Non-smooth dynamics of a SIR model with nonlinear state-dependent impulsive control

1.   Introduction

The mathematical model of infectious diseases has always played an important role in the prevention and control of infectious diseases, for example the classical SIR model is usually used to describe the transmission dynamics of infectious diseases among humans such as measles, chickenpox, whooping cough, mumps, etc ^[1,2,3,4,5]. It not only provides important dynamic descriptions for the evolution of infectious diseases, but also provides important quantitative tools for evaluating the effectiveness of various prevention and control strategies. The above important roles have been more important in the three years of the COVID-19 outbreak. Cui et al. consider the impact of limited medical resources or treatment capacity on infectious disease transmission to develop the SIR model ^[6,7,8] with nonlinear term, as shown in Figure 1, which illustrates the relationships between $S$, $I$ and $R$, where $S, I, R$ represent the density of susceptible, infected and recovered population, respectively. The corresponding model is as follows:

where

● $\Lambda$ is the constant recruitment rate of susceptible population, $\mu$ is natural death rate, $\beta$ is the transmission rate;

● $d$ represents the death rate caused by disease. $v$ is the recovery rate without hospital treatment. The funtion $h(I) = cI/(b+I)$ is the recovered population with hospital treatment, where $c$ gives the maximum recovery rate and $b$ is the infected size at which there is 50% saturation ($h(b) = c/2$).

Due to the first two equations being independent of the third one, we can only study the following reduced model:

where $\gamma = \mu+d+v$.

Infectious disease prevention and control has been a hot topic in recent years, usually by controlling the source of infection, cutting off the transmission route and protecting susceptible people. In practice, we often respond to infectious diseases by vaccinating susceptible populations and isolating infected individuals for treatment, which often result in the right function of the dynamical system to be unsmooth or even discontinuous. However the continuous dynamical systems like the above cannot describe the aforementioned situation. By establishing appropriate mathematical model, we can quantify these control measures and analyze the effectiveness of the measures qualitatively or quantitatively, and the state-dependent impulsive model is one type of mathematical model that can well characterize infectious disease control ^[9,10,11]. The model assumes that no control measures are implemented when the number of susceptible population is within a certain range, and that comprehensive measures including immunization of the susceptible population and treatment of infected persons are taken when the size of the susceptible population reaches or exceeds the control threshold. Subsequently, numerous scholars have developed a large number of state-dependent impulsive models for different types of infectious disease characteristics and discussed the impact of state-dependent impulsive control strategies on the dynamic behaviors, such as disease elimination and epidemics.

Zhang et al.^[12] and Cheng et al.^[13] qualitatively analyzed the state-dependent impulsive models under different control measures. There have been relatively systematic studies on single-threshold state-dependent impulsive models ^{[14,15,16,17,18,19,20]}, but it is of concern that if only the size of the susceptible population is used as the basis for control, the following two situations may occur: one in which the size of the susceptible population is small but the growth rate is large, and the other in which there is a larger susceptible population but its growth rate is relatively small. Then in the first case, due to the delay effect, it may not be possible to achieve the desired prevention and control goal; similarly, in the second case, if the growth rate is small, there is no need for infectious disease prevention and control ^{[21,22,23,24]}. Therefore, this paper integrates the case of threshold control for a convex combination of the size of the susceptible population and its growth rates, which is more complex but closer to reality. Thus, based on model (1.2), we propose the following state-dependent feedback control SIR model:

Here, non-negative constant $AT$ represents the action threshold and non-negative constants $\alpha_1$ and $\alpha_2$ are the weight parameter in the threshold condition and satisfies $\alpha_1+\alpha_2 = 1$. $p\in[0, 1]$ denotes the vaccination rate of the susceptible population and $q\in[0, 1]$ the isolated ratio of the infected population. Considering the practical significance, we assume that the initial value $(S_0, I_0)$ of system (1.3) comes from the domain $\Omega$, where

Otherwise, the initial values are taken after an integrated control strategy application ^[25,26].

The main purpose of this study is to investigate the dynamics of the proposed state-dependent impulsive model, including but not limited to the periodic solutions and the bifurcations. The organization of the rest part of the paper is as follows: In Section 2, we first give the basic definitions of the state-dependent impulsive model and some lemmas on the stability of the disease-free periodic solution (DFPS). The main properties of the ordinary differential equation (ODE) are also introduced in this section. In Section 3, we address the existence and stability of the periodic solution including DFPS and the non-trivial periodic solution by analyzing the properties of ODE and the Poincaré map. In Section 4, the transcritical bifurcations have been investigated with respect to two interesting parameters ^{[27,28,29,30,31,32,33,34,35,36,37,38,39]}. Finally, we summarize the whole work and give some discussions in the last section.

2.   Preliminaries and notations

2.1.   Planar impulsive semi-dynamic systems and preliminaries

In this section, we will give a brief summary about the main results used in the following section. Consider the following generalized planar impulsive semi-dynamic system

Here, $(x, y)\in R^2_+ = \{(x, y)|x\geq0, y\geq 0\}$, $\triangle x = x^+-x$ and $\triangle y = y^+-y$. $F_1, \; F_2, \; \bar{\alpha}, \; \bar{\beta}$ are continuous functions from $R_+^2$ into $R$. The impulsive function $H:R_+^2\rightarrow R_+^2$ is defined as

and $N^+ = (x^+, y^+)$ is called an impulsive point of $M = (x, y)$. We can define the planar impulsive semi-dynamic system and an order-$k$ periodic solution of model (2.1) in the following based on the notation and definition presented in literatures ^[17,40,41].

Definition 2.1. A solution $(x(t), y(t))$ of an impulsive system is said to be an order-$k$ periodic solution with period $T$, if $T$ is the smallest positive number satisfying $(x(t+kT), y(t+kT)) = (x(t), y(t))$ for all $k\geq0$ and $t\geq0$, and the trajectory $(x(t), y(t))$ pulses $k$ times within period $T$.

Further, the following analogue of Poincaré criterion can be used to analyze the local stability of an order-$k$ periodic solution.

Lemma 2.1. (^[42]) The solution $(x(t), y(t)) = (\xi(t), \eta(t))$ with T-periodic of the system (2.1) is orbitally asymptotically stable if the Floquet multiplier $\mu_2$ satisfies the condition $\mid\mu_2\mid < 1$, where

with

and $F_1, \; F_2, \; \frac {\partial\bar{\alpha}}{\partial x}, \; \frac {\partial\bar{\alpha}}{\partial y}, \; \frac {\partial\bar{\beta}}{\partial x}, \; \frac {\partial\bar{\beta}}{\partial y}, \; \frac {\partial\phi}{\partial x}, \; \frac {\partial\phi}{\partial y}$ can be calculated at the point $(\xi(\tau_k), \eta(\tau_k))$, and $F_1^+ = F_1(\xi(\tau_k^+), \eta(\tau_k^+))$, $F_2^+ = F_2(\xi(\tau_k^+), \eta(\tau_k^+))$. Here $\phi(x, y)$ is sufficiently smooth such that grad $\phi(x, y)\neq0$, and $\tau_k(k\in N)$ is the moment of the k-th impulse effect.

In order to discuss the bifurcation of the Poincaré map, we introduce the following lemma:

Lemma 2.2. (Transcritical bifurcation ^[43]) Let $:U\times \Theta\longrightarrow \mathbb{R}$ define a one-parameter family of maps $\mathcal{P}(I, \alpha)$, where $\mathcal{P}$ is $C^r$ with $r\geq2$, and $U$, $\Theta$ are open intervals of the real line containing 0. Assume that

Then there are $\alpha_1 < 0 < \alpha_2$ and $\varepsilon > 0$ such that:

(i) If $\alpha_1 < \alpha < 0$, $\mathcal{P}_\alpha$ has two fixed points, 0 and $I_{1\alpha} > 0$ in $(-\varepsilon, \varepsilon)$. The origin is asymptotically stable, the other fixed point is unstable.

(ii) If $0 < \alpha < \alpha_2$, $\mathcal{P}_\alpha$ has two fixed points, 0 and $I_{1\alpha} < 0$ in $(-\varepsilon, \varepsilon)$. The origin is unstable, the other fixed point is asymptotically stable.

Here again, the case $\frac{\partial^2 \mathcal{P}}{\partial I\partial \alpha}(0, 0) < 0$ is handled by making the change of parameter $\alpha\rightarrow -\alpha$.

2.2.   The main properties of ODE system

Firstly, we focus on the ODE system:

It's easy to know that system (2.4) has one disease-free equilibrium at $E_0 = (\frac{\Lambda}{\mu}, 0)$. By calculating the Jacobian matrix at $E_0$, we have the following lemma.

Lemma 2.3. $E_0$ is locally asymptotically stable if $R_0: = \frac{\Lambda b \beta}{\mu (b\gamma +c)}\leq 1$ and unstable if $R_0 > 1$.

Then we focus on the existence and stability of the endemic equilibrium. Let

then

where $a_1 = \frac{\beta \gamma b+\mu \gamma +c\beta-\Lambda \beta }{\beta \gamma}, a_2 = \frac{\mu\gamma b+c\mu-\Lambda \beta b}{\beta\gamma} = \frac{\Lambda b}{\gamma}(\frac{1}{R_0}-1)$. The positive root of $I^2+a_1I+a_2 = 0$ implies the existence of the endemic equilibrium. So it's easy to prove the following lemma:

Lemma 2.4. For the existence of the endemic equilibrium, we have the following conclusion:

1. If $R_0 > 1$, then there exists a unique endemic equilibrium.

2. If $R_0 = 1$, then there is no endemic equilibrium when $a_1\geq 0$ and there exists a unique endemic equilibrium when $a_1 < 0$.

3. If $R_0 < 1$ and $a_1\geq 0$, then there is no endemic equilibrium.

4. If $R_0 < 1$ and $a_1 < 0$, then we have $R^*_0 = (\frac{a_1^2 \gamma}{4\Lambda b}+1)^{-1}$ and there is no endemic equilibrium when $R_0 < R^*_0$, otherwise there exists a unique endemic equilibrium of multiplicity two or two endemic equilibria when $R_0 = R^*_0$ or $R_0 > R^*_0$, respectively.

If $R^*_0 < R_0 < 1$ and $a_1 < 0$, $E^*_1(S^*_1, I^*_1)$ and $E^*_2(S^*_2, I^*_2)$ are the corresponding equilibria, where

and $\Delta = a_1^2-4a_2.$

The Jacobian of the system (2.4) at $E^*(S^*, I^*)$ is

and the characteristic equation is given by

where

Therefore, we have the following lemmas about the stability of the endemic equilibrium:

Lemma 2.5 (^[7]). If $R_0 > 1, b > \frac{\mu}{\beta}$, then the endemic equilibrium $E^*$ is a stable node or focus when $H(I^*) > 0$; $E^*$ is an unstable node or focus when $H(I^*) < 0$; $E^*$ is a center when $H(I^*) = 0$.

Lemma 2.6 (^[7]). If $R^*_0 < R_0 < 1, b > \frac{\mu}{\beta}$ and $a_1 < 0$, then the endemic equilibrium $E^*_2$ is a saddle; $E^*_1$ is a stable node or focus when $H(I^*) > 0$; $E^*_1$ is an unstable node or focus when $H(I^*) < 0$; $E^*_1$ is a center when $H(I^*) = 0$.

2.3.   Notations

We firstly denote some essential curves for the further study in the next section.

The vertical and horizontal isoclines of system (1.2) are shown below,

which intersect with S-axis at $(\frac{\Lambda}{\mu}, 0)$ and $(\frac{1}{\beta}(\frac{c}{b}+\gamma), 0)$, respectively. The positional relationship of the two points is determined by $R_0$, that is, $\frac{\Lambda}{\mu} > \frac{1}{\beta}(\frac{c}{b}+\gamma)$ when $R_0 > 1$ and $\frac{\Lambda}{\mu} < \frac{1}{\beta}(\frac{c}{b}+\gamma)$ when $R_0 < 1$. In addition, the intersection point of $L_1$ and $L_2$ in the first quadrant is the endemic equilibrium of system (1.2).

In the phase plane, we can define the impulsive curve $L_M$ and the phase curve $L_N$, which once the trajectory intersects with $L_M$, it will impulse to $L_N$. And we can get the expression of $L_M$ and $L_N$ by $\alpha_1 S(t)+\alpha_2 \frac{dS(t)}{dt} = AT$ and impulse functions ^{[21,22,23,24]}.

When $\alpha_1 = 1$, the impulsive curve $L_M$ and the phase curve $L_N$ of system (1.3) are straight lines $S = AT$ and $S = (1-p)AT$, respectively.

When $\alpha_1 \in [0, 1)$, the impulsive curve $L_M$ of system (1.3) is a curve $I = L_M(S)$, where

which follows from $\alpha_1 S(t)+\alpha_2 \frac{dS(t)}{dt} = AT$. $L_M$ intersects with S-axis and $L_1$ at $M_0(S_v, 0)$ and $M_1(S_L, L_1(S_{l1}))$, where

respectively. Based on the biological significance of system (1.3), here

And thus function $I = L_M(S)$ monotonically increases with respect to $S$. The phase curve $L_N$ is represented by an increasing function $I = L_N(S)$ correspondingly, where

By solving $L_M(S) = L_N(S)$, we conclude that curves $L_M$ and $L_N$ have a unique intersection point, denoted as $(S_{mn}, L_M(S_{mn}))$, where $S_{mn} = \frac{1-(1-p)(1-q)}{q}S_{v} > S_{v}$.

We define some auxiliary functions as follows (see more details in ^[44,45]),

Thus,

represent the normal vectors of the impulsive curve $L_M$ and the phase curve $L_N$, respectively. Let $(\frac{dS}{dt}, \frac{dI}{dt})$ denote the tangent vector of the phase orbit of system (1.2). Denote

which are useful in section 3. Note that for a point $(S, I)$ satisfying $P(S, I) = 0$,

● if $\sigma_M(S, I) > 0$, a trajectory pulses at point $(S, I)$ and pulses to point $((1-p)S, (1-q)I)$;

● if $\sigma_M(S, I) = 0$, a trajectory is tangent to the impulsive curve $L_M$ and pulses to point $((1-p)S, (1-q)I)$;

● if $\sigma_M(S, I) < 0$, a trajectory passes through point $(S, I)$ and no pulse occurs at this point.

For a point $(S, I)$ satisfying $Q(S, I) = 0$,

● if $\sigma_N(S, I) > 0$, a trajectory passes through point $(S, I)$ from above curve $L_N$ to below;

● if $\sigma_N(S, I) = 0$, a trajectory is tangent to the phase curve $L_N$ at $(S, I)$;

● if $\sigma_N(S, I) < 0$, a trajectory passes through point $(S, I)$ from below curve $L_N$ to above.

3.   Existence and stability of periodic solution

3.1.   Disease-free periodic solution

Let $I(0) = 0$ in system (1.3), then $I(t)\equiv0$ and $S(t)$ satisfies the following equations:

where $S_v = \frac{AT-\alpha_2\Lambda}{\alpha_1-\alpha_2\mu} > 0$ since both $\alpha_1-\alpha_2\mu$ and $AT-\alpha_2\Lambda$ are positive in this article.

Solving the first equation of system (3.1) with the initial condition $S(0) = (1-p)S_v$, we have

Obviously, if $S_v < \frac{\Lambda}{\mu}$, then the model (3.1) has a periodic solution $\tilde{S}(t)$ with period $T$, here

and

It follows from the relationship of system (1.3) and system (3.1) that the following theorem holds true naturally.

Theorem 3.1. If $S_v < \frac{\Lambda}{\mu}$, then system (1.3) has a DFPS $(\tilde{S}(t), 0)$ with period $T$.

Using Lemma 2.1, the stabilities of the DFPS are shown in the following theorems.

Theorem 3.2. If $R_0\leq1$ and $S_v < \frac{\Lambda}{\mu}$, then the DFPS $(\tilde{S}(t), 0)$ of system (1.3) is locally orbitally asymptotically stable.

Proof. Under the assumptions $R_0\leq1$ and $S_v < \frac{\Lambda}{\mu}$, we claim that the DFPS $(\tilde{S}(t), 0)$ of system (1.3) is orbitally asymptotically stable. For system (1.3), there are $F_1(S, I) = \Lambda -\mu S-\beta SI$, $F_2(S, I) = \beta SI-\gamma I-\frac{cI}{b+I}$, $\bar{\alpha}(S, I) = -pS$, $\bar{\beta}(S, I) = -qI$, $\phi(S, I) = \alpha_1 S+\alpha_2(\Lambda-\mu S-\beta SI)-AT$, $(\tilde{S}(T), 0) = (S_v, 0)$, and $(\tilde{S}(T^+), 0) = ((1-p)S_v, 0)$. Thereby,

and

Taking the above two equations into formula (2.2) yields

where $(1-q)\exp\left(-\frac{p\beta S_v}{\mu}\right)\in[0, 1)$. Moreover, It follows from $S_v < \frac{\Lambda}{\mu}$ and $R_0 = \frac{\Lambda b \beta}{\mu (b\gamma +c)}\leq1$ that $\left(\frac{\Lambda-\mu S_v}{\Lambda-\mu (1-p)S_v}\right)^{\frac{b\mu\gamma+c\mu-\Lambda b \beta}{b\mu^2}}\in(0, 1]$. Therefore, when $S_v < \frac{\Lambda}{\mu}$ and $R_0\leq1$ there is $|\mu_2| < 1$, i.e., $(\tilde{S}(t), 0)$ is orbitally asymptotically stable. □

The attraction domain of DFPS when $R_0\leq1$ and $S_v < \frac{\Lambda}{\mu}$ will be introduced in the next subsection (see Theorem 3.4). With regard to $R_0 > 1$, we have the following conclusion:

Theorem 3.3. If $R_0 > 1$ and $S_v\leq (\gamma+\frac{c}{b})/\beta$ hold true, then the DFPS is locally orbitally asymptotically stable.

Proof. It follows from Eq (3.5) that

Let

It follows from $R_0 > 1$ that

which indicates that $f(s)$ is increasing and $f(\frac{1}{\beta}(\gamma+\frac{c}{b})) = 0$. Thus, if $S_v\leq \frac{1}{\beta}(\gamma+\frac{c}{b})$ then $f(s)\leq0$ for all $s\in[(1-p)S_v, S_v]$ and $0 < \mu_2|_{q = 0}\leq1$. Moreover, it follows from the formula of $\mu_2$ that we have

Therefore, if $R_0 > 1$ and $S_v\leq \frac{1}{\beta}(\gamma+\frac{c}{b})$, i.e., $0 < \mu_2|_{q = 0}\leq1$, then $|\mu_2| = (1-q)(\mu_2|_{q = 0}) < 1$ holds for all $q\in (0, 1]$, which means that the DFPS is locally orbitally asymptotically stable. □

Corollary 3.1. If $R_0 > 1$, $(\gamma+\frac{c}{b})/\beta\leq(1-p)S_v < S_v < \frac{\Lambda}{\mu}$ and $q = 0$ hold true, then the DFPS is unstable.

Remark 3.1. When $R_0 > 1$, $(1-p)S_v < (\gamma+\frac{c}{b})/\beta < S_v < \frac{\Lambda}{\mu}$ and $q = 0$ hold true, $\mu_2 = 1$ may occur, which means that the bifurcation phenomenon may exist near the DFPS with respect to the critical parameters (see Section 4).

3.2.   Non-trivial periodic solution

In this section, we first define the Poincaré map of system (1.3), and then discuss its main properties, which help us to discuss the non-trivial periodic solution of the system.

3.2.1.   Formation of Poincaré map $\mathcal{P}_{\mathcal{M}}$

Suppose $\mathcal{N}_0$ is the phase set and $\mathcal{M}_0$ is the impulsive set. If the solution starting from $N^+_n(S^+_n, I^+_n)\in \mathcal{N}_0 \subset L_N$ will arrive at the threshold line $L_M$ for the first time after a finite time, then the intersection point can be marked as $M_{n+1}(S_{n+1}, I_{n+1})$, as shown in Figure 2. Point $M_{n+1} \in \mathcal{M}_0$ and it will pulse to point $N^+_{n+1}(S^+_{n+1}, I^+_{n+1})\in \mathcal{N}_0$. The relation between $N^+_n$ and $N^+_{n+1}$ is determined by the solution of the ODE system. Thus we define

where

Therefore we have the following difference equations:

i.e., we have

Then a Poincaré map can be defined as follows:

According to the existence of the endemic equilibrium, we discuss the existence and stability of system (1.3) by analyzing the properties of Poincaré map $\mathcal{P}_{\mathcal{M}}$ in the following cases. It is worth reiterating that we assume that the initial value $(S_0, I_0)$ comes from the domain $\Omega$ where

3.2.2.   The system has no endemic equilibrium

Suppose $R_0 \leq 1$ and $a_1\geq 0$ or $R_0 < 1$ and $R_0 < R_0^*$ in this subsection. Lemma 2.4 shows that under the above conditions system (1.2) has a unique disease-free equilibrium $(\frac{\Lambda}{\mu}, 0)$, which is globally stable. Combining it with $(1-p)S_v < \frac{\Lambda}{\mu}$, we can conclude that any trajectory of system (1.3) undergoes finitely times impulses or is free from pulse effect, and then tends to $(\frac{\Lambda}{\mu}, 0)$ when $\frac{\Lambda}{\mu}\leq S_v < \frac{1}{\beta}(\frac{c}{b}+\gamma)$. When $S_v\geq \frac{1}{\beta}(\frac{c}{b}+\gamma)$, by calculating the function $\sigma_M(S, L_M(S))$ defined in (2.13), we can easily get that $\sigma_M(S, L_M(S)) < 0$ for all $S\geq S_v$, that is, system (1.3) has no impulsive effect in this case. Thus, system (1.3) may experience infinitely many impulsive effects only if $S_v < \frac{\Lambda}{\mu}$. So we discuss the dynamics of system (1.3) under the condition $S_v < \frac{\Lambda}{\mu}$ in the following text.

Firstly, the curves mentioned in Section 2.3, $L_1, L_2, L_M, L_N$, divide the plane into several parts. For example, if the relative positions of the impulsive curve and the isoclinic lines are shown in the Figure 2, we marked the parts as $V_1, V_2, ..., V_9$ (see Figure 2).

What's more, by calculating the function $\sigma_M(S, L_M(S))$, we can get that there exists at least one point $\bar{M}(S_\sigma, L_M(S_\sigma))$ satisfying $\sigma_M(\bar{M}) = 0$. Therefore, the precise impulsive set is $\mathcal{M}_0 = \{(S, L_M(S))|S \in \mathbb{D})\}$ and the precise phase set is $\mathcal{N}_0 = \{(S, L_N(S))|S \in (1-p)\mathbb{D}\}$, where

$S_\sigma\in(S_v, +\infty)$ is the root of $\sigma_M(S, L_M(S)) = 0$, and $S_{mn}$ is the unique root of $L_M(S)-L_N(S) = 0$.

We first discuss the following in the situation that $S_{mn} > (1-p)S_{\sigma}$ (see Figure 2). Any trajectory from initial point in part $V_1$, part $V_2$, part $V_3$ and part $V_5$ will stay in part $V_4$ after a finite number of impulsive effects. However, we have $\frac{dI}{dt} < 0$ in part $V_4$, and the impulsive effect will decrease $S$ and $I$. Thus, the trajectory will decreasingly converge to the disease free periodic solution and will not have a non-trivial periodic solution (Figure 2). The domain of attraction of DFPS $(\tilde{S}(t), 0)$ is $\Omega$. Then if $S_{mn} < (1-p)S_{\sigma}$, we denote the trajectory passing through the intersection of $L_M$ and $L_N$, $P_{mn}$, as $\Gamma_{mn}$, which divides $\Omega$ as $\Omega_1$(above) and $\Omega_2$(below). Apparently, any trajectory from an initial point in $\Omega_1$ will converge to DFPS $(\tilde{S}(t), 0)$ with the impulsive effect and any trajectory from an initial point in $\Omega_2$ undergoes at most one pulse and then tends to the boundary equilibrium $(\frac{\Lambda}{\mu}, 0)$. So we have the following theorem:

Theorem 3.4. Suppose $R_0 \leq 1$ and $a_1\geq 0$ or $R_0 < 1$ and $R_0 < R_0^*$. When $S_v\geq \frac{\Lambda}{\mu}$ then $(\frac{\Lambda}{\mu}, 0)$ is globally stable on $\Omega$. When $S_v < \frac{\Lambda}{\mu}$ then $(\frac{\Lambda}{\mu}, 0)$ and DFPS $(\tilde{S}(t), 0)$ are bistable and their domains of attraction are $\Omega_1$ and $\Omega_2$, respectively.

3.2.3.   The system has one endemic equilibrium

Based on the analysis in Section 2.2 (Lemma 2.4) and the relative position between the isoclinic lines and the impulsive curve, we will discuss the following cases:

● case (A): $R_0 > 1$

  – case ($a_1$): $S_v > \frac{\Lambda}{\mu}$

  – case ($a_2$): $\frac{1}{\beta}(\gamma+\frac{c}{b}) < S_v < \frac{\Lambda}{\mu}$

  – case ($a_3$): $S_v < \frac{1}{\beta}(\gamma+\frac{c}{b})$

● case (B): $R_0 < 1, a_1 < 0, R_0 = R^*_0$

In case ($a_1$), the trajectory of system (1.3) has no impulsive effect since $\sigma_M(S, L_M(S)) < 0$ for all $S\geq S_v$, that is, the solution of system (1.3) is determined by the ODE system (2.4) completely.

In case ($a_2$), it's easy to calculate that $\sigma_M(M_0) > 0$ and $\sigma_M(M_1) < 0$, where $M_0(S_v, 0)$ and $M_1(S_{l1}, L_1(S_{l1}))$ are the intersections of pulse curve $L_M$ and S-axis or $L_1$, respectively (see Figure 3(a)). So there exists at least one point $\bar{M}$ satisfying $\sigma_M(\bar{M}) = 0$. Assume that $\bar{M}(S_\sigma, L_M(S_\sigma))$ is the only point within the first quadrant that satisfies $\sigma_M(\bar{M}) = 0$. Here, $S_\sigma\in(S_v, S_{l1})$. Under this assumption, the precise impulsive set is $\mathcal{M}_0 = \{(S, L_M(S))|S \in \mathbb{D})\}$ and the precise phase set is $\mathcal{N}_0 = \{(S, L_N(S))|S \in (1-p)\mathbb{D}\}$, where $\mathbb{D}$ is shown in Eq (3.11).

Assume $b > \frac{\mu}{\beta}$ and $H(I^*) > 0$ in case ($a_2$), which ensures that $E^*$ is a globally stable focus or node of system (1.2) (see Lemma 2.5). Then there is a unique point $M_a$ either on the Ⅰ-axis or curve $L_M$ and the trajectory of system (1.2) starting from $M_a$ first reaches $L_M$ after a finite time $t > 0$ and intersects at point $\bar{M}$ when $S_\sigma < \frac{S_{mn}}{1-p}$. The trajectory $\overset{\frown}{M_a\bar{M}}$ divides $\Omega$ as $\Omega_3$ and $\Omega_4$, where $\Omega_3$ (resp. $\Omega_4$) at the left (resp. right) of trajectory $\overset{\frown}{M_a\bar{M}}$. Any trajectory starting from $\Omega_3$ will be free from impulsive effects and tend to $E^*$. Any trajectory starting from $\Omega_4$ will pulse finite (infinite) times. Then by discussing the properties of the Poincaré map $\mathcal{P}_{\mathcal{M}}$ defined in Eq (3.9), we have the following main results:

Theorem 3.5. Suppose $R_0 > 1, \frac{1}{\beta}(\gamma+\frac{c}{b}) < S_v < \frac{\Lambda}{\mu}$, $b > \frac{\mu}{\beta}$, $H(I^*) > 0$, $S_\sigma$ is the unique root of $\sigma_M(S, L_M(S)) = 0$ on interval $(S_v, S_{l1})$ and $S_\sigma < \frac{S_{mn}}{1-p}$.

(i) When $\overset{\frown}{M_a\bar{M}}$ and $L_N$ have at least one intersection point, denoting the smaller one is $N_m^+(S_m, L_N(S_m))$, then the domain of map $\mathcal{P}_{\mathcal{M}}$ is $\mathbb{D}_1 = [(1-p)S_v, S_m]$ and $\mathcal{P}_{\mathcal{M}}$ are increasing and continuous, that is, system (1.3) does not have order-$k$ ($k\geq2$) periodic solution. Moreover, if $S_m < (1-p)S_\sigma$, map $\mathcal{P}_{\mathcal{M}}(S)$ has at least one fixed point on interval $\mathbb{D}_1$ (see Figure 3(b)), that is, system (1.3) has at least one order-$1$ non-trivial periodic solution.

(ii) When $\overset{\frown}{M_a\bar{M}}$ and $L_N$ have no intersection point, then the domain of map $\mathcal{P}_{\mathcal{M}}$ is $\mathbb{D}_2 = [(1-p)S_v, (1-p)S_\sigma]$ and $\mathcal{P}_{\mathcal{M}}(S)$ are increasing and continuous, that is, system (1.3) does not have order-$k$ ($k\geq2$) periodic solution.

Proof. Firstly, under the assumptions of Theorem 3.5, it follows from Lemma 2.5 that $E^*$ is a globally stable focus or node of system (1.2).

It follows from the vector field of system (1.2), the Poincaré map $\mathcal{P}_{\mathcal{M}}$ is well defined on interval $\mathbb{D}_1$ for $(i)$ and on interval $\mathbb{D}_2$ for $(ii)$. Moreover, as previously analyzed, when $S_\sigma < \frac{S_{mn}}{1-p}$, the precise impulsive set of system (1.3) is $[S_v, S_\sigma]$. Therefore, the domain of map $\mathcal{P}_{\mathcal{M}}(S)$ as shown in Theorem (3.5). And to discuss the existence of the periodic solution of system (1.3), we only need to discuss the properties of the Poincaré map $\mathcal{P}_{\mathcal{M}}$ defined on interval $\mathbb{D}_1$ (resp. $\mathbb{D}_2$) for $(i)$ (rsep. for $(ii)$).

According to the vector field and the uniqueness of the solution of ODE (1.2), we can conclude that map $\mathcal{P}_{\mathcal{M}}$ is increasing on its domain. And the continuity of $\mathcal{P}_{\mathcal{M}}$ can be confirmed by using the theorem of continuity of the solution of an ODE with respect to its initial value. For a one-dimensional monotonically increasing discrete map, it is evident that there is no $k$-periodic point ($k\geq2$), which indicates that system (1.3) does not have order-$k$ ($k\geq2$) periodic solution.

Finally, we prove that the last part of $(i)$ holds. Theorem 3.3 shows that the DFPS is locally stable, that is, $(1-p)S_v = \mathcal{P}_{\mathcal{M}}((1-p)S_v)$ and $.\frac{d\mathcal{P}_{\mathcal{M}}(S_0^+)}{d S_0^+}|_{S_0^+ = (1-p)S_v} = \mu_2 < 1$. Combining the above with condition $S_m < (1-p)S_\sigma = \mathcal{P}_{\mathcal{M}}(S_m)$ for increasing map $\mathcal{P}_{\mathcal{M}}$, we can conclude that $\mathcal{P}_{\mathcal{M}}$ has at least one fixed point on interval $((1-p)S_v, S_m)$, as shown in Figure 3(b), which indicates that system (1.3) has at least one order-$1$ non-trivial periodic solution. □

Remark 3.2. When the unique endemic equilibrium $E^*$ is an unstable focus or unstable node or center, the domain and continuity of map $\mathcal{P}_{\mathcal{M}}$ are relatively complex. Thus for this case, the properties of map $\mathcal{P}_{\mathcal{M}}$ and the existence of periodic solutions of system (1.3) are not discussed in detail in this work, but it is worth further consideration in the future.

In case ($a_3$), we consider two special cases that the trajectory is tangent to $L_N$ and $L_M$, respectively. Firstly, there must be a point $\bar{N}\in L_N$ satisfying $\sigma_N(\bar{N}) = 0$ because $\sigma_N(N_1)\cdot\sigma_N(N_2) < 0$. The same exists for $\bar{M}\in L_M$ so that $\sigma_M(\bar{M}) = 0$.

Assume that $\bar{N}$ is the unique point in first quadrant satisfying $\sigma_N(S, L_N(S)) = 0$. If the trajectory from the initial point $\bar{N}$ can reach the impulsive curve $L_M$, we mark the intersection as $M_m$. Then the precise impulsive set is $\mathcal{M}_0 = \{(S, L_M(S))|S \in [S_v, S_{M_m}]\}$, where $S_{M_m}$ is the horizontal coordinate of $M_m$ (Figure 4(a)). Further, we mark the parts that are worth discussing as $V_1, V_2, V_3$ if the relative positions of the impulsive curve and the isoclinic lines are shown in the Figure 4(a). Based on the analysis of the ODE system, $\frac{dI}{dt} < 0$ holds for $V_1$ and $V_3$ while $\frac{dI}{dt} > 0$ for $V_2$. And the impulsive effect will always decrease the $S$ and $I$, so the periodic solution will not appear only in part $V_3$. Once the orbit crosses from $V_3$ to $V_2$, there may be an order-1 periodic solution.

Assume that $\bar{M}$ is the unique point in first quadrant satisfying $\sigma_M(S, L_M(S)) = 0$. Moreover, assuming a reverse trajectory starting from $\bar{M}$ intersects with $L_N$ and first intersects at point $N_{m}^+(S_m, L_N(S_m))$ (see Figure 4(b)). Thus, the precise impulsive set is $\mathcal{M}_0 = \{(S, L_M(S))|S \in [S_v, S_\sigma]\}$, where $S_\sigma$ is the horizontal coordinate of $\bar{M}$. Further, we mark the parts that are worth discussing as $V_1, V_2$ if the relative positions of the impulsive curve and the isoclinic lines are shown in the Figure 4(b). Based on the analysis of the ODE system, $\frac{dI}{dt} < 0$ holds for $V_1$ while $\frac{dI}{dt} > 0$ for $V_2$. And the impulsive effect will always decrease the $S$ and $I$, so if the orbit crosses from $V_1$ to $V_2$, there may be an order-1 periodic solution.

Theorem 3.6. Suppose $R_0 > 1$ and $S_v < \frac{1}{\beta}(\gamma+\frac{c}{b})$, the system (1.3) may have an order-1 periodic solution.

In case ($B$), $L_M$ and $L_2$ have a unique intersection point, denoted as $M_{L_2}$, where $S_{M_{L_2}}$ is the horizontal coordinate of $M_{L_2}$. It follows from the vector field of system (1.2) that $\sigma_M(S, L_M{S}) < 0$ for all $S\geq S_{M_{L_2}}$. Thus, if a trajectory starting from $\Omega$ experiences pulse effect, it must pulse at point $(S_{w1}, L_M(S_{w1}))$, where $S_{w1} < S_{M_{L_2}}$, and then pulse to point $((1-p)S_{w1}, (1-q)L_M(S_{w1}))$, which in the part $\frac{dI}{dt} < 0$. Thus, after a finite time, this trajectory will decrease and reach point $(S_{w2}, L_M(S_{w2}))$ on the pulse curve $L_M$. Here, $L_M(S_{w2}) < (1-q)L_M(S_{w1}) < L_M(S_{w1})$. From this, it can be concluded that the system (1.3) does not have order-1 non-trivial periodic solution, which also indicates that the system (1.3) does not have an order-$k$ ($k\geq1$) periodic solution except for the DFPS. The specific conclusion is as follows.

Theorem 3.7. Suppose $R_0 < 1, a_1 < 0, R_0 = R^*_0$, the system (1.3) will not have an order-$k$ periodic solution ($k\geq1$) except for the DFPS.

3.2.4.   The system has two endemic equilibria

Same as case ($B$), it's easy to know that the non-trivial periodic solutions are only possible when $S_1^* < \frac{AT}{\alpha_1} < S_2^*$, where $S_1^*$ and $S_2^*$ are the horizontal coordinates of $E_1^*$ and $E_2^*$, respectively. So we just focus on this situation (see Figure 5).

Similar to the above analysis, there must be an $\bar{M}$ satisfying $\sigma_M(\bar{M}) = 0$. So the precise impulsive set is $\mathcal{M}_0 = \{(S, L_M(S))|S \in [S_v, S_\sigma]\}$, where $S_\sigma$ is the horizontal coordinate of $\bar{M}$. What's more, when the trajectory crosses from the part satisfying $\frac{dI}{dt} < 0$ to the part $\frac{dI}{dt} > 0$, there may be an order-1 periodic solution.

Theorem 3.8. Suppose $R_0 < 1$, $a_1 < 0$, $R_0 > R^*_0$ and $S_1^* < \frac{AT}{\alpha_1} < S_2^*$, where $S_1^*$ and $S_2^*$ are the horizontal coordinates of $E_1^*$ and $E_2^*$, respectively. The system (1.3) may have an order-1 periodic solution.

Corollary 3.2. Replace the inequalities $R_0 > 1$ and $\frac{1}{\beta}(\gamma+\frac{c}{b}) < S_v < \frac{\Lambda}{\mu}$ in Theorem 3.5 by $R_0 < 1$, $a_1 < 0$, $R_0 > R^*_0$ and $S_1^* < \frac{AT}{\alpha_1} < S_2^*$. Then the conclusion of Theorem 3.5 still holds.

4.   Bifurcations when $q=0$

Based on Remark 3.1, in order to discuss the bifurcation near the DFPS, we assume that $R_0 > 1$, $(1-p)S_v < \frac{1}{\beta}(r+\frac{c}{b}) < S_v < \frac{\Lambda}{\mu}$ and $q = 0$ always hold true in this section.

On the phase space $\{(S, I)|0\leq I < L_1(S), 0 < S < \frac{\Lambda}{\mu}\}$, considering the following scalar differential equation

where $I_n^+\in[0, \epsilon)$, $S_n^+ = L_N^{-1}(I_n^+)$ and $\epsilon$ is small enough. We can solve $I$ with respect to $S$ as the following:

Thus, when $q = 0$ and the definition of point $(S_{n+1}, I_{n+1})\in L_M$ is shown in Subsection 3.2.1, the Poincaré map can be also represented as

Here, $I_n^+\in[0, \epsilon)$ is the variable and $\alpha\in\Theta$ is the parameter of map $\mathcal{P}(I_n^+, \alpha)$. Thus, $\mathcal{P}$ is defined as a one-parameter-family of maps from $[0, \epsilon)\times \Theta$ to $\mathbb{R}$.

Referring to Lemma 2.2, for the purpose of bifurcation analysis about map $\mathcal{P}(I_n^+, \alpha)$, we first calculate that

Let

Utilizing the variable formula, we have

Further, we denote

where $(S_n^+, I_n^+)\in L_N$ and $(S_{n+1}, I_{n+1})\in L_M$. Then we calculate its first-order derivative and second-order derivative with respect to $I_n^+$ as follows:

where

and

Because $I_{n+1} = L_M(S_{n+1})$ and $I_{n}^+ = L_M(S_{n}^+)$, we have

and

When $I_n^+ = 0$, then $S_n^+ = (1-p)S_v$, $I_{n+1} = 0$, $S_{n+1} = S_v = \frac{AT-\alpha_2\Lambda}{\alpha_1-\alpha_2\Lambda}$, $h(S_n^+, I_n^+) = h((1-p)S_v, 0) = 0$, $h(S_{n+1}, I_{n+1}) = h(S_v, 0) = 0$, and thus

Denote

and

Then

$f(s) = 0$ has a unique root $s = \frac{1}{\beta}(\gamma+\frac{c}{b})$ and $f^{'}(s) = (\Lambda\beta-\mu(\gamma+\frac{c}{b}))/(\Lambda-\mu s)^2 > 0$ when $R_0 > 1$.

Therefore, when $(1-p)S_v < \frac{1}{\beta}(\gamma+\frac{c}{b}) < S_v < \frac{\Lambda}{\mu}$, there are $f\big((1-p)S_v\big) < 0 < f(S_v)$ and

for all $S\in((1-p)S_v, S_v)$. Moreover, if $\mu_2 = 1$ then $0 < \exp(\int_{(1-p)S_v}^{S}f(s)ds) < 1$ and $\theta > 0$. Thus when $R_0 > 1$, $(1-p)S_v < \frac{1}{\beta}(\gamma+\frac{c}{b}) < S_v < \frac{\Lambda}{\mu}$ and $q = 0$, there is

4.1.   Bifurcations with respect to $p$

Now, under the assumptions of $q = 0$ and $\frac{1}{\beta}(\gamma+\frac{c}{b}) < S_v < \frac{\Lambda}{\mu}$, we consider the bifurcations near the DFPS $(\tilde{S}(t), 0)$ with respect to parameter $p$.

Taking $\mu_2$ as a function of $p$ with $p\in[0, 1]$ and taking the derivative of $\mu_2$ with respect to $p$ yields

Thus $p = \tilde{p}$ with $\tilde{p} = 1-\frac{1}{\beta S_v}(\gamma+\frac{c}{b})\in(0, 1)$ is the unique root of $\frac{d \mu_2(p)}{d p} = 0$ and $\frac{d \mu_2(p)}{d p} > 0$ for $p\in(0, \tilde{p})$, $\frac{d \mu_2(p)}{d p} < 0$ for $p\in(\tilde{p}, 1)$. Furthermore, we have

Therefore,

● if $\mu_2\big|_{p = 1} \geq 1$, then there is no $p^*\in (0, 1)$ satisfying $\mu_2\big|_{p = p^*} = 1$.

● if $\mu_2\big|_{p = 1} < 1$, then there is a unique $p^*\in (\tilde{p}, 1)$ satisfying $\mu_2\big|_{p = p^*} = 1$ and $\frac{d \mu_2}{d p}\big|_{p = p*} < 0$.

Theorem 4.1. When $q = 0$, $\frac{1}{\beta}(\gamma+\frac{c}{b}) < S_v < \frac{\Lambda}{\mu}$ and $\mu_2\big|_{p = 1} < 1$, map $\mathcal{P}$ undergoes the transcritical bifurcation at $p = p^*$, where $p^*$ is the unique root of $\mu_2(p) = 1$. Moreover, an unstable non-trivial fixed point (non-trivial order-1 periodic solution of system (1.3)) appears if $p\in(p^*, p^*+\epsilon)$ with $\epsilon > 0$ is small enough.

Proof. According to Eqs (4.14) and (4.18), we already have

What's more, it's easy to know that $\mathcal{P}(0, p) = 0$ for all $p\in[0, 1].$ Thus, to prove the Theorem 4.1 by using Lemma 2.2, we just need to verify the sign of $\frac{\partial^2 \mathcal{P}}{\partial I_n^+ \partial p}(0, p^*)$, where $\frac{\partial^2 \mathcal{P}}{\partial I_n^+ \partial p}(0, p^*) = \frac{\partial^2 I_{n+1}}{\partial I_n^+ \partial p}\big|_{I_n^+ = 0}^{p = p^*}$.

From Eq (4.7) it follows that

where $W(I_n^+ = 0) = 1$ and

We have $\frac{\partial I_n^+}{\partial p} = 0$ because $I_n^+$ and $p$ are independent. Thus, we have the following equation

By solving Eq (4.22), we have

In order to calculate $\frac{\partial I_{n+1}}{\partial S_n^+}$, we first calculate $\frac{\partial I(S; S_n^+, I_n^+)}{\partial S_n^+}$. That is

Using the variational formula we have

Thus

Then we can verify the symbol of the following equations

and

Therefore, we can conclude that $(1-p^*)S_v < \frac{1}{\beta}(\gamma+\frac{c}{b}) < S_v < \frac{\Lambda}{\mu}$, $f((1-p^*)S_v) < 0 < f(S_v)$ and

Combining Eq (4.18) and Lemma 2.2, we complete the proof. □

4.2.   Bifurcations with respect to AT

When $R_0 > 1$ and $q = 0$, we consider the bifurcations near the DFPS $(\tilde{S}(t), 0)$ with respect to parameter $AT$, where $AT\in(\alpha_2\Lambda, \alpha_1\frac{\Lambda}{\mu})$ since $0 < S_v = \frac{AT-\alpha_2\Lambda}{\alpha_1-\alpha_2\mu} < \frac{\Lambda}{\mu}$.

Taking $\mu_2$ as a function of $AT$, we have

where

The sign of $\frac{d \mu_2(AT)}{d AT}$ is determined by $g(S_v)$. Let $g(S_v) = 0$, we have

with

Let $D(p) = \frac{(p-2)^2}{1-p}$, then we have $D(0) = 4$ and $D^{'}(p) > 0$ for all $p\in(0, 1)$.Thus $D(p) > 4$ holds for all $p\in(0, 1)$. This means $\Delta > 0$ when $R_0 > 1$. Hence $g(S_v) = 0$ has two roots, $S_{v1}$ and $S_{v2}$, which satisfy $0 < S_{v1} < \frac{\Lambda}{\mu} < S_{v2}$. Thus $g(S_v)$ is decreasing on $(0, S_{v1})$ and increasing on $(S_{v1}, \frac{\Lambda}{\mu})$, which indicates that $\mu_2(AT)$ is decreasing on $(\alpha_2\Lambda, (\alpha_1-\alpha_2\mu)S_{v1}+\alpha_2\Lambda)$ and increasing on $((\alpha_1-\alpha_2\mu)S_{v1}+\alpha_2\Lambda, \alpha_1\frac{\Lambda}{\mu})$.

Furthermore, there are

Therefore, there is a unique $S_v^*\in(S_{v_1}, \frac{\Lambda}{\mu})$ such that $\mu_2\big|_{S_v = S_v^*} = 1$. That is to say, there is a unique $AT^* = (\alpha_1-\alpha_2\mu)S_v^*+\alpha_2\Lambda\in((\alpha_1-\alpha_2\mu)S_{v1}+\alpha_2\Lambda, \alpha_1\frac{\Lambda}{\mu})\subset(\alpha_2\Lambda, \alpha_1\frac{\Lambda}{\mu})$ satisfying $\mu_2\big|_{AT = AT^*} = 1$. Similarly, we have the following theorem:

Theorem 4.2. When $R_0 > 1$ and $q = 0$, map $\mathcal{P}$ undergoes the transcritical bifurcation at $AT = AT^*$, where $AT^*\in(\alpha_2\Lambda, \alpha_1\Lambda/\mu)$ is the unique root of $\mu_2(AT) = 1$. Moreover, an unstable non-trivial fixed point (non-trivial order-1 periodic solution of system (1.3)) appears if $AT\in(AT^*-\epsilon, AT^*)$ with $\epsilon > 0$ small enough.

Proof. Same as the proof of Theorem 4.1, we already have

And it's easy to know that $\mathcal{P}(0, AT) = 0$ for all $AT\in(\alpha_2\Lambda, \alpha_1\Lambda/\mu)$. Then we just need to verify the sign of $\frac{\partial^2 \mathcal{P}}{\partial I_n^+ \partial AT}(0, AT^*)$, where $\frac{\partial^2 \mathcal{P}}{\partial I_n^+ \partial AT}(0, AT^*) = \frac{\partial^2 I_{n+1}}{\partial I_n^+ \partial AT}\big|_{I_n^+ = 0}^{AT = AT^*}$. Firstly, we can calculate that

where

Similarly, we have the following equation

Therefore,

Then we figure out the symbol of the following equations

and

According to equations (4.31) to (4.35), it can be concluded that

□

5.   Discussion and conclusions

In previous state-dependent feedback control SIR models ^[12,13], the timing of implementing control measures was only related to whether the number of susceptible individuals reached the threshold, ignoring the impact of growth rates on control thresholds. This will inevitably increase the risk of effective control of infectious diseases. In this work, we propose a SIR model with nonlinear state-dependent feedback control, in which the control measures, such as isolation and vaccination, are adopted when the convex combinations of the size of the susceptible population and their growth rates reach the action threshold. The control form adopted is more in line with the development laws of the population. And we add a non-linear term to the classical SIR model to describe the impact of limited medical resources or treatment capacity on infectious disease transmission. To analyze the dynamical behavior of the proposed model, we first analyze the relevant properties of its ODE system (2.4), including the existence and stability of the equilibria. And then the analytical methods are developed to address the existence of order-$k$ ($k\geq1$) non-trivial periodic solutions, the existence and stability of a DFPS and its bifurcations.

In Section 3, we first prove the existence and the asymptotical stability of DFPS in Section 3.1. Then we analyse the direction of the phase plane solution trajectories for different positional relationships between the impulsive curves and the isoclinic lines, and using the defined sigma function, we find the maximum set of impulsive effect in each case, thus giving the domain of definition of the Poincaré map. In Section 3.2.2, we discuss the precise impulsive set when the system has no endemic equilibrium and figure out the attractive domain of the DFPS, which is not mentioned in Section 3.1. In Sections 3.2.3 and 3.2.4, we analyse the existence of the non-trivial periodic solutions in some special cases. Moreover, we give the sufficient conditions for the existence of the order-1 non-trivial periodic solution and the non-existence of the order-$k$ ($k\geq1$) periodic solution when the endemic equilibrium $E^*$ is a stable focus or node. However, as shown in Remark 3.2, when endemic equilibrium $E^*$ is an unstable focus or node, the domain, continuity and convexity of map $\mathcal{P}_{\mathcal{M}}$ are relatively complex. Thus, discussing the existence of non-trivial periodic solutions of system (1.3) in this situation is challenging, which is worth further consideration in the future.

In Section 4, we calculate the transcritical bifurcation with respect to the parameters $p$ and $AT$ in the case $q = 0$. From a biological perspective, $q = 0$ means that when the threshold condition is reached, we only vaccinate susceptible individuals without isolating infected individuals. Mathematically, the reason for the calculation at $q = 0$ is that when calculating the relevant derivatives according to the Lemma 2.2, we can only get the inequality (4.17) at $q = 0$, which means that $q = 0$ is a sufficient condition. Then we conclude that the transcritical bifurcation around the DFPS exists with respect to the parameter $p$ or $AT$ (see Theorem 4.1, 4.2).

When the endemic equilibrium of the system (2.4) is unstable, the dynamic behavior of the pulse system (1.3) is complex and rich, and thus it is well worth following up. The analytical techniques developed here not only can be applied to analyze epidemic models with nonlinear state-dependent impulsive control, but also can be extended to other fields including integrated pest management and tumor control.

Use of AI tools declaration

The author declares she has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 12031010) and National Science Basic Research Plan in Shaanxi Province of China (No. 2021JQ-215).

Conflicts of interest

We declare that we have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.