparing Figs. 9(c) and 9(e), we find that under the same parameters in Fig. 9(c), the solution trajectories of the system (3) ultimately tend to the disease-free equilibrium $P_{S}^{0}$. In Fig. 9(e), the solution trajectories of the system (4) ultimately tend to the pseudo-equilibrium $P_{S}^{+}$, indicating that when medical resources are sufficient, the disease will eventually disappear. By comparing Figs. 9(d) and 9(e), we find that under the same parameters, in Fig. 9(d), the solution trajectories of the system eventually tend to the endemic equilibrium $P_{S2}^{+}$ of subsystem $S_{2}$. This means that considering the latent compartment stabilizes the number of infected individuals at a realistic level (i.e., $I_{c}$).

Figure 9

(a)–(b): The time series graphs with different latent period $1/\sigma$. (c)–(e): The phase diagrams are for the smooth SEIS ODE model (i.e., model (3)), the Filippov SIS model without considering the latent compartment, and the Filippov SEIS model (i.e., model (4)), respectively. (f): Fitting of bacterial and amoebic dysentery cases in China

Meanwhile, we will simulate the number of existing cases of bacterial and amoebic dysentery cases in China from 2015 to 2022 [47]. As is well known, bacterial and amoebic dysentery are infectious diseases with the latent period. Figure 9(f) displays the temporal progression of current infection cases, with corresponding parameters presented in Table 1. The smooth SEIS ODE model (i.e., model (3)) is depicted by the cyan line, the non-smooth Filippov SIS model without the latent compartment is represented by the green line, and the non-smooth Filippov SEIS model accounting for the latent compartment is shown by the blue line (i.e., model (4)). We found that the simulation results of model (4) fit well with the actual data, which means that considering the latent compartment and segmented treatment is more in line with the true situation of disease transmission.

Table 1 Parameters of Fig. 9(f)

6 Conclusion

This paper proposes a non-smooth three-dimensional Filippov SEIS epidemic model with a segmented treatment function depending on the number of infected individuals. At the same time, we introduce a latent compartment and the segmented treatment function to explore the influence of limited medical resources on disease transmission, which is different from many previous studies on smooth ODE models [19, 41] and planar Filippov models [25, 28].

In particular, we intentionally studied the dynamical behaviors of model (4) when the threshold values changes, i.e., $R_{01}$, $R_{02}$ and 1. It is worth noting that: when $1 > {R_{01}} > {R_{02}}$, according to Fig. 6(a-b), We find that the solution trajectories of the system (4) tend to the endemic equilibrium $P_{S2}^{2+}$ or the disease-free equilibrium $P_{S1}^{0+}$. In the biological meaning, it indicates that infectious diseases would persist when medical resources are insufficient, even the basic reproduction numbers in two subsystems are less than 1. Figure 6 (c) shows that when the threshold function $I_{c}=4.5$, the solution trajectories of system (4) tend to the disease-free equilibrium $P_{S1}^{0+}$, which means that the disease would disappear. According to Figs. 6(d-e)-8, when ${R_{01}} > 1 > {R_{02}}$ or ${R_{01}} > {R_{02}} > 1$, the system (4) stabilizes at the endemic equilibria or at the pseudo-equilibrium, depending on the threshold $I_{c}$, which indicates the importance of the choice of the threshold strategy. Therefore, selecting appropriate control parameters to determine whether to initiate intervention in the spread of infectious diseases is crucial. Meanwhile, we investigate the impact of latent compartment and medical resources on system (4). According to the Fig. 9(a)–(e), when the latent compartment is not considered, it overestimates the number of infected individuals, which is not conducive to disease control. At the same time, we find that sufficient medical resources can ultimately eliminate the disease. Figure 9 (f) confirms that model (4) is more in line with the reality of disease transmission.

The main results suggest that a suitable threshold $I_{c}$ should be selected to determine treatment strategies in the control of diseases spread. If the number of infected individuals exceeds the threshold $I_{c}$, due to the limitation of medical resources, emergency medical services, such as isolation treatment, personal protection, and medical observation, should be taken immediately to control the disease spread as soon as possible. These results not only could help us to assess how to minimize the number of infected individuals in the presence of limited medical resources, but also could be beneficial for prevention of emergency infection.