A network model of social contacts with small-world and scale-free features, tunable connectivity, and geographic restrictions

1.   Introduction

With the development of society, the number of deaths caused by influenza infection has had gradually increased. For example, the global pandemic in 1918 caused more deaths in the world than in the First World War^[1,2]. According to statistical analysis, the global influenza pandemic is caused by H2N2 virus, which is called "Asian influenza" because it first occurred in Asia. The incidence rate was about 15–30% ^[3,4]. Recently, COVID-19 also has a great impact on human life. According to the latest real-time statistics of the World Health Organization, as of October 10, 2022, the cumulative number of confirmed cases worldwide exceeded 600 million, and the cumulative number of deaths was about 6.5 million^[5]. Therefore, in order to better control and reduce the number of deaths, how to effectively and quickly control the spread of disease is worthy of further study.

In the face of the global pandemic, many countries have responded to and mitigated the impact of influenza on human life through media reports, vaccination, disinfection, the use of protective equipment (such as masks), quarantine and other measures. In addition, some biological mathematicians have also established many mathematical models in order to better understand influenza infection and quantify the effectiveness of various control measures ^{[6,7,8,9,10,11,12,13]}. For example, considering social factors such as vaccination, media reports and protective measures, ^{[14,15,16,17,18,19,20,21]} established some different forms of disease models to analyze the impact of disease transmission. Practice shows that frequent hand washing, disinfection or wearing protective masks, as well as reducing or avoiding close contact with infected persons, can effectively reduce transmission ^{[22,23,24,25,26,27,28,29]}. However, media reports on influenza cases and deaths may have a great impact on the public, because the public will take some protective measures after media reports. In addition, quarantine and other measures will greatly reduce the effective contact rate between vulnerable people and infected people, thus reducing the spread of disease.

Recently, many different control strategies have been proposed to apply to the control of some disease models. For example, by using multiple optimal control, Ndii and Adi ^[12] have studied the effects of individual awareness and vector controls on Malaria transmission dynamics. By using a non-smooth control strategy, Li et al.^[13] have considered the bifurcations and dynamics of a plant disease system. By using the two thresholds control strategy, Li et al.^[30] have considered the global dynamics of a Filippov predator-prey model. Chen et al. ^[31] have considered a two-thresholds policy for a Filippov model in combating influenza. Zhou et al. ^[32] have discussed a two-thresholds policy to interrupt transmission of West Nile Virus to birds. Meanwhile, based on the above mentioned transmission factors, some scholars have also done a lot of meaningful work. For example, Dong et al. ^[33] studied a kind of nonlinear incidence Filippov epidemic model to describe the impact of media in the process of epidemic transmission. They found that choosing an appropriate threshold value and control intensity can prevent the outbreak of infectious diseases, and media coverage can reduce the burden of disease outbreaks and shorten the duration of disease outbreaks. Can et al.^[25] have studied a Filippov model describing the effects of media coverage and quarantine on the spread of human influenza. Xiao et al. ^[28] have discussed a media impact switching surface during an infectious disease outbreak. In real life, when the influenza started to spread, that is, when the epidemic was not serious, the public paid little attention to the influenza. Generally, only when the number of people who have already felt it reaches and exceeds a certain threshold level will the mass media begin to coverage in large numbers and the public begin take some countermeasures ^{[11,17,19,20]}. Currently, many control managements have been developed, including threshold ^[30], harvesting control management^[34,35,36], impulsive control management^[37,38] and other controls ^{[39,40,41,42,43,44,45]}. However, the implementation of this measure may cause some social impacts, such as socioeconomic decline and psychological impact on the quarantined people. It is important to consider when to take these control measures. Therefore, an appropriate threshold policy is needed to deal with the influenza outbreak, or at least reduce the number of infected people to an acceptable level. As far as we know, under logistic source and broken line control strategy, the sliding dynamics and bifurcations of a human influenza system have been seldom reported in existing work.

Motivation and inspiration come from the discussion above. In this paper, we propose a human influenza system under logistic source and broken line control strategy. The main contributions of this paper include three points: First, choose two thresholds $I_{T}$ and $S_{T}$ as a broken line control strategy: Once the number of infected people exceeds $I_{T}$, the media influence comes into play, and when the number of susceptible individuals is greater than $S_{T}$, the control by quarantine of susceptible individuals control is open. Furthermore, by choosing different thresholds $I_{T}$ and $S_{T}$, the existence and stability of all possible equilibria considered, and then the Filippov system tends to the pseudo-equilibrium on sliding mode domain or one endemic equilibrium or bistability endemic equilibria under some conditions. It is worth pointing out that in our paper there exists a new phenomenon of two real equilibrium points co-existing.

In fact, from the perspective of control strategy, our paper puts forward a broken line control strategy. Different from the previous single-stage control strategy ^{[30,34,35,37,38]}, our control strategy is divided into two stages and can control and simulate the real life situation well.

This paper is structured as follows. In Section 2, we propose a non-smooth model under logistic source and broken line control strategy. By changing the infection threshold values and susceptibility threshold values, in Section 3, we consider the global dynamics of the system under Case 1: $S_T < S_1^* < S_{2}^{*} = S_{3}^{*}$. In Section 4, we study the global dynamics of the system under Case 2: $S_1^* < S_T < S_{2}^{*} = S_{3}^{*}$. Section 5 considers the global dynamics of the system under Case 3: $S_{2}^{*} = S_{3}^{*} < S_T$.The regular/virtulal equilibrium bifurcations are given in Section 6. Finally, we summarize the main results of this paper and discuss some biological conclusions in Section 7.

2.   Model description and basic knowledge

2.1.   Model description

In ^[25], a Filippov human influenza model with effects of media coverage and quarantine was considered. Because a logistic source can better depict the real situation, motivated by the above discussion and the existing multiple threshold conditions ^{[25,30,31,32,45]}, we consider the logistic source factor with the susceptible population, and then the new human influenza model can be described by

where $S$ is susceptible individuals, $I$ is infected individuals, and $R$ is recovered individuals. $\gamma$ is an intrinsic growth rate of susceptible individuals, and $K$ is carrying capacity. $\beta$ is the transmission rate. $d$ is the natural death rate. $\delta$ is the death rate caused by disease, and $r$ is the recovered rate.

To better control the spread of the disease, we give the following broken line control strategy. If the infected populations is lower than $I_T$, no control is required. When the infected populations is greater than $I_T$, the media coverage control is open. That is, the mass media being coverage of information about the disease, including the route of transmission, the number of infected cases and the number of deaths.

Therefore, the public realizes the harm of the disease, and they changed their behavior, leading to a decline in the contact rate $\beta$. In this paper, we consider that the positive number $v_1 (0\leq v_1\leq 1)$ is the reduction amount of contact rate. In addition, we consider the quarantine control as follows. If the number of susceptible individuals is less than $S_ {T}$, we do not quarantine susceptible individuals. If $S > S_ {T}$ holds, the quarantine control is open, and this time we assume that the positive number $v_2(0\leq v_2\leq 1)$ is the quarantine rate of susceptible individuals. Figure 1 shows the broken line control strategy. Based on system (2.1) and the broken line control strategy, in this paper, we propose a Filippov influenza system with media control and quarantine of susceptible populations as follows:

with

Notice that the third equation $R$ does not contain the variables $S$ and $I$ of (2.2), so we can not consider the third equation $R$ of the system. Then, in this paper, we investigate a Filippov system

Then, the first quadrant is divided by the following five regions:

The non-smooth system in region $\Gamma_i$ for $i = 1, 2, 3$ is described by

2.2.   Basic knowledge

The normal vector of the system is defined as $n_{1} = (0, 1)^{T}$ in $\Pi_1$, while the normal vector of the system is $n_{2} = (1, 0)^{T}$ in $\Pi_2$. Denote the right-hand side of system (2.4) by $f$. The following definitions are necessary in this paper^[14,46,47].

Definition 1. ^[14] A point $E$ is called a real equilibrium of system (2.4) if there exists $i \in\{1, 2, 3\}$ such that $F_{i}\left(E^{}\right) = 0$ and $E\in \Gamma_{i}$, denoted by $E^R$.

Definition 2. ^[14] A point $E$ is called a virtual equilibrium of system (2.4) if there exists $i \in\{1, 2, 3\}$ such that $F_{i}\left(E^{}\right) = 0$ and $E \notin \overline{\Gamma_{i}}$, where $\overline{\Gamma_{i}}$ is the closure of $\Gamma_{i}$, denoted by $E^V$.

Denote the equations that describe the sliding mode dynamics on the sliding mode domain $\ell_{ i} \subset \Pi_i$ by $F_{s i}(S, I), i \in\{1, 2\}$.

Definition 3. ^[14] A point $E$ is called a pseudo-equilibrium of system (2.4) if point $E$ is an equilibrium of $F_{s i}(S, I)$ on sliding mode domain $\ell$. That is, $F_{s i}\left(E^{}\right) = 0$, and point $E \in \ell \subset \Pi_i, i \in\{1, 2\}$.

2.3.   Positiveness, boundedness and dynamics analysis of (2.4) in $\Gamma_i$(i=1, 2, 3)

Lemma 1. The solution $(S(t), I(t))$ of system (2.4) with the positive initial value $S(0)$ and $I(0)$ satisfy $S(t) > 0$ and $I(t) > 0$ for $t \in[0, +\infty)$.

Proof. First, we prove that $S(t) > 0$. If not, we assume that there is a time $t_{1}$ satisfying the solution $S(t_{1}) \leq 0$. Then we know that there exists the other time $t^{*} > 0$ such that the solution $S\left(t^{*}\right) = 0$ and $S(t) > 0$ for $t \in\left[0, t^{*}\right)$. From the first equation of system (2.4), one has

Further, we obtain

which is a contradiction with $S\left(t^{*}\right) = 0$. Thus, $S(t) > 0$, for all $t > 0$.

Next, we prove that $I(t) > 0$. If not, we assume there is a time $t_{2}$ satisfying the solution $I\left(t_{2}\right) \leq 0$. Then, there exists the other time $\tilde{t} > 0$ such that the solution $I(\tilde{t}) = 0$, and $I(t) > 0$ for time $t \in[0, \tilde{t})$. From the second equation of system (2.4), one gets

Then, if $t = \tilde{t}$, we have

which is a contradiction with $I(\tilde{t}) = 0$. Thus, $I(t) > 0$, for all $t > 0$. In a word, we have that $(S(t), I(t))$ of (2.4) with the positive initial value $S(0)$ and $I(0)$ satisfies $S(t) > 0$ and $I(t) > 0$ for $t \in[0, +\infty)$. The proof is finished.

Lemma 2. The solution $(S, I)$ of system (2.4) is bounded.

Proof. Notice the first equation of system (2.4). Then,

Hence, there is a positive number $T$ such that the solution $S(t) < K$ for $t\geq T$.

Let $N = S+I$, and then for $t\geq T$, we obtain

Moreover,

Therefore, under Lemma 1, we know that the solution of (2.4) is bounded. The proof is finished.

Next, we give the basic reproduction number $R_{0i}, i = 1, 2, 3$, of the system. For example, in region $\Gamma_1$, the basic reproduction number is defined as $R_{01} = \frac{K\beta}{d+\delta+r}$. In region $\Gamma_2$, the basic reproduction number is defined as $R_{02} = \frac{K\beta(1-p)}{d+\delta+r}.$ In region $\Gamma_3$, the basic reproduction number is defined as $R_{03} = \frac{(\gamma-p)K\beta(1-p)}{\gamma(d+\delta+r)}$.

For system (2.4), clearly, in region $\Gamma_1$, (2.4) has three equilibria, i.e., $E_{10} = (0, 0)$, $E_{11} = (K, 0)$ and $E_{1} = (S_1^*, I_1^*) = (\frac{d+\delta+r}{\beta}, \frac{\gamma }{\beta}(1-\frac{d+\delta+r}{K\beta}))$, which is a stable node if $R_{01} > 1$.

In region $\Gamma_2$, system (2.4) has three equilibria, i.e., $E_{20} = (0, 0)$, $E_{21} = (K, 0)$ and $E_{2} = (S_2^*, I_2^*) = (\frac{d+\delta+r}{\beta(1-p)}, \frac{\gamma}{\beta(1-p)}(1-\frac{d+\delta+r}{K\beta(1-p)}))$, which is a stable node (focus) if $R_{02} > 1$.

In region $\Gamma_3$, system (2.4) has three equilibria, i.e., $E_{30} = (0, 0)$, $E_{31} = (K\frac{\gamma-q}{\gamma}, 0)$ and $E_{3} = (S_3^*, I_3^*) = (\frac{d+\delta+r}{\beta(1-p)}, \frac{\gamma}{\beta(1-p)}(1-\frac{d+\delta+r}{K\beta(1-p)}-\frac{q}{r}))$, which is a stable node (focus) if $R_{03} > 1$.

Theorem 1. Suppose that $R_{0i} < 1$, and the disease free equilibrium $E_{i1}(i = 1, 2, 3)$ of (2.4) is globally asymptotically stable. In addition, the endemic equilibrium $E_{i}(i = 1, 2, 3)$ of (2.4) is globally asymptotically stable if $R_{0i} > 1$.

Proof. Since $R_{0i} < 1, i = 1, 2, 3$, a Lyapunov function is considered as

Applying LaSalle's invariance principle^[13,14], we know that $E_{i1}(i = 1, 2, 3)$ of system (2.4) is globally asymptotically stable.

When $R_{0i} > 1, i = 1, 2, 3$, the following Lyapunov function is considered:

Using LaSalle's invariance principle ^[13,14], we know that points $E_{i}(i = 1, 2, 3)$ of system (2.4) areglobally asymptotically stable. The proof is finished.

This paper only considers the global dynamics of (2.4) under case $R_{0i} > 1$ $(i = 1, 2, 3)$. Next, we aim to address the richness of the possible equilibria and sliding modes on $\Pi_1$ and $\Pi_2$ that the system with can exhibit.

From (2.4), when $S_1^* < S_2^* = S_3^*$ and $I_{3}^{*} < I_{2}^{*}$, we consider the following three cases: $S_{T} < S_{1}^{*}, S_{1}^{*} < S_{T} < S_{2}^{*} = S_{3}^{*},$ and $S_{2}^{*} = S_{3}^{*} < S_{T}$ with varied $I_{T}$. Further, according to the dynamics in each case, the biological phenomena of (2.4) are described in this section.

Throughout the paper, the $S$-nullclines and $I$-nullclines of (2.4) are represented by the dashed curves and dash-dot lines, respectively. Thus, $S = S_{1}^{*}, S_{2}^{*}$ and $S_{3}^{*}$ are the $I$-nullclines of $F_{1}, F_{2}$ and $F_{3}$, denoted by $L_{12}, L_{22}$ and $L_{32}$, respectively. That is, the curves

and

are the $S$-nullclines of systems $F_{1}, F_{2}$ and $F_{3}$, denoted by $L_{11}, L_{21}$ and $L_{31}$, respectively.

3.   Sliding dynamics and bifurcations of (2.4) under Case 1: $S_T < S_1^* < S_{2}^{*}=S_{3}^{*}$

In this part, we first consider sliding mode dynamics of (2.4) on $\Pi_1$ under Case 1: $S_T < S_1^* < S_{2}^{*} = S_{3}^{*}$. Second, the sliding mode dynamics on $\Pi_2$ are also given under Case 1: $S_T < S_1^* < S_{2}^{*} = S_{3}^{*}$. In addition, we investigate the bifurcations of (2.4) under Case 1: $S_T < S_1^* < S_{2}^{*} = S_{3}^{*}$. Finally, some numerical simulations are displayed to confirm the results.

3.1.   Sliding mode dynamics of (2.4) on $\Pi_1$ under Case 1: $S_T < S_1^* < S_{2}^{*}=S_{3}^{*}$

This part investigates the existence of the sliding mode region on $\Pi_1$. Based on Definition 3, if $\langle n_1, F_1\rangle > 0$ and $\langle n_1, F_3\rangle < 0$ hold, then we know that there is the sliding mode region $\ell_1$, which is expressed as

Using the Filippov convex method ^[13,14], we obtain

So, we have the differential equations describing the sliding mode dynamics along the manifold $\ell_1$ for system (2.4):

Next, we analyze the existence of the positive equilibriums on $\ell_1$ of (3.1). Let

Proposition 1. For varied $I_{T}$, we have the following results.

● If $I_T > I_T^*$, then system (3.1) has no equilibrium.

● If $I_T^* > I_T > \frac{q}{\beta p}$, system (3.1) has two positive equilibria $E_{s1}^{\pm} = (S_{s1}^{\pm}, I_T)$, where $S_{s1}^{\pm} = \frac{(\gamma-\frac{q}{p})\pm\sqrt \Delta_1}{2\gamma}K$.

● If $I_T < \frac{q}{\beta p}$, then system (3.1) has a unique positive equilibrium $E_{s2} = (S_{s2}, I_T)$, where $S_{s2} = \frac{(\gamma-\frac{q}{p})+\sqrt \Delta_1}{2\gamma}K$.

In addition, when the sliding mode $\ell_1$ has a pseudo-equilibrium, we have

Proposition 2. Under the condition $S_{T}^{*} < S_{1}^{*}$, $E_{s 1}^{-} \notin \ell_1$, and the following results are given.

● If $I_T < I_{3}^*$, we have $E_{s 1}^{+} \notin \ell_1$.

● If $I_{3}^* < I_T < I_{1}^*$, we have $E_{s 1}^{+} \in \ell_1$.

● If $I_T > I_{1}^*$, we have $E_{s 1}^{+} \notin \ell_1$.

Proposition 3. Under the condition $S_{1}^{*} < S_{T}^{*} < S_{3}^{*}$, the following results hold.

(1) Assume that $I_{1}^* < I_{3}^*$, and then

● if $I_{1}^* < I_T < I_{3}^*$, we have $E_{s 1}^{-} \in \ell_1$, $E_{s 1}^{+} \notin \ell_1$;

● if $I_{3}^* < I_T < I_{T}^*$, we have $E_{s 1}^{-} \in \ell_1$, $E_{s 1}^{+} \in \ell_1$.

(2) Assume that $I_{1}^* > I_{3}^*$, and then

● if $I_{3}^* < I_T < I_{1}^*$, we have $E_{s 1}^{-} \notin \ell_1$, $E_{s 1}^{+} \in \ell_1$;

● if $I_{1}^* < I_T < I_{T}^*$, we have $E_{s 1}^{-} \in \ell_1$, $E_{s 1}^{+} \in \ell_1$.

Proposition 4. Under the condition $S_{T}^{*} > S_{3}^{*}$, $E_{s 1}^{+} \notin \ell_1$, and the following conclusions hold.

● If $I_T < I_{1}^*$, we have $E_{s 1}^{-} \notin \ell_1$.

● If $I_{1}^* < I_T < I_{3}^*$, we have $E_{s 1}^{-} \in \ell_1$.

● If $I_T > I_{3}^*$, we have $E_{s 1}^{-} \notin \ell_1$.

Theorem 2. If $I_T^* > I_T > \frac{q}{\beta p}$, then a stable pseudo-equilibriums $E_{s1}^+$ is located on the sliding mode $\ell_1$, and the unstable pseudo-equilibriums $E_{s1}^-$ is located on the sliding mode $\ell_1$.

Proof. Notice that

Therefore, the point $E_{s1}^+$ is attracting, and the point $E_{s1}^-$ is repelling.

Proposition 5. Assume that $\frac{\gamma-\frac{q}{p}}{\gamma}K > \frac{d+\delta+r}{\beta(1-p)}$, and then the pseudo-equilibrium $E_{s 2}$ is not located on $\ell_1$.

Proof. From Proposition 1, Eq (3.2) and the condition of Proposition 5, one has

which implies that $S_{s 2} \geq 2 S_{T}^{*} > S_{3}^{*}$. Based on the definition of sliding mode region $\ell_1$, we know that $S_{s 2} \notin \ell_1$. Then, $\ell_1$ does not have a pseudo-equilibrium $E_{s 2}$. The proof is completed.

3.2.   Sliding mode dynamics on $\Pi_2$ under Case 1: $S_T < S_1^* < S_{2}^{*}=S_{3}^{*}$

Let

Clearly $J_1 < J_2$. A subset of $\Pi_2$ is a sliding mode domain if $\langle F_{2}, n_{2}\rangle\langle F_{3}, n_{2}\rangle < 0$. If $I_{T} > J_2$, there is no sliding mode domain on $\Pi_2$. If $I_{T} < J_2$, then the sliding mode domain of (2.4) on $\Pi_2$ is given as

Using the Filippov convex method^[13], we have the differential equations describing the sliding mode dynamics along the manifold $\ell_2$ for system (2.4) with (2.3):

Clearly, there is not a positive equilibrium. Thus, if there exists a sliding domain $\ell_2$ on $\Pi_2$, we know the system does not have a pseudo-equilibrium.

3.3.   Bifurcations of (2.4) under Case 1: $S_T < S_1^* < S_{2}^{*}=S_{3}^{*}$

Due to $S_T < S_1^* < S_{2}^{*} = S_{3}^{*}$, we know that $E_{2}\notin\Gamma_2$ is a virtual equilibrium, denoted by $E_{2}^V$. However, point $E_{1}$ and point $E_{3}$ may exist depending on $I_T$, and we have the following three Propositions:

Proposition 6. Assume that $S_{T}^{*} < S_{1}^{*}$, and the following assertions hold.

● If $I_T < I_{3}^*$, we have $E_{s 1}^{-} \notin \ell_1, E_{s 1}^{+} \notin \ell_1, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$.

● If $I_{3}^* < I_T < I_{1}^*$, we have $E_{s 1}^{-} \notin \ell_1, E_{s 1}^{+} \in \ell_1, E_{1}\notin \Gamma_1, E_{3}\notin \Gamma_3$.

● If $I_{1}^* < I_T < I_{T}^*$, we have $E_{s 1}^{-} \notin \ell_1, E_{s 1}^{+} \notin \ell_1, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

● If $I_T > I_{T}^*$, we have $E_{s 1}^{-} \mathit{{\text{and}}} E_{s 1}^{+} \mathit{{\text{don't exist}}}, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

Proposition 7. Under the condition $S_{1}^{*} < S_{T}^{*} < S_{3}^{*}$, the following assertions hold.

(1) Assume that $I_{1}^* < I_{3}^*$, and further,

● if $I_T < I_{1}^*$, we have $E_{s 1}^{-} \notin \ell_1, E_{s 1}^{+} \notin \ell_1, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_{1}^* < I_T < I_{3}^*$, we have $E_{s 1}^{-} \in \ell_1, E_{s 1}^{+} \notin \ell_1, E_{1}\in \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_{3}^* < I_T < I_{T}^*$, we have $E_{s 1}^{-} \in \ell_1$, $E_{s 1}^{+} \in \ell_1, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

● if $I_T > I_{T}^*$, we have $E_{s 1}^{-} \mathit{{\text{and}}} E_{s 1}^{+} \mathit{{\text{don't exist}}}, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

(2) Assume that $I_{1}^* > I_{3}^*$, and further,

● if $I_T < I_{3}^*$, we have $E_{s 1}^{-} \notin \ell_1, E_{s 1}^{+} \notin \ell_1, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_{3}^* < I_T < I_{1}^*$, we have $E_{s 1}^{-} \notin \ell_1, E_{s 1}^{+} \in \ell_1, E_{1}\notin \Gamma_1, E_{3}\notin \Gamma_3$;

● if $I_{1}^* < I_T < I_{T}^*$, we have $E_{s 1}^{-} \in \ell_1$, $E_{s 1}^{+} \in \ell_1, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

● if $I_T > I_{T}^*$, we have $E_{s 1}^{-} \mathit{{\text{and}}} E_{s 1}^{+} \mathit{{\text{don't exist}}}, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

Proposition 8. Under the condition $S_{T}^{*} > S_{3}^{*}$, the following assertions hold.

● If $I_T < I_{1}^*$, we have $E_{s 1}^{-} \notin \ell_1, E_{s 1}^{+} \notin \ell_1, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$.

● If $I_{1}^* < I_T < I_{3}^*$, we have $E_{s 1}^{-} \in \ell_1, E_{s 1}^{+} \notin \ell_1, E_{1}\in \Gamma_1, E_{3}\in \Gamma_3$.

● If $I_{3}^* < I_T < I_{T}^*$, we have $E_{s 1}^{-} \notin \ell_1, E_{s 1}^{+} \notin \ell_1, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

● If $I_T > I_{T}^*$, we have $E_{s 1}^{-} \mathit{{\text{and}}} E_{s 1}^{+} \mathit{{\text{don't exist}}}, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

Based on Propositions 6–8, we have the following summary:

B1. Let $E_{s1}^{-} \notin \ell_1, E_{s1}^{+} \notin \ell_1, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $B_{1-1}$. Then, we conclude that there does not exist a pseudo-equilibrium, and all trajectories of the system (2.4) will converge to $E_{3}^R$, as shown in Figure 2(a), where

B2. Let $E_{s1}^{-} \notin \ell_1, E_{s1}^{+} \in \ell_1, E_{1}\notin \Gamma_1, E_{3}\notin \Gamma_3$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $B_{2-1}$. Then, we know that $E_{s1}^{+}\in \ell_1\subset \Pi_1$ is a stable pseudo-equilibrium, and all solutions of the system will approach $E_{s1}^{+}$, as shown in Figure 2(b), where

B3. Let $E_{s1}^{-} \notin \ell_1, E_{s1}^{+} \notin \ell_1, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $B_{3-1}\cup B_{3-2}\cup B_{3-3}$. Then, we conclude that the system (2.4) does not have a pseudo-equilibrium, and all trajectories of the system (2.4) will converge to the equilibrium point $E_{1}^R$, as shown in Figure 2(c), where

B4. Let $E_{s1}^{-} \in \ell_1, E_{s1}^{+} \notin \ell_1, E_{1}\in \Gamma_1, E_{3}\in \Gamma_3$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $B_{4-1}$. Then, we know that $E_{s1}^{-}\in \ell_1\subset \Pi_1$ is an unstable pseudo-equilibrium, and all solutions will approach $E_{1}^R$ or $E_{3}^R$. The result of this numerical simulation is shown in Figure 2(d), where

B5. Let $E_{s1}^{-} \in \ell_1, E_{s1}^{+} \in \ell_1, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $B_{5-1}$. Then, we conclude that $E_{s1}^{-}\in \ell_1\subset \Pi_1$ is an unstable pseudo-equilibrium, and all solutions of the system (2.4) will tend to $E_{1}^R$ or $E_{s1}^{+}$. The result of this numerical simulation is shown in Figure 2(e), where

For the B1 of case 1, when system (2.4) has a unique equilibrium $E_{3}^{R}$, we have the following.

Theorem 3. If $S_T < S_1^* < S_3^* = S_2^*$ and $I_T < min \{I_{1}^*, I_{3}^{*} \}$, then the point $E_{3}^{R}$ of system (2.4) is globally asymptotically stable.

Proof. Suppose that the system (2.4) has a closed orbit $U$ (shown in Figure 3(a)) that surrounds the real equilibrium $E_{32}^{R}$ and the sliding mode $\ell_{1}$. Define $U = U_{1}+U_{2}+U_{3}$, where $U_{i} = U \cap \Gamma_{i}, i = 1, 2, 3$. Let $\Omega$ be the bounded region delimited by $U$ and $\Omega_{i} = \Omega \cap \Gamma_{i}$ for $i = 1, 2, 3$. Considering the Dulac function $D = \frac{1}{S I}$. Three Steps are given as follows:

Step 1: System (2.4) does not have a closed orbit in region $\Gamma_{i}, i = 1, 2, 3$.

where $f_{1}$ is the first component of $f$, and $f_{2}$ is the second component of $f.\; F_{i 1}$ is the first component of $F_{i}$ and $F_{i 2}$ is the second component of $F_{i}, i = 1, 2, 3$. Let $\tilde{\Omega}_{i}$ be the region bounded by $\tilde{U}_{i}, {P}_{i}$ and ${Q}_{i}$, where $\tilde{\Omega}_{i}$ and $\tilde{U}_{i}$ depend on $\epsilon$ and converge to $\Omega_{i}$ and $U_{i}$ as $\epsilon$ approaches 0.

Step 2: System (2.4) does not have a closed trajectory in region $\Gamma$.

We can get

Since $d S = F_{11} dt$ and $d I = F_{12} dt$ along $\tilde{U}_{1}$ and $d I = 0$ along ${P}_{1}$, by using Green's theorem, for region $\tilde{\Omega}_{1}$, we have

Similarly, we obtain

and

From Eqs (3.5)–(3.7), we have

Denote the intersection points of the closed trajectory $U$ and the line $I = I_{T}$ by $A_{1}$ and $A_{2}$ and the intersection point of $U$ and line $S = S_{T}$ if $I > I_{T}$ by $A_{3}$. In addition, denote the intersection point of the line $I = I_{T}$ and the line $S = S_{T}$ by $E_{T}$. Note that $A_{1S} < S_{T} < A_{2S}$ and $A_{3I} > I_{T}$. Then, Eq (3.6) becomes

which is a contradiction. Therefore, we know that there does not have a closed orbit $U$ surrounding the sliding mode $\ell_{1}$ and the real equilibrium $E_{3}^{R}$.

Step 3: System (2.4) does not have a closed trajectory in regions $\Gamma_i$ and $\Gamma_j$($i\neq j$). With a similar proof procedure to Step 2, it is easy to that there is no closed trajectory in $\Gamma_1$ and $\Gamma_2$ (see Figure 3(d)), $\Gamma_1$ and $\Gamma_3$ (see Figure 3(b)), $\Gamma_2$ and $\Gamma_3$ (see Figure 3(c)), respectively.

Therefore, based on Steps 1–3, we know that the point $E_{3}^{R}$ of (2.4) is globally asymptotically stable if $S_T < S_1^* < S_3^* = S_2^*$ and $I_T < {\text{min}}\{I_{1}^*, I_{3}^{*} \}$. This completes this theorem.

With a similar proof procedure to Theorem 3, we have the following.

Theorem 4. If $S_T < S_1^* < S_{2}^{*} = S_{3}^{*}$ and the conditions of (B3) hold, then the point $E_{1}^{R}$ of system (2.4) is globally asymptotically stable.

4.   Sliding dynamics and bifurcations of (2.4) under Case 2: $S_1^* < S_T < S_{2}^{*}=S_{3}^{*}$

In this part, we first consider sliding mode dynamics of (2.4) on $\Pi_1$ under Case 2: $S_1^* < S_T < S_{2}^{*} = S_{3}^{*}$. Second, we study the sliding mode dynamics on $\Pi_2$ under Case 2: $S_1^* < S_T < S_{2}^{*} = S_{3}^{*}$. In addition, the bifurcations of (2.4) are investigated under Case 2: $S_1^* < S_T < S_{2}^{*} = S_{3}^{*}$. Finally, some numerical simulations are displayed to confirm the results.

4.1.   Sliding mode dynamics of system (2.4) on $\Pi_1$ under Case 2: $S_1^* < S_T < S_{2}^{*}=S_{3}^{*}$

For Case 2, $S_1^* < S_T < S_{2}^{*} = S_{3}^{*}$, two sliding domains on $\Pi_{1}$ are given as

The dynamics on $\ell_3$ are governed by

Now, we investigate the existence of a positive equilibrium on $\ell_3$ of system (4.1). Let

Proposition 9. For varied $I_{T}$, we have the following results.

(1) If $I_T > I_T^{*'}$, then system (4.1) does not have an equilibrium;

(2) If $0 < I_T < I_T^{*'}$, system (4.1) has two positive equilibria $E_{s3}^{\pm} = (S_{s3}^{\pm}, I_T)$, where $S_{s3}^{\pm} = \frac{\gamma\pm\sqrt \Delta_2}{2\gamma}K$.

Next, we find the conditions of the pseudo-equilibrium on the sliding mode $\ell_3$. Let

Proposition 10. Under the condition $S_{T}^{*'} < S_{1}^{*} < S_{T}$, $E_{s 3}^{-} \notin \ell_3$, and the following assertions hold.

(1) If $I_T > I_1^*$, we have $E_{s 3}^{+} \notin \ell_3$;

(2) If $H_{1} < I_T < I_1^*$, we have $E_{s 3}^{+} \in \ell_3$;

(3) If $I_T < H_{1}$, we have $E_{s 3}^{+} \notin \ell_3$.

Proposition 11. Under the condition $S_{1}^{*} < S_{T}^{*'} < S_{T}$, we have the following results.

(1) If $I_T < min \{I_1^*, H_{1}\}$, we have $E_{s 3}^{\pm} \notin \ell_3$.

(2) If $min \{I_1^*, H_{1}\} < I_T < \mathit{{\text{max}}} \{I_1^*, H_{1}\}$, we have

● $E_{s 3}^{-} \in \ell_3, E_{s 3}^{+} \notin \ell_3 \mathit{{\text{if}}} I_1^* < H_{1}$;

● $E_{s 3}^{-} \notin \ell_3, E_{s 3}^{+} \in \ell_3 \mathit{{\text{if}}} I_1^* > H_{1}$.

(3) If $\mathit{{\text{max}}} \{I_1^*, H_{1}\} < I_T < I_T^{*'}$, we have $E_{s 3}^{\pm} \in \ell_3$.

Proposition 12. Under the condition $S_{1}^{*} < S_T < S_{T}^{*'}$, $E_{s 3}^{+} \notin \ell_3$, and the following assertions hold.

(1) If $I_T < I_1^*$, we have $E_{s 3}^{-} \notin \ell_3$;

(2) If $I_1^* < I_T < H_{1}$, we have $E_{s 3}^{-} \in \ell_3$;

(3) If $I_T > H_{1}$, we have $E_{s 3}^{-} \notin \ell_3$.

Theorem 5. If $0 < I_T < I_T^{*'}$, then a stable pseudo-equilibrium $E_{s 3}^{+}$ of (2.4) is located on the sliding mode $\ell_3$, and the unstable pseudo-equilibrium $E_{s 3}^{-}$ of (2.4) is located on the sliding mode $\ell_3$.

Proof. Notice that

Therefore, the point $E_{s3}^+$ is attracting, and the point $E_{s3}^-$ is repelling.

The dynamics on region $\ell_4$ are described by (3.1). Let

From Proposition 1, we have the following.

Proposition 13. Under the condition $S_{T}^{*} < S_T < S_{3}^{*}$, $E_{s 1}^{-} \notin \ell_4$, and the following assertions hold.

(1) If $I_T < I_3^*$, we have $E_{s 1}^{+} \notin \ell_4$;

(2) If $I_3^* < I_T < H_{2}$, we have $E_{s 1}^{+} \in \ell_4$;

(3) If $I_T > H_{2}$, we have $E_{s 1}^{+} \notin \ell_4$.

Proposition 14. Under the condition $S_{T} < S_{T}^{*} < S_{3}^{*}$, we have the following results.

(1) If $I_T < min \{I_3^*, H_{2}\}$, we have $E_{s 1}^{\pm} \notin \ell_4$.

(2) If $min \{I_3^*, H_{2}\} < I_T < \mathit{{\text{max}}} \{I_3^*, H_{2}\}$, we have

● $E_{s 1}^{-} \in \ell_4, E_{s 1}^{+} \notin \ell_4 \mathit{{\text{if}}} I_3^* > H_{2}$;

● $E_{s 1}^{-} \notin \ell_4, E_{s 1}^{+} \in \ell_4 \mathit{{\text{if}}} I_3^* < H_{2}$.

(3) If $\mathit{{\text{max}}} \{I_3^*, H_{2}\} < I_T < I_T^*$, we have $E_{s 1}^{\pm} \in \ell_4$.

Proposition 15. Under the condition $S_T < S_{3}^{*} < S_{T}^{*}$, $E_{s 1}^{+} \notin \ell_4$, and the following assertions hold.

(1) If $I_T < H_{2}$, we have $E_{s 1}^{-} \notin \ell_4$;

(2) If $H_{2} < I_T < I_3^*$, we have $E_{s 1}^{-} \in \ell_4$;

(3) If $I_T > I_3^*$, we have $E_{s 1}^{-} \notin \ell_4$.

Theorem 6. If $0 < I_T < I_T^{*'}$, then a stable pseudo-equilibrium $E_{s1}^+$ is located on $\ell_4$, and the unstable pseudo-equilibrium $E_{s1}^-$ is located on $\ell_4$.

4.2.   Sliding mode dynamics of (2.4) on $\Pi_2$ under Case 2: $S_1^* < S_T < S_{2}^{*}=S_{3}^{*}$

If $S_1^* < S_T < S_{2}^{*} = S_{3}^{*}$, the sliding mode dynamics on region $\Pi_2$ are the same as Section 3.2. We omit it here.

4.3.   Bifurcations of system (2.4) under Case 2: $S_1^* < S_T < S_{2}^{*}=S_{3}^{*}$

For this case, we conclude that the point $E_{2}$ is a virtual equilibrium. $E_{1}$ and $E_{3}$ are changeable depending on $I_T$, and we have the following five situation.

Proposition 16. Under the conditions $S_{T}^{*'} < S_{1}^{*} < S_{T}$ and $S_{T}^{*} < S_T < S_{3}^{*}$, the following assertions hold.

(1) If $I_T < I_3^*$, we have $E_{s3}^{+} \notin \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$;

(2) If $I_3^* < I_T < H_{2}$, we have $E_{s3}^{+} \notin \ell_3, E_{s1}^{+} \in \ell_4, E_{1}\notin \Gamma_1, E_{3}\notin \Gamma_3$;

(3) If $H_{2} < I_T < H_{1}$, we have $E_{s3}^{+} \notin \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\notin \Gamma_3$;

(4) If $H_{1} < I_T < I_1^*$, we have $E_{s3}^{+} \in \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\notin \Gamma_3$;

(5) If $I_1^* < I_T < I_T^*$, we have $E_{s3}^{+} \notin \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

Proposition 17. Under the conditions $S_{1}^{*} < S_{T}^{*'} < S_{T}$ and $S_{T}^{*} < S_T < S_{3}^{*}$, the following assertions hold.

(1) Assume that $I_1^* > H_{1}$, and further,

● if $I_T < I_3^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_3^* < I_T < H_{2}$, we have $E_{s3}^{-} \notin \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{+} \in \ell_4, E_{1}\notin \Gamma_1, E_{3}\notin \Gamma_3$;

● if $H_{2} < I_T < H_{1}$, we have $E_{s3}^{-} \notin \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\notin \Gamma_3$;

● if $H_{1} < I_T < I_1^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s3}^{+} \in \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\notin \Gamma_3$;

● if $I_1^* < I_T < I_{T}^{*'}$, we have $E_{s3}^{-} \in \ell_3, E_{s3}^{+} \in \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

(2) Assume that $H_{2} < I_1^* < H_{1}$, and further,

● if $I_T < I_3^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_3^* < I_T < H_{2}$, we have $E_{s3}^{-} \notin \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{+} \in \ell_4, E_{1}\notin \Gamma_1, E_{3}\notin \Gamma_3$;

● if $H_{2} < I_T < I_1^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\notin \Gamma_3$;

● if $I_1^* < I_T < H_{1}$, we have $E_{s3}^{-} \in \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

● if $H_{1} < I_T < I_{T}^{*'}$, we have $E_{s3}^{-} \in \ell_3, E_{s3}^{+} \in \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

(3) Assume that $I_3^* < I_1^* < H_{2}$, and

● if $I_T < I_3^*$, then $E_{s3}^{-} \notin \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_3^* < I_T < I_1^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{+} \in \ell_4, E_{1}\notin \Gamma_1, E_{3}\notin \Gamma_3$;

● if $I_1^* < I_T < H_{2}$, we have $E_{s3}^{-} \in \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

● if $H_{2} < I_T < H_{1}$, we have $E_{s3}^{-} \in \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

● if $H_{1} < I_T < I_{T}^{*'}$, we have $E_{s3}^{-} \in \ell_3, E_{s3}^{+} \in \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

(4) Assume that $I_1^* < I_3^*$, and further,

● if $I_T < I_1^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_1^* < I_T < I_3^*$, we have $E_{s3}^{-} \in \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_3^* < I_T < H_{2}$, we have $E_{s3}^{-} \in \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

● if $H_{2} < I_T < H_{1}$, we have $E_{s3}^{-} \in \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

● if $H_{1} < I_T < I_{T}^{*'}$, we have $E_{s3}^{-} \in \ell_3\, E_{s3}^{+} \in \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

Proposition 18. Under the conditions $S_{1}^{*} < S_T < S_{T}^{*'}$ and $S_{T}^{*} < S_T < S_{3}^{*}$, we have the following results.

(1) Assume that $I_1^* > H_{2}$, and further,

● if $I_T < I_3^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_3^* < I_T < H_{2}$, we have $E_{s3}^{-} \notin \ell_3, E_{s1}^{+} \in \ell_4, E_{1}\notin \Gamma_1, E_{3}\notin \Gamma_3$;

● if $H_{2} < I_T < I_1^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\notin \Gamma_3$;

● if $I_1^* < I_T < H_{1}$, we have $E_{s3}^{-} \in \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

● if $H_{1} < I_T < I_{T}^{*'}$, we have $E_{s3}^{-} \notin \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

(2) Assume that $I_3^* < I_1^* < H_{2}$, and further,

● if $I_T < I_3^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_3^* < I_T < I_1^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s1}^{+} \in \ell_4, E_{1}\notin \Gamma_1, E_{3}\notin \Gamma_3$;

● if $I_1^* < I_T < H_{2}$, we have $E_{s3}^{-} \in \ell_3, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

● if $H_{2} < I_T < H_{1}$, we have $E_{s3}^{-} \in \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

● if $H_{1} < I_T < I_{T}^{*'}$, we have $E_{s3}^{-} \notin \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

(3) Assume that $I_1^* < I_3^*$, and further,

● if $I_T < I_1^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_1^* < I_T < I_3^*$, we have $E_{s3}^{-} \in \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_3^* < I_T < H_{2}$, we have $E_{s3}^{-} \in \ell_3, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

● if $H_{2} < I_T < H_{1}$, we have $E_{s3}^{-} \in \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

● if $H_{1} < I_T < I_{T}^{*'}$, we have $E_{s3}^{-} \notin \ell_3, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

Proposition 19. Under the conditions $S_{1}^{*} < S_T < S_{T}^{*'}$ and $S_{T} < S_{T}^{*} < S_{3}^{*}$, we have the following results.

(1) Assume that $I_3^* > H_{1}$, $I_1^* < H_{2} < H_{1} < I_3^*$, and

● if $I_T < I_1^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s 1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_1^* < I_T < H_{2}$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\in \Gamma_3$;

● if $H_{2} < I_T < H_{1}$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \in \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\in \Gamma_3$;

● if $H_{1} < I_T < I_3^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s 1}^{-} \in \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_3^* < I_T < I_{T}^{*}$, we have $E_{s3}^{-} \notin \ell_3, E_{s 1}^{-} \in \ell_4, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

(2) Assume that $H_{2} < I_3^* < H_{1}$, $I_1^* < H_{2} < I_3^* < H_{1}$, and further,

● if $I_T < I_1^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s 1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_1^* < I_T < H_{2}$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\in \Gamma_3$;

● if $H_{2} < I_T < I_3^*$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \in \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_T > I_3^*$, then

  ◇ if $I_{T}^{*} > H_{1}$, and further,

    ◇ if $I_3^* < I_T < H_{1}$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \in \ell_4, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

    ◇ if $H_{1} < I_T < I_{T}^{*}$, we have $E_{s3}^{-} \notin \ell_3, E_{s 1}^{-} \in \ell_4, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

  ◇ if $I_{T}^{*} < H_{1}$, and

    ◇if $I_3^* < I_T < I_{T}^{*}$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \in \ell_4, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

    ◇ if $I_{T}^{*} < I_T < H_{1}$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

    ◇ if $H_{1} < I_T < I_{T}^{*'}$, we have $E_{s3}^{-} \notin \ell_3, E_{s 1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

(3) Assume that $I_1^* < I_3^* < H_{2}$, $I_1^* < I_3^* < H_{2} < H_{1}$, and

● if $I_T < I_1^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s 1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_1^* < I_T < I_3^*$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_3^* < I_T < H_{2}$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \notin \ell_4, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

● if $I_T > H_{2}$,

  ◇ if $I_{T}^{*} > H_{1}$, and further,

    ◇ if $H_{2} < I_T < H_{1}$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \in \ell_4, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

    ◇ if $H_{1} < I_T < I_{T}^{*}$, we have $E_{s3}^{-} \notin \ell_3, E_{s 1}^{-} \in \ell_4, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

  ◇ if $I_{T}^{*} < H_{1}$, and further,

    ◇ if $H_{2} < I_T < I_{T}^{*}$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \in \ell_4, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

    ◇ if $I_{T}^{*} < I_T < H_{1}$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

    ◇ if $H_{1} < I_T < I_{T}^{*'}$, we have $E_{s3}^{-} \notin \ell_3, E_{s 1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

(4) Assume that $I_3^* < I_1^*$, further, $I_3^* < I_1^* < H_{2} < H_{1}$, and

● if $I_T < I_3^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s 1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_3^* < I_T < I_1^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s 1}^{-} \notin \ell_4, E_{s1}^{+} \in \ell_4, E_{1}\notin \Gamma_1, E_{3}\notin \Gamma_3$;

● if $I_1^* < I_T < H_{2}$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \notin \ell_4, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

● if $I_T > H_{2}$, and further,

  ◇ if $I_{T}^{*} > H_{1}$, and

    ◇if $H_{2} < I_T < H_{1}$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \in \ell_4, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

    ◇if $H_{1} < I_T < I_{T}^{*}$, we have $E_{s3}^{-} \notin \ell_3, E_{s 1}^{-} \in \ell_4, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

  ◇ if $I_{T}^{*} < H_{1}$, and further

    ◇ if $H_{2} < I_T < I_{T}^{*}$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \in \ell_4, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

    ◇ if $I_{T}^{*} < I_T < H_{1}$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

    ◇ if $H_{1} < I_T < I_{T}^{*'}$, we have $E_{s3}^{-} \notin \ell_3, E_{s 1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

Proposition 20. Suppose $S_{1}^{*} < S_T < S_{T}^{*'}$ and $S_T < S_{3}^{*} < S_{T}^{*}$, $E_{s3}^{+} \notin \ell_3$ and $E_{s 1}^{+} \notin \ell_4\subset \Pi_1$, we have the following results.

(1) Assume that $I_3^* > H_{1}$, and further,

● if $I_T < I_1^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s 1}^{-} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_1^* < I_T < H_{2}$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\in \Gamma_3$;

● if $H_{2} < I_T < H_{1}$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\in \Gamma_3$;

● if $H_{1} < I_T < I_3^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s 1}^{-} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_3^* < I_T < I_{T}^{*'}$, we have $E_{s3}^{-} \notin \ell_3\, E_{s 1}^{-} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

(2) Assume that $I_3^* < H_{1}$, and further,

● if $I_T < I_1^*$, we have $E_{s3}^{-} \notin \ell_3, E_{s 1}^{-} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_1^* < I_T < H_{2}$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\in \Gamma_3$;

● if $H_{2} < I_T < I_3^*$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\in \Gamma_3$;

● if $I_3^* < I_T < H_{1}$, we have $E_{s3}^{-} \in \ell_3, E_{s 1}^{-} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$;

● if $H_{1} < I_T < I_{T}^{*'}$, we have $E_{s3}^{-} \notin \ell_3, E_{s 1}^{-} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$.

Based on Propositions 16–20, the following summary is given.

C1. Let $E_{s3}^{-} \notin \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\in \Gamma_3$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $C_{1-1}$. Then, we conclude that the system (2.4) does not have a pseudo-equilibrium, and all trajectories of the system (2.4) will converge to the equilibrium point $E_{3}^R$. The result of this numerical simulation is shown in Figure 4(a), where

C2. Let $E_{s3}^{-} \notin \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{-} \notin \ell_4, E_{s1}^{+} \in \ell_4, E_{1}\notin \Gamma_1, E_{3}\notin \Gamma_3$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $C_{2-1}$. Then, we know that $E_{s1}^{+}\in \ell_4\subset \Pi_1$ is a stable pseudo-equilibrium, and all solutions of the system (2.4) will approach the point $E_{s1}^{+}$, as shown in Figure 4(b), where

C3. Let $E_{s3}^{-} \notin \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\notin \Gamma_1, E_{3}\notin \Gamma_3$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $C_{3-1}$. Then, we conclude that the system (2.4) does not have a pseudo-equilibrium, and all trajectories of the system (2.4) will converge to the point $E_T = (S_T, I_T)$. The result of this numerical simulation is shown in Figure 4(c), where

C4. Let $E_{s3}^{-} \notin \ell_3, E_{s3}^{+} \in \ell_3, E_{s1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4\subset \Pi_1, E_{1}\notin \Gamma_1, E_{3}\notin \Gamma_3$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $C_{4-1}$. Then, we know that the point $E_{s3}^{+}\in \ell_3$ is a stable pseudo-equilibrium, and all solutions of the system (2.4) will approach the point $E_{s3}^{+}$, as shown in Figure 4(d), where

C5. Let $E_{s3}^{-} \notin \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $C_{5-1} \cup C_{5-2} \cup C_{5-3}\cup C_{5-4}$. Then, we know that the system (2.4) does not have a pseudo-equilibrium, and all trajectories of the system (2.4) will converge to the point $E_{1}^R$. The result of this numerical simulation is shown in Figure 4(e), where

C6. Let $E_{s3}^{-} \in \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\in \Gamma_3$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $C_{6-1}$. Then, we conclude that $E_{s3}^{-}\in \ell_3$ is an unstable pseudo-equilibrium, and the solution of the system (2.4) will approach the point $E_{1}^R$ or $E_{3}^R$ or $E_T$. The result of this numerical simulation is shown in Figure 4(f), where

C7. Let $E_{s3}^{-} \in \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{-} \notin \ell_4, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $C_{7-1}$. Then, we know that $E_{s3}^{-}\in \ell_3$ is an unstable pseudo-equilibrium, where $E_{s1}^{+} \in \ell_4$ is stable. The solution of the system (2.4) will tend to $E_{s1}^{+}$ or $E_{1}^R$ or $E_T$. The result of this numerical simulation is shown in Figure 5(a), where

C8. Let $E_{s3}^{-} \in \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $C_{8-1} \cup C_{8-2} \cup C_{8-3}\cup C_{8-4}$. Then, we conclude that $E_{s3}^{-}\in \ell_3$ is an unstable pseudo-equilibrium. The solution of the system (2.4) will approach the equilibrium point $E_{1}^R$ or $E_T$, as shown in Figure 5(b), where

C9. Let $E_{s3}^{-} \in \ell_3, E_{s3}^{+} \in \ell_3, E_{s1}^{-} \notin \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to $C_{9-1}$. Then, we conclude that $E_{s3}^{-}\in \ell_3$ is a unstable pseudo-equilibrium, where $E_{s3}^{+} \in \ell_3$ is stable. The solution of the system (2.4) will tend to the equilibrium point $E_{s3}^{+}$ or $E_{1}^R$ or $E_T$. The results of this numerical simulation are shown in Figure 5(c), where

C10. Let $E_{s3}^{-} \in \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{-} \in \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\in \Gamma_3$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $C_{10-1} \cup C_{10-2}$. Then, we conclude that $E_{s3}^{-}\in \ell_3$ is an unstable pseudo-equilibrium. The solution of the system (2.4) will converge to $E_{1}^R$ or $E_{3}^R$ or $E_T$, as shown in Figure 5(d), where

C11. Let $E_{s3}^{-} \notin \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{-} \in \ell_4, E_{s1}^{+} \notin \ell_4, E_{1}\in \Gamma_1, E_{3}\in \Gamma_3$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $C_{11-1} \cup C_{11-2}$. Then, we show that $E_{s1}^{-}\in \ell_4$ is an unstable pseudo-equilibrium. The solution of the system (2.4) will tend to $E_{1}^R$ or $E_{3}^R$ or $E_T$, as shown in Figure 5(e), where

C12. Let $E_{s3}^{-} \in \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{-} \in \ell_4, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$, $\text { if }S_{1}^{*} < S_T < S_{T}^{*'} {\text{ and }} S_{T} < S_{T}^{*} < S_{3}^{*}$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $C_{12-1}$. Then, we know that the point $E_{s3}^{-}\in \ell_3$ and the value $E_{s1}^{-} \in \ell_4$ are unstable pseudo-equilibriums, where $E_{s1}^{+} \in \ell_4$ is stable. The solution of the system (2.4) will approach $E_{s1}^{+}$ or $E_{1}^R$ or $E_T$. The result of this numerical simulation is shown in Figure 5(f), where

C13. Let $E_{s3}^{-} \notin \ell_3, E_{s3}^{+} \notin \ell_3, E_{s1}^{-} \in \ell_4, E_{s1}^{+} \in \ell_4, E_{1}\in \Gamma_1, E_{3}\notin \Gamma_3$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $C_{13-1}$. Then, we conclude that $E_{s1}^{-}\in \ell_4$ is an unstable pseudo-equilibrium, where $E_{s1}^{+} \in \ell_4\subset \Pi_1$ is stable. The solution of the system (2.4) will converge to $E_{s1}^{+}$ or $E_{1}^R$ or $E_T$, as shown in Figure 6, where

5.   Sliding dynamics and bifurcations of (2.4) under Case 3: $S_{2}^{*}=S_{3}^{*} < S_T$

In this part, we first consider sliding mode dynamics of (2.4) on $\Pi_1$ under Case 3: $S_{2}^{*} = S_{3}^{*} < S_T$. Second, the sliding mode dynamics on $\Pi_2$ are given under Case 3: $S_{2}^{*} = S_{3}^{*} < S_T$. In addition, we investigate the bifurcations of (2.4) under Case 3: $S_{2}^{*} = S_{3}^{*} < S_T$. Finally, some numerical simulations are displayed to confirm the results.

5.1.   Sliding mode dynamics on $\Pi_1$ under Case 3: $S_{2}^{*}=S_{3}^{*} < S_T$

If $\langle n_1, F_1\rangle > 0$ and $\langle n_1, F_2\rangle < 0$ on $\ell_5$, then $\ell_5$ is described as

Next, the conditions of a pseudo-equilibrium on the sliding mode $\ell_5$ are given as follows.

Proposition 21. Under the condition $S_{T}^{*'} > S_{2}^{*}$, $E_{s 3}^{+} \notin \ell_5$ and the following assertions hold.

(1) If $I_T < I_1^*$, we have $E_{s 3}^{-} \notin \ell_5$;

(2) If $I_{1}^{*} < I_T < I_{2}^{*}$, we have $E_{s 3}^{-} \in \ell_5$;

(3) If $I_T > I_{2}^{*}$, we have $E_{s 3}^{-} \notin \ell_5$.

Proposition 22. Under the condition $S_{1}^{*} < S_{T}^{*'} < S_{2}^{*}$, the following assertions hold.

(1) Assume that $I_{1}^* < I_{2}^*$, and further,

● if $I_{1}^* < I_T < I_{2}^*$, we have $E_{s 3}^{-} \in \ell_5$, $E_{s 3}^{+} \notin \ell_5$;

● if $I_{2}^* < I_T < I_{T}^{*'}$, we have $E_{s 3}^{-} \in \ell_5$, $E_{s 3}^{+} \in \ell_5$.

(2) Assume that $I_{1}^* > I_{2}^*$, and further,

● if $I_{2}^* < I_T < I_{1}^*$, we have $E_{s 3}^{-} \notin \ell_5$, $E_{s 3}^{+} \in \ell_5$;

● if $I_{1}^* < I_T < I_{T}^{*'}$, we have $E_{s 3}^{-} \in \ell_5$, $E_{s 3}^{+} \in \ell_5$.

Proposition 23. Under the condition $S_{T}^{*'} < S_{1}^{*}$, $E_{s 3}^{-} \notin \ell_5$ and the following assertions hold.

(1) If $I_T < I_{2}^{*}$, we have $E_{s 3}^{+} \notin \ell_5$;

(2) If $I_{2}^{*} < I_T < I_{1}^{*}$, we have $E_{s 3}^{+} \in \ell_5$;

(3) If $I_T > I_1^*$, we have $E_{s 3}^{+} \notin \ell_5$.

Theorem 7. If $0 < I_T < I_T^{*'}$, the sliding mode $\ell_5$ has a stable pseudo-equilibrium $E_{s 3}^{+}$, and the sliding mode $\ell_5$ $E_{s 3}^{-}$ has an unstable pseudo-equilibrium.

5.2.   Sliding mode dynamics on $\Pi_2$ under Case 3: $S_{2}^{*}=S_{3}^{*} < S_T$

When $S_{2}^{*} = S_{3}^{*} < S_T$, the sliding mode dynamics on $\Pi_2$ are the same as Section 3.2. We omit it here.

5.3.   Bifurcations of system (2.4) under Case 3: $S_{2}^{*}=S_{3}^{*} < S_T$

For this Case 3, the point $E_{3}$ is a virtual equilibrium, denoted by $E_{3}^V$. Points $E_{1}$ and $E_{2}$ are changeable depending on $I_T$, and then we have the following.

Proposition 24. Under the condition $S_{T}^{*'} > S_{2}^{*}$, the following assertions hold.

(1) If $I_T < I_{1}^*$, we have $E_{s 3}^{-} \notin \ell_5, E_{s 3}^{+} \notin \ell_5, E_{1}\notin \Gamma_1, E_{2}\in \Gamma_2$;

(2) If $I_{1}^* < I_T < I_{2}^*$, we have $E_{s 3}^{-} \in \ell_5, E_{s 3}^{+} \notin \ell_1, E_{1}\in \Gamma_1, E_{2}\in \Gamma_2$;

(3) If $I_{2}^* < I_T < I_{T}^{*'}$, we have $E_{s 3}^{-} \notin \ell_5, E_{s 3}^{+} \notin \ell_5, E_{1}\in \Gamma_1, E_{2}\notin \Gamma_2$;

(4) If $I_T > I_{T}^{*'}$, we have that $E_{s 3}^{-} \mathit{{\text{and}}} E_{s 3}^{+} \mathit{{\text{do not exist}}}, E_{1}\in \Gamma_1, E_{2}\notin \Gamma_2$.

Proposition 25. Under the condition $S_{1}^{*} < S_{T}^{*'} < S_{2}^{*}$, the following assertions hold.

(1) Assume that $I_{1}^* < I_{2}^*$, and further,

● if $I_T < I_{1}^*$, we have $E_{s 3}^{-} \notin \ell_5, E_{s 3}^{+} \notin \ell_5, E_{1}\notin \Gamma_1, E_{2}\in \Gamma_2$;

● if $I_{1}^* < I_T < I_{2}^*$, we have $E_{s 3}^{-} \in \ell_5, E_{s 3}^{+} \notin \ell_5, E_{1}\in \Gamma_1, E_{2}\in \Gamma_2$;

● if $I_{2}^* < I_T < I_{T}^{*'}$, we have $E_{s 3}^{-} \in \ell_1$, $E_{s 3}^{+} \in \ell_5, E_{1}\in \Gamma_1, E_{2}\notin \Gamma_2$;

● if $I_T > I_{T}^{*'}$, we have that $E_{s 1}^{-} \mathit{{\text{and}}} E_{s 1}^{+} \mathit{{\text{do not exist}}}, E_{1}\in \Gamma_1, E_{2}\notin \Gamma_2$.

(2) Assume that $I_{1}^* > I_{2}^*$, and further,

● if $I_T < I_{2}^*$, we have $E_{s 3}^{-} \notin \ell_5, E_{s 3}^{+} \notin \ell_5, E_{1}\notin \Gamma_1, E_{2}\in \Gamma_2$;

● if $I_{2}^* < I_T < I_{1}^*$, we have $E_{s 3}^{-} \notin \ell_5, E_{s 3}^{+} \in \ell_5, E_{1}\notin \Gamma_1, E_{2}\notin \Gamma_2$;

● if $I_{1}^* < I_T < I_{T}^{*'}$, we have $E_{s 3}^{-} \in \ell_5$, $E_{s 3}^{+} \in \ell_5\, E_{1}\in \Gamma_1, E_{2}\notin \Gamma_2$;

● if $I_T > I_{T}^{*'}$, we have that $E_{s 3}^{-} \mathit{{\text{and}}} E_{s 3}^{+} \mathit{{\text{do not exist}}}, E_{1}\in \Gamma_1, E_{2}\notin \Gamma_2$.

Proposition 26. Under the condition $S_{T}^{*'} < S_{1}^{*}$, the following assertions hold.

(1) If $I_T < I_{2}^*$, we have $E_{s 3}^{-} \notin \ell_5, E_{s 3}^{+} \notin \ell_5, E_{1}\notin \Gamma_1, E_{2}\in \Gamma_2$;

(2) If $I_{2}^* < I_T < I_{1}^*$, we have $E_{s 3}^{-} \notin \ell_5, E_{s 3}^{+} \in \ell_5, E_{1}\notin \Gamma_1, E_{2}\notin \Gamma_2$;

(3) If $I_{1}^* < I_T < I_{T}^{*'}$, we have $E_{s 3}^{-} \notin \ell_5, E_{s3}^{+} \notin \ell_5, E_{1}\in \Gamma_1, E_{2}\notin \Gamma_2$;

(4) If $I_T > I_{T}^{*'}$, we have that $E_{s 3}^{-} \mathit{{\text{and}}} E_{s 3}^{+} \mathit{{\text{do not exist}}}, E_{1}\in \Gamma_1, E_{2}\notin \Gamma_2$.

Based on Propositions 24–26, the following summary is given.

D1. Let $E_{s3}^{-} \notin \ell_5, E_{s3}^{+} \notin \ell_5, E_{1}\notin \Gamma_1, E_{2}\in \Gamma_2$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $D_{1-1}$. Then, we conclude that the system (2.4) does not have a pseudo-equilibrium, and all trajectories of the system (2.4) will converge to $E_{2}^R$. The result of this numerical simulation is shown in Figure 7(a), where

D2. Let $E_{s3}^{-} \in \ell_5, E_{s3}^{+} \notin \ell_5, E_{1}\in \Gamma_1, E_{2}\in \Gamma_2$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $D_{2-1}$. Then, we show that $E_{s3}^{-}\in \ell_5$ is an unstable pseudo-equilibrium. The solution of the system (2.4) will approach $E_{1}^R$ or $E_{2}^R$, as shown in Figure 7(b), where

D3. Let $E_{s3}^{-} \notin \ell_5, E_{s3}^{+} \notin \ell_5, E_{1}\in \Gamma_1, E_{2}\notin \Gamma_2$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $D_{3-1}\cup D_{3-2}\cup D_{3-3}$. Then, we know that the system (2.4) does not have a pseudo-equilibrium, and all trajectories of the system (2.4) will converge to $E_{1}^R$, as shown in Figure 7(c), where

D4. Let $E_{s3}^{-} \in \ell_5, E_{s3}^{+} \in \ell_5, E_{1}\in \Gamma_1, E_{2}\notin \Gamma_2$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $D_{4-1}$. Then, we show that $E_{s3}^{-}\in \ell_5$ is an unstable pseudo-equilibrium, and the solution of system (2.4) will approach $E_{1}^R$ or $E_{s3}^{+}$, as shown in Figure 7(d), where

D5. Let $E_{s3}^{-} \notin \ell_5, E_{s3}^{+} \in \ell_5, E_{1}\notin \Gamma_1, E_{2}\notin \Gamma_2$, and the value $\Upsilon = \left(S_{T}, I_{T}\right)$ belongs to the set $D_{5-1}$. Then, we conclude that $E_{s3}^{+}\in \ell_5$ is a stable pseudo-equilibrium. All solutions of the system (2.4) will tend to $E_{s3}^{+}$. The result of this numerical simulation is shown in Figure 7(e), where

Remark 1. For smooth system (2.1), we have discussed the three equilibrium points, that is, $(0, 0)$, the disease free equilibrium point $E_{i1}(i = 1, 2, 3)$ and the endemic equilibrium point $E_{i}(i = 1, 2, 3)$ of (2.4). By constructing a Lyapunov function, we obtain the global stability of system (2.1) in Theorem 1. For the non-smooth system (2.4), we investigate the non-smooth system (2.4) with two threshold control strategies. Using a Filippov analysis method, Green's formula, the comparison theorem and numerical simulation method, the rich dynamics of the system are given, such as the bistability phenomenon, the globally stable pseudo-equilibrium and the regular/virtulal equilibrium bifurcations. Through the two control strategies, we can control the disease individuals to the appropriate balance. In particular, Theorems 3, 6 and 7 in this paper cannot appear in the smooth system (2.1); please see Figure 4(b), (c). There is bistability in the system (2.4).

Remark 2. In ^[25], a Filippov model describing the effects of media coverage and quarantine on the spread of human influenza was considered, and the threshold conditions for stability switches were obtained analytically. The discontinuous system (2.4) considered in our paper is a logistic source, and ^[25] considered a linear source. Second, the dynamics are different. Our paper employs the Green's theorem and a Dulac function. Then, we show that two real equilibria occur simultaneously in our paper. Using numerical simulation methods, the sliding dynamics and bifurcations of a human influenza system under logistic source and broken line control strategy are given. The results of this paper are new with respect to^[25].

6.   The regular/virtulal equilibrium bifurcations

Based on the previous discussion, it is shown that system (2.4) will exhibit multiple equilibriums and sliding modes. In order to better construct the bifurcation diagram, we choose $\gamma$ and $I_{T}$ as bifurcation parameters, and the other parameters are fixed as shown in Figure 8. With the expressions of equilibria found in Section 2.3, the lines to divide the relevant parameter plane are given as follows:

The three solid lines $l_{1}$, $l_{2}$ and $l_{3}$ divide the $\gamma-I_{T}$ two-dimensional plane space into four regions in the first quadrant. Suppose that the control value $I_{T}$ satisfies $I_{3}^{*} < I_{T} < I_{2}^{*}$ and $I_{1}^{*} < I_{3}^{*}$ (that is, regions $\Omega_{2}^{*}$ and region $\Omega_{3}^{*}$; see Figure 8), the points $E_{2}$ and $E_{3}$ are virtual equilibria points (denoted by $E_{2}^{v}$, and $E_{3}^{v}$, respectively), and $E_{s1}^{-}$ exists with the sliding mode domain. If the control value $I_{T}$ satisfies $I_{T} > I_{2}^{*}$ (that is, $\Omega_{1}^{*}$, as shown Figure 8), $E_{2}$ is a regular equilibrium, while the point $E_{3}$ is a virtual equilibrium (denoted by the equilibrium $E_{2}^{R}$ and the equilibrium $E_{3}^{v}$, respectively), and point $E_{s1}^{-}$ does not exist with the sliding mode domain. If $I_{T} < I_{3}^{*}$ (that is, $\Omega_{4}^{*}$; see Figure 8), $E_{3}$ is a regular equilibrium, while point $E_{2}$ is a virtual equilibrium (denoted by $E_{3}^{R}$ and $E_{2}^{v}$), and point $E_{s1}^{-}$ does not exist with the sliding mode domain.

Next, we choose $q$ and $I_{T}$ as bifurcation parameters, and the other parameters are fixed. From Proposition 1 in this paper, the lines to divide the relevant parameter plane are given as follows:

The two solid lines $l_{4}$ and $l_{5}$ divide the $q-I_{T}$ two-dimensional plane space into three regions in the first quadrant $\mathbb{R}^{+}$. Suppose that the control value $I_{T}$ satisfies $I_T > I_T^*$ (that is, region $\Omega_{7}$; see Figure 9(a)), then system (3.1) has no equilibrium. If the control value $I_{T}$ satisfies $I_T^* > I_T > \frac{q}{\beta p}$ (that is, region $\Omega_{6}$, as shown in Figure 9(a)), system (3.1) has two positive equilibria $E_{s1}^{+} = (S_{s1}^{+}, I_T)$ and $E_{s1}^{-} = (S_{s1}^{-}, I_T)$, where $S_{s1}^{\pm} = \frac{(\gamma-\frac{q}{p})\pm\sqrt \Delta_1}{2\gamma}K$. When the control value $I_{T}$ satisfies $0 < I_T < \frac{q}{\beta p}$ (that is, region $\Omega_{5}$; see Figure 9(a)), system (3.1) has a unique positive equilibrium $E_{s2} = (S_{s2}, I_T)$, where $S_{s2} = \frac{(\gamma-\frac{q}{p})+\sqrt \Delta_1}{2\gamma}K$.

We choose $\gamma$ and $I_{T}$ as bifurcation parameters, and the other parameters are fixed. With Proposition 9 in this paper, the line to divide the relevant parameter plane is given as

The solid line $l_{6}$ divides the $\gamma-I_{T}$ two-dimensional plane space into two regions in the first quadrant $\mathbb{R}^{+}$. Suppose that the control value $I_{T}$ satisfies $I_T > I_T^{*'}$ (that is, region $\Omega_{8}$, as shown in Figure 9(b)), and then system (4.1) does not have an equilibrium. If the control value $I_{T}$ satisfies $0 < I_T < I_T^{*'}$ (that is, region $\Omega_{9}$, as shown in Figure 9(b)), system (4.1) has two positive equilibria $E_{s3}^{+} = (S_{s3}^{+}, I_T)$ and $E_{s3}^{-} = (S_{s3}^{-}, I_T)$, where $S_{s3}^{\pm} = \frac{\gamma\pm\sqrt \Delta_2}{2\gamma}K$.

Remark 3. Notice that^[30] considered the global dynamics of a Filippov predator-prey model with two thresholds for integrated pest management. By using Filippov theory, the sliding mode dynamics and global dynamics were established. Different from ^[30], our paper shows the dynamic behavior of the Filippov model with respect to all possible equilibria. It is shown that the Filippov system tends to the pseudo-equilibrium on sliding mode domain or one endemic equilibrium or two endemic equilibria under some conditions. Second, although both this paper and ^[30] discuss the global dynamics of a Filippov model with two thresholds, this paper first gives different control strategies. In particular, the two real equilibria occur simultaneously using methods such as Green's theorem and a Dulac function.

7.   Discussion

In this paper, we have established a non-smooth system to determine whether it is necessary to adopt the control strategy of media coverage and quarantine of susceptible individuals according to the number of infected and susceptible individuals. Media coverage changes the transmission mode of influenza. Further, in order to reduce the spread of influenza, when the number of cases exceeds the larger infection threshold $I_T$, and the number of susceptible individuals is greater than $S_T$, we will quarantine the susceptible individuals. It is worth noting that there are two difficulties in this paper. First, the traditional continuity theory cannot be applied due to the non-smooth system with the broken line control strategy. For example, when proving the global stability of discontinuous systems, the traditional Lyapunov function cannot be similarly constructed. Second, Green's formula of continuous systems cannot be used to prove the existence of global stability of the pseudo equilibriums in discontinuous systems. In this paper, by choosing different thresholds $I_{T}$ and $S_{T}$ and using Filippov theory, we study the dynamic behavior of the Filippov model with respect to all possible equilibria. The regular/virtulal equilibrium bifurcations are given. It is shown that the Filippov system tends to the pseudo-equilibrium on sliding mode domain or one endemic equilibrium or two endemic equilibria under some conditions.

Next, we summarize the corresponding biological results of Tables 1–3. These results show that the choice of values of $I_T$ and $S_T$ is very important, and it determines whether to adopt control strategies.

● In Table 1, we know that the infection threshold value $I_T$ is chosen to be small enough, i.e., $I\ll I_{T}$, and then the number of infected individuals will reach the equilibrium $E_1^R$ of system (2.4).

● From Table 2, system (2.4) has a unique globally asymptotically stable pseudo-equilibrium $E_{s1}^+$ or $E_{s3}^+$ if $I = I_T$ or admits a unique globally asymptotically stable equilibrium when $I < I_T$. Our control goal can be achieved finally, and there is no need to adjust the threshold strategy.

● In Table 3, the solution of system (2.4) will converge to a locally asymptotically stable equilibrium if $I < I_{T}$ or tends to a locally asymptotically stable equilibrium $E_{3}^{R}$ when $I > I_{T}$ or pseudo-equilibrium $E_{s1}^+, E_{s3}^+$ if $I = I_{T}$. We show that it may be necessary to adjust the threshold policy according to the initial number of susceptible individuals and infected individuals. The results obtained have certain guiding significance for choosing thresholds and designing a corresponding threshold strategy.

Next, we consider the effect of key parameters in the subsystem on the basic regeneration number $R_{0i}$ as follows.

The three-dimensional diagram of the parameter space $(\delta, d, R_{01})$ is shown in Figure 10(a) under the parameter values of $K = 4$, $\beta = 0.5$, $r = 0.2$. It observe when the parameter $\delta = 0.8, d$ increase from $0.83$ to $1$, the basic reproduction number $R_{01}$ decreases correspondingly and is less than unit 1. The trajectory of the subsystem (2.4) will converge globally to the free equilibrium (see Theorem 1), implying that the infected individuals extinct and then a stable free steady state occurs.

The three-dimensional diagram of the parameter space $(\beta, p, R_{02})$ is shown in Figure 10(b) under the parameter values of $K = 2$, $d = 0.1$, $r = 0.05$, $\delta = 0.05$. It is easy to observe when fixing the $p = 0.5$, $\beta$ increase from $0.78$ to $1$, $R_{02}$ decreases correspondingly and is less than unit 1. By using Theorem 1, the infected individuals persist and the trajectory of the subsystem (2.4) will converge globally to the endemic steady state.

The three-dimensional diagram of the parameter space $(\beta, \delta, R_{01})$ is shown in Figure 10(c) under the parameter values of $K = 2$, $d = 0.1$, $r = 0.1$. We observe when the parameters $\delta, \beta$ increase from $0.6$ to $1$, $R_{01}$ also increases correspondingly and is greater than the unit 1. By using Theorem 1, the infected individuals persist and the trajectory of the subsystem (2.4) will converge globally to the endemic steady state.

The three-dimensional diagram of the parameter space $(\beta, p, R_{03})$ is shown in Figure 10(d) under the parameter values of $K = 1$, $\gamma = 1.8$, $r = \frac{1}{27}$, $\delta = \frac{1}{27}$, $d = \frac{1}{27}$. It is easy to observe when fixing the parameter $p = 0.2,$ with a transmission rate $\beta$ increase from $0.81$ to $1$, $R_{03}$ increases correspondingly and is greater than the unit 1. By using Theorem 1, the infected individuals persist, and the trajectory of the subsystem (2.4) will converge globally to the endemic steady state. When fixing the parameters $p = 0.6,$ transmission rate $\beta$ increase from $0.81$ to $1$, $R_{03}$ decreases correspondingly and is less than the unit 1. The trajectory of the subsystem (2.4) will converge globally to the free equilibrium (see Theorem 1 of our paper), implying that the infected individuals become extinct, and then a stable free steady state occurs.

In addition, this model has not been validated by actual influenza data, and we only have analyzed theoretically. In the next stage, we will verify and simulate the validity of the conclusions in actual time from some websites and statistics of health departments. However, the paper has studied the non-smooth system of two threshold control strategies and has validated the correctness of the theory through numerical simulation. Due to the serious lack of current influenza data from SARS-CoV-2 infections, the verification of the work is extremely difficult. However, the theory of this paper can provide appropriate guidance for the current influenza by SARS-CoV-2 infection. In this paper, we only consider the dynamics of the system (2.4) if the basic reproduction number $R_{0i} > 1$. However, under the saturation rate $\frac{\beta SI}{1+I}$, and the conditions $R_{02} < R_{01} < 1$ and $R_{02} < 1 < R_{01}$, the dynamical behaviors of system (2.4) and the method of proving global stability are not yet fully clear and would be our further topic.

Acknowledgments

We sincerely thank the anonymous referees for their very detailed and helpful comments on which improved the quality of this paper. This work is supported in part by the Yunnan Fundamental Research Projects (No: 202101BE070001-051).

Conflict of interest

The authors have no conflict of interest to declare in carrying out this research work.