- Research
- Open Access
Nonlinear state-dependent feedback control strategy in the SIR epidemic model with resource limitation
- Zhi Long He^{1}Email author,
- Ji Gang Li^{1},
- Lin Fei Nie^{2} and
- Zhen Zhao^{3}
https://doi.org/10.1186/s13662-017-1229-8
© The Author(s) 2017
- Received: 23 December 2016
- Accepted: 4 June 2017
- Published: 27 July 2017
Abstract
In present work, in order to avoid the spread of disease, the impulse control strategy is implemented to keep the density of infections at a low level. The SIR epidemic model with resource limitation including a nonlinear impulsive function and a state-dependent feedback control scheme is proposed and analyzed. Based on the qualitative properties of the corresponding continuous system, the existence and stability of positive order-k (\(k\in\mathbf{Z}^{+}\)) periodic solution are investigated. By using the Poincaré map and the geometric method, some sufficient conditions for the existence and stability of positive order-1 or order-2 periodic solution are obtained. Moreover, the sufficient conditions which guarantee the nonexistence of order-k (\(k\geq3\)) periodic solution are given. Some numerical simulations are carried out to illustrate the feasibility of our main results.
Keywords
- SIR model
- nonlinear state-dependent impulsive function
- periodic solution
- orbital stability
- chaotic solution
1 Introduction
Infectious diseases have had a tremendous influence on human life and have brought huge panic and disaster to mankind once out of control from antiquity to the present. Every year millions of human beings suffer from or die of various infectious diseases. For example, measles, dengue, tuberculosis, cholera, Ebola and avian influenza have had a tremendous influence on human health during the last few years. Therefore, epidemiological modeling of infectious disease transmission has an increasing influence on the theory and practice of disease management and control. It is well known that one of the most important concerns in the analysis of mathematical modeling of the spread of infectious diseases is the efficacy of vaccination programmes. In Central and South America [1, 2] and UK [3], vaccination strategies have a positive effect on the prevention of diseases such as measles, tetanus, diphtheria, pertussis, and tuberculosis. Long-term clinical data show that the vaccination strategies lead to infectious disease eradication if the proportion of the successfully vaccinated individuals is larger than a certain critical value, for example, approximately equal to 95% for measles. The effectiveness of vaccination has been widely studied and verified for the influenza vaccine [4], the human papilloma virus (HPV) vaccine [5], the chicken pox vaccine [6], and others. Generally, there are two types of vaccination strategies: continuous vaccination strategy (CVS) and pulse vaccination strategy (PVS). For certain kinds of infectious diseases, PVS is more affordable and easier to implement than CVS. Recently, PVS has gained prominent achievement as a result of its highly successful application in the control of poliomyelitis and measles throughout Central and South America. In viewing of this, epidemiological models with PVS have been set up and investigated in many literature works (see, e.g., [7–13] and the references therein). Particularly, Agur et al. [7] first proposed a mathematical model with PVS, which consists of periodical repetitions of impulsive vaccinations of all the age cohorts in a population, which has been confirmed as an important and effective strategy for the elimination of infectious diseases.
In a real world application, however, the eradication of a disease is sometimes difficult both practically and economically in a short time. In order to prevent and control the spread of an infectious disease, impulsive vaccination and pulse treatment are important and effective methods. So far, it is necessary to keep the density of infections at a low level to avoid the spread of the disease. In many practical problems, impulse control strategy often occurs at state-dependent time, and it is more reasonable to take a strategy of state-dependent feedback control to model the issues of real world phenomena. Recently, the state-dependent feedback control, which is modeled by the impulsive semi-dynamic systems, has become a hot topic and has been applied in other fields and science [14–18], and it suggests that the control tactics should only be applied once the states of model reach a prescribed given threshold.
Following this idea, many epidemic models with state-dependent feedback control (including PVS and pulse treatment strategy (PTS)) have been proposed and analyzed by a number of authors in recent years. The SIR model with state-dependent PVS and PTS has been studied by Tang et al. [14] and later by Nie et al. [19]. Further, Nie et al. [20] proposed the SIRS model with state-dependent PVS and PTS, and analyzed the existence and stability of the periodic solution using the Poincaré map and the method of qualitative analysis. In these epidemic models with state-dependent pulse control strategies, it is usually assumed that the pulse vaccination rate p of the susceptibles and the pulse treatment rate q of the infectives are constants, which implies that the medical resources such as drugs, vaccines, hospital beds are very sufficient for infectious disease. In reality, however, every community or country has an appropriate or limited capacity for treatment, especially for emerging infectious diseases, and so understanding resource limitation is critical to effective management.
To the best of our knowledge, no work has been done for the effects of resource limitation on the SIR model with state-dependent PTS. In order to investigate the effect of limited vaccine and treatment availability on the spread of infectious disease, a saturation phenomenon of limited medical resources is considered. That is, we will study the dynamic behavior of the SIR epidemic model with state-dependent nonlinear PTS. This paper is structured as follows. In Section 2, the SIR epidemic model with resource limitation including state-dependent feedback control strategies and nonlinear impulsive function is constructed, some basic definitions, preliminaries and lemmas are given. In Section 3, we discuss the SIR model with the nonlinear impulsive vaccination as state-dependent feedback control, the existence and stability of positive periodic solution of this model. Finally, some numerical simulations are given to illustrate our results and suitability of state feedback control.
2 Model formulation and preliminaries
We assume, throughout this paper, that \(\epsilon=0\). That is to say, the disease-related mortality is so very small that can be ignored. Motivated by the previous works [14, 15, 19, 20, 22], we propose a state-dependent nonlinear pulse vaccination for the susceptible at control threshold value. That is, when the number of the infected individuals reaches the higher hazardous threshold value RL at time \(t_{i}(\mathit{RL})\) at the ith time, vaccination is taken and the number of susceptible and recovered individuals turns very suddenly to a great degree to \((1-p(t))S(t_{i}(\mathit{RL}))\) and \(R(t_{i}(\mathit{RL}))+p(t)S(t_{i}(\mathit{RL}))\), respectively, where \(p(t)\in(0,1)\) is the vaccination rate.
Parameter definitions, values and source of model ( 2.2 )
Parameter | Definition | Value | Source |
---|---|---|---|
Λ | Recruitment rate of the population \((year^{-1})\) | 16 | [21] |
β | Probability of transmission per contact \((year^{-1})\) | 0.002∼0.008 | [21] |
k | The parameter measuring the psychological or inhibitory effect | 0.01 | [21] |
d | Nature death rate of the population \((year^{-1})\) | 0.1 | [21] |
γ | Natural recovery rate of the infective individuals \((year^{-1})\) | 0.01 | [21] |
α | The maximal medical resources supplied per unite time | 6 | [21] |
ω | The half-saturation constant | Assumption | - |
RL, \(\mathit{RL}'\) | The number of infectives such that control actions must be taken in order to avoid economic and social damages | Assumption | - |
\(p_{\max}\) | The maximal vaccination proportion | [0,1) | |
θ | The half-saturation constant | Assumption | - |
It is then seen that the initial value problems for models (2.2) are biologically well posed in the sense that the trajectories of model (2.2) with the initial condition \((S(t_{0}),I(t_{0}), R(t_{0}))\in {\mathbb {R}}_{3}^{+}\) are positivity preserving.
Lemma 2.1
Suppose that \((S(t),I(t),R(t))\) is a solution of model (2.2) with the initial condition \((S(t_{0}),I(t_{0}),R(t_{0}))\in {\mathbb {R}}_{+}^{3}\). Then \((S(t),I(t),R(t))\in {\mathbb {R}}_{+}^{3}\) for all \(t\geq0\).
Proof
- (a)
The solution intersects with \(I=\mathit{RL}\) infinitely many times, at time instances \(t_{k}\), \(k=1,2,\ldots\) , and \(t_{k}\rightarrow\infty\). In this case, if the conclusion of Lemma 2.1 is false, we then obtain that there exists a positive integer n and \(t^{*}\in (t_{n-1},t_{n})\) such that \(\min\{S(t^{*}),I(t^{*}),R(t^{*})\}=0\), and \(S(t)>0\), \(I(t)>0\), \(R(t)>0\) for all \(t\in[t_{0}, t^{*})\).
The first possibility is that \(S(t^{*})=0\), \(I(t^{*})>0\), \(R(t^{*})>0\). For this case, it follows from the first and fourth equations of model (2.2) thatwhich contradicts the fact \(S(t^{*})=0\).$$S \bigl(t^{*} \bigr)>\prod_{i=1}^{n-1} \biggl(1- \frac{p_{\max}S(t_{i})}{S(t_{i})+\theta } \biggr)S(t_{0})\exp \biggl(- \int_{t_{0}}^{t^{*}} \biggl(\frac{\beta I(\tau )}{1+kI(\tau)+d}\,\mathrm {d}\tau \biggr) \biggr)>0, $$The second possibility is that \(S(t^{*})>0\), \(I(t^{*})=0\), \(R(t^{*})>0\). For this case, it follows from the second and fifth equations of model (2.2) thatwhich contradicts the fact \(I(t^{*})=0\), where \(I(t_{1})=\cdots =I(t_{n-1})=\mathit{RL}\), \(i=0,1,\ldots,n-1\).$$I \bigl(t^{*} \bigr)=I(t_{i})\exp \biggl(- \int_{t_{0}}^{t^{*}} \biggl(d+\gamma+\varepsilon+ \frac {\alpha}{\omega+I(\tau)} \biggr)\,\mathrm {d}\tau \biggr)>0 , $$The third possibility is that \(S(t^{*})>0\), \(I(t^{*})>0\), \(R(t^{*})=0\). For this case, it follows from the third and sixth equations of model (2.2) thatwhich contradicts \(R(t^{*})=0\).$$R \bigl(t^{*} \bigr)>R(t_{0})\exp \bigl(-d \bigl(t^{*}-t_{0} \bigr) \bigr)+\sum_{i=1}^{n-1} \biggl( \frac{p_{\max }S^{2}(t_{i})}{S(t_{i})+\theta}\exp \bigl(-d \bigl(t^{*}-t_{i} \bigr) \bigr) \biggr)>0, $$ - (b)
The solution intersects with \(I=\mathit{RL}\) finitely many times. In this case, since the endemic equilibrium \((S_{*},I_{*},R_{*})\) is globally asymptotically stable, so \(S(t)>0\), \(I(t)>0\), and \(R(t)>0\) for all \(t\geq0\).
Let \((X,\Pi, {\mathbb {R}}_{+})\) or \((X,\Pi)\) be a semi-dynamic model, where \(X=R_{+}^{2}\) is a metric space, and \({\mathbb {R}}_{+}\) is the set of all nonnegative reals. For any \(z\in X\), the function \(\Pi_{z}:{\mathbb {R}}_{+}\rightarrow X\) defined by \(\Pi_{z}(t)=\Pi(z,t)\) is clearly continuous such that \(\Pi(z,0)=z\) for all \(z\in X\), and \(\Pi(\Pi(z,t),s)=\Pi(z,t+s)\) for all \(z\in X\) and \(t,s\in {\mathbb {R}}_{+}\). The set \(C^{+}(z)=\{\Pi(x,t)|t\in {\mathbb {R}}\}\) is called the positive orbit of Z. For any set \(\mathcal{M}\subset X\), let \(\mathcal {M}^{+}(z)=C^{+}(z)\cap\mathcal{M}-{z}\), where \(G(z,t)=\{w\in X|\Pi(w,t)=z\} \) is the attainable set of z at \(t\in {\mathbb {R}}_{+}\). Based on the above notations, we need the following definitions and lemma.
Definition 2.1
An impulsive semi-dynamic model \((X,\Pi; \mathcal{M}, \mathcal{I})\) consists of a continuous semi-dynamic model \((X,\Pi)\) together with a nonempty closed subset \(\mathcal{M}\) ( or impulsive set) of \({\mathbb {R}}^{2}\) and a continuous function \(\mathcal{I}: \mathcal{M}\rightarrow X\) such that the following property holds: No point \(z\in X\) is a limit point of \(\mathcal{M}(z)\); \(\{t|G(z,t)\cap\mathcal{M}\neq\emptyset\}\) is a closed subset of \({\mathbb {R}}\).
Throughout the paper, we denote the points of discontinuity of \(\Pi_{z}\) by \({z_{n}^{+}}\) and define a function \(\Phi:X\rightarrow {\mathbb {R}}_{+}\cup{\infty}\) for any \(z\in X\). If \(\mathcal{M}^{+}(z)=\emptyset\), we set \(\Phi (z)=\infty\); otherwise, \(\mathcal{M}^{+}(z)\neq\emptyset\), and we set \(\Phi(z)=s\), where \(\Pi(z,t)\notin\mathcal{M}\) for \(0< t< s\) but \(\Pi (z,s)\in\mathcal{M}\).
Definition 2.2
A trajectory \(\Pi_{z}\) in \((X,\Pi,\mathcal{M},\mathcal{I})\) is said to be periodic of period \(T_{k}\) and order k if there exist nonnegative integers \(m\geq0\) and \(k\geq1\) such that k is the smallest integer for which \(z_{m}^{+}=z_{m+k}^{+}\) and \(T_{k}=\sum_{i=m}^{m+k-1}\Phi(z_{i})= \sum_{i=m}^{m+k-1}s_{i}\).
For more details on the concepts and properties of impulsive semi-dynamic model (2.4), see [24–26, 28]. For simplicity, we denote a periodic trajectory of period \(T_{k}\) and order-k by an order-k periodic solution. The local stability of an order-k periodic solution can be determined by using the following analogue of Poincaré criterion [25].
Lemma 2.2
Analogue of Poincaré criterion [25]
For model (2.2), we assume that the conditions \(\mathcal {R}_{0}>1\) and \(\alpha\leq\omega^{2}(\beta+dk)+\omega^{2}k(d+\gamma)+\omega (2d+\alpha k)\) hold. That is to say, model (2.2) without impulsive effects has a unique globally asymptotically stable endemic equilibrium \((S_{*},I_{*}, R_{*})\).
3 Main results
3.1 The case of \(\mathit{RL}\leq I_{*}\)
Theorem 3.1
For the case \(\mathit{RL}\leq I_{*}\), if \(p_{\max}\in(p^{*},1)\), then model (2.5) has a positive order-1 periodic solution.
Proof
By (3.2), (3.3) and (3.5), it follows that the Poincaré map (2.7) has a fixed point, that is, model (2.5) has a fixed point, thus, model (2.5) has a positive order-1 periodic solution. This completes the proof of this theorem. □
Next, we state and prove our result on the existence and stability of positive order-k (\(k=1,2\)) periodic solutions of model (2.5).
Theorem 3.2
For case \(\mathit{RL}\leq I_{*}\), if \(p_{\max}\in(p^{*},1)\), then model (2.5) has a positive order-1 or order-2 periodic solution, which is orbitally asymptotically stable. Further, model (2.5) has no order-k solution (\(k\geq3\), \(k\in\mathbb{N}\)).
Proof
- (i)
If \(S_{0}^{+}=S_{1}^{+}\), model (2.5) has a positive order-1 periodic solution;
- (ii)
If \(S_{0}^{+}\neq S_{1}^{+}\), without loss of generality, suppose that \(S_{0}^{+}< S_{1}^{+}\), it follows from (3.6) that \(S_{2}^{+}< S_{1}^{+}\). Furthermore, if \(S_{2}^{+}=S_{0}^{+}\), then model (2.5) has a positive order-2 periodic solution;
- (iii)If \(S_{0}^{+}\neq S_{1}^{+}\neq S_{2}^{+}\neq\cdots\neq S_{k}^{+}\) \((k\geq3)\) and \(S_{0}^{+}=S_{k}^{+}\), then model (2.5) has a positive order-k periodic solution. In fact, this is impossible. If \(S_{1}^{+}< S_{0}^{+}\), then from disjointness of any two trajectories and (2.7), we have \(S_{1}^{+}< S_{2}^{+}\) and then \(S_{1}^{+}< S_{2}^{+}< S_{0}^{+}\) or \(S_{1}^{+}< S_{0}^{+}< S_{2}^{+}\). If \(S_{0}^{+}< S_{1}^{+}\), we have \(S_{2}^{+}< S_{1}^{+}\) and then \(S_{0}^{+}< S_{2}^{+}< S_{1}^{+}\) or \(S_{2}^{+}< S_{0}^{+}< S_{1}^{+}\). Therefore, there are four relations among \(S_{0}^{+}\), \(S_{1}^{+}\) and \(S_{2}^{+}\) given byNow, we consider each case, respectively.$$\begin{gathered} \mathrm{(a)}\quad S_{1}^{+}< S_{2}^{+}< S_{0}^{+},\qquad\mathrm{(b)}\quad S_{1}^{+}< S_{0}^{+}< S_{2}^{+}, \\ \mathrm{(c)}\quad S_{0}^{+}< S_{2}^{+}< S_{1}^{+},\qquad\mathrm{(d)}\quad S_{2}^{+}< S_{0}^{+}< S_{1}^{+}. \end{gathered} $$
- (a)If \(S_{1}^{+}< S_{2}^{+}< S_{0}^{+}\), it follows that \(S_{1}^{+}< S_{3}^{+}< S_{2}^{+}\) by (3.6). Repeating the above process, we haveSimilar to case (a), we have$$ 0< S_{1}^{+}< S_{3}^{+}< \cdots< S_{2k+1}^{+}< \cdots< S_{2}^{+}< S_{0}^{+}< G(\mathit{RL}). $$(3.7)
- (b)If \(S_{1}^{+}< S_{0}^{+}< S_{2}^{+}\), then$$ 0< \cdots< S_{2k+1}^{+}< \cdots< S_{3}^{+}< S_{1}^{+}< S_{0}^{+}< S_{2}^{+}< \cdots < S_{2k}^{+}< \cdots< G(\mathit{RL}). $$(3.8)
- (c)If \(S_{0}^{+}< S_{2}^{+}< S_{1}^{+}\), then$$ 0< S_{0}^{+}< S_{2}^{+}< \cdots< S_{2k}^{+}< \cdots< S_{3}^{+}< S_{1}^{+}< G(\mathit{RL}). $$(3.9)
- (d)If \(S_{2}^{+}< S_{0}^{+}< S_{1}^{+}\), then$$ 0< \cdots< S_{2k}^{+}< \cdots< S_{2}^{+}< S_{0}^{+}< S_{1}^{+}< S_{3}^{+}< \cdots < S_{2k+1}^{+}< \cdots< G(\mathit{RL}). $$(3.10)
- (a)
If there exists an order-k periodic solution \((k\geq3, k\in\mathbb {N})\) in model (2.5), then \(S_{0}^{+}\neq S_{1}^{+}\neq S_{2}^{+}\neq \cdots\neq S_{k-1}^{+}\), and \(S_{0}^{+}=S_{k}^{+}\), which contradicts (3.7)-(3.10). So we conclude that model (2.5) has no period-k \((k\geq3,k\in\mathbb{N})\) solution with \(p_{\max}\in(p^{*},1)\).
Further, for \(k\in\mathbb{N}\), the sequences \(\{S_{2k}\}\) and \(\{ S_{2k+1}\}\) are convergent. Therefore, there exist \(S_{1}^{*}\) and \(S_{2}^{*}\) such that \(\lim_{k\rightarrow\infty}S_{2k}^{+}=S_{1}^{*}\), \(\lim_{k\rightarrow \infty}S_{2k+1}^{+}=S_{2}^{*}\). Using the Poincaré (2.7), we have \(S_{1}^{*}=\mathcal{F}_{1}(S_{2}^{*},p_{\max}, \theta)\), \(S_{2}^{*}=\mathcal {F}_{1}(S_{1}^{*},p_{\max}, \theta)\). Therefore, model (2.5) has an orbitally asymptotically stable positive periodic solution. Due to the vector field of (2.5), the positive periodic solution is order-1 in the cases of (a) and (c) and it is an order-2 periodic solution in the cases of (b) and (d). □
3.2 The case of \(I_{*}<\mathit{RL}\)
Theorem 3.3
For the case \(I_{*}<\mathit{RL}\), if \(p^{**}< p_{\max}<1\), then model (2.5) has a positive order-1 or order-2 periodic solution, which is orbitally asymptotically stable. Further, if \(0< p_{\max}<\widetilde {p}^{**}\), then model (2.5) has no positive order-k \((k\geq 1)\) periodic solution.
The proof of Theorem 3.2 is similar to the proof of Theorem 3.3, we therefore omit it here.
4 Numerical simulation
5 Discussion
In order to control the infected individuals, the SIR epidemic model with resource limitation including nonlinear impulsive function and state-dependent feedback control is studied both theoretically and numerically. From Theorems 3.1 and 3.2, we obtain sufficient conditions for the existence and stability of positive order-1 or order-2 periodic solution. It is assumed that \(\Lambda=16\), \(\alpha=10\), \(\beta=0.004\), \(k=0.01\), \(d=0.1\), \(\omega=35\), \(\gamma =0.01\), \(\theta=0.8\), and we choose the economic threshold \(\mathit{RL}=20< I^{*}\). It is easy to know that the conditions of Theorems 3.1 and 3.2 are satisfied, then the solution of model (4.1) initiating from \((S_{0},I_{0},R_{0})=(59.34,20,62.11)\) tends to the orbitally asymptotically stable positive order-1 periodic solution (see Figure 2). However, the numerical simulations show that the dynamic behavior of model (4.1) will become more complex if the conditions of Theorem 3.2 are unsatisfied (see Figure 3). Further, we choose the economic threshold \(\mathit{RL}=28>I^{*}\). From Theorem 3.3, the solution of model (4.1) initiating from \((S_{0},I_{0})=(65,12)\) tends to a stable positive order-1 periodic solution when \(p_{\max}>p^{**}\). On the other hand, the solution of model (4.1) initiating from \((S_{0},I_{0})=(65,12)\) tends to the endemic equilibrium \((S_{*},I_{*})\) when \(p_{\max}<\tilde{p}^{**}\)(see Figure 4). Our main results imply that we can choose proper control parameters to maintain the density of infections at a low level for preventing the spread of the disease. At the same time, some numerical simulations also show that model (2.5) has richer dynamic behaviors because of the effects of state-dependent impulse control strategies.
Declarations
Acknowledgements
The article is supported by the National Natural Science Foundation of China (11461067, 11402223 and 11271312).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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