Dynamics of deterministic and stochastic multi-group MSIRS epidemic models with varying total population size
© Wang et al.; licensee Springer. 2014
Received: 18 September 2014
Accepted: 15 October 2014
Published: 21 October 2014
In this paper, we extend the deterministic single-group MSIRS epidemic model to a multi-group model, and we also extend the deterministic multi-group framework to a stochastic one and formulate it as a stochastic differential equation. In the deterministic multi-group model, the basic reproduction number is a threshold that completely determines the persistence or extinction of the disease. By using Lyapunov function techniques, we show that if , then the disease will prevail, the infective condition persists and the endemic state is asymptotically stable in a feasible region. If , then the infective condition disappears and the disease dies out. For the stochastic version, we perform a detailed analysis on the asymptotic behavior of the stochastic model, which also depends on the value of , when , we determine the asymptotic stability of the endemic equilibrium by measuring the difference between the solution and the endemic equilibrium of the deterministic model in time-averaged data. Numerical methods are used to illustrate the dynamic behavior of the model and to solve the systems.
In this MSIRS model, the flow of disease transmission is as follows. A mother has been infected, and some IgG antibodies are transferred across the placenta, so that her new-born infant has temporary passive immunity to an infection. The class M contains infants with passive immunity. After the maternal antibodies disappear from the body, the infant moves to the susceptible class S. Infants who do not have any passive immunity, because their mothers were never infected, also enter the class S of susceptible individuals; next the susceptible enters the I class while they are infectious and then move to the recovered class R upon temporary recovery. The MSIRS model for infections that do not confer permanent immunity (i.e., an infection does not leave a long-lasting immunity, and thus individuals who have recovered return to being susceptible), the individual enters the susceptible class S (for a detailed introduction, see ).
Because the natural births and deaths are not balanced, that is, , the total population of the model is of an exponentially changing size. Thus, it is more difficult to analyze mathematically because the population size is an additional variable that is governed by a differential equation. In accordance with Guo et al. , we investigate the asymptotic behavior of system (1.2). When studying epidemic systems, we are interested in two problems: one is when the disease will die out, and the other is when the disease will prevail and persist in a population. For a deterministic system, we solve the problems by determining the stability of the two equilibria under different conditions. However, note that because of environmental noises, the deterministic approach has some limitations in the mathematical modeling of the transmission of an infectious disease; as a result, several authors have begun to consider the effect of white noise in epidemic models, which involves a parameter perturbation and perturbations around the positive endemic equilibrium of the epidemic models [15–18]. Beretta et al. proved the stability of the epidemic model using stochastic time delays influenced by the probability under certain conditions . Such stochastic perturbations were first proposed in [19, 20] and later were successfully used in many other papers for many different systems (see, e.g., [21–25]). Yuan et al.  and Yu et al.  both investigated epidemic models with fluctuations around the positive equilibrium, and they proved the locally stochastically asymptotic stability of the positive equilibrium. Ji et al. also considered a multi-group SIR model with stochastic perturbation and deduced the globally asymptotic stability of the disease-free equilibrium when , which means the disease will die out; the determined that when , the disease will prevail, which is measured through the difference between the solution and the endemic equilibrium of the deterministic model using time-averaged data . Imhof and Walcher  considered a stochastic chemostat model and they proved that the stochastic model led to extinction, even though the deterministic counterpart predicted persistence. In our previous work, we considered an SEIR epidemic model with constant immigration and random fluctuation around the endemic equilibrium, and we performed a detailed analysis on the asymptotic behavior of the stochastic model ; we also investigated a two-group epidemic model with distributed delays and random perturbation . Because of the similarity between the transmission of human infectious diseases and the transmission of malicious objects in a computer network, we used the epidemic models to describe the transmission of malicious objects in the cyber world . In the current paper, to examine the influence of white noise on system (1.2), we also consider a stochastic version of the MSIRS model by perturbing the deterministic system (1.2) using white noise and assuming that the perturbations are around the positive endemic equilibrium of the epidemic models. While some papers study the effect of stochastic perturbation on epidemic models, we are not aware of any literature addressing this issue in MSIRS epidemic models. This paper is an attempt to fill this gap.
This paper is organized as follows. We begin in Section 2 by providing the necessary background with respect to the deterministic multi-group MSIRS model and introduce some results of the graph theory used by Guo et al. in epidemic models. We establish the global dynamics of the disease-free state by using the basic reproduction number and present one of our main results (Theorem 3.2) in Section 3. We derive the asymptotic stability of a unique epidemic state in Section 4 (see Theorem 4.4). In Section 5, we derive the stochastic version from the deterministic model (1.2) and perform an analysis of the asymptotic behavior of the stochastic model by means of the method of Lyapunov functions and graph theory in Theorem 5.4. Numerical methods are used to simulate the dynamic behavior of the model. The effect of the rate of immunity loss is also analyzed in the deterministic models and the corresponding stochastic models in Section 6. Finally, we provide the conclusion of our article in Section 7.
2 Deterministic multi-group MSIRS models
To investigate the dynamical behavior, the first concern is whether the solution has a global existence. Moreover, for a population dynamics model, whether the value is nonnegative is also considered. Hence in this section we first show that the solution of system (1.2) is global and nonnegative.
Summary of notation
Passively immune infants in k th group
Susceptibles in the k th group
Infectives in the k th group
Recovered people with immunity in the k th group
Total population size in the k th group
Fraction at which new-borns of group k have the passive immunity
Rate of disease transmission between susceptible individuals in the k th group and infectious individuals in the j th group
, , ,
Mortality rates of susceptible, infectious and recovered individuals in the k th group, respectively
The rate of immunity loss in the k th group
Rate of the transfer out of the passively immune class in the k th group
Recovery rate of infectious individuals in the k th group
We assume that , , , and that the rest of the parameters be nonnegative for all k. It is clear that the population size changes in an exponentially increasing manner. In particular, if there is no transmission of the disease between compartments and .
Then it is easy to verify that the trivial solution of system (2.2) is given by , where , . It is clear that . In epidemiology it is called the disease-free equilibrium, at which the population remains in the absence of disease. Nontrivial solutions of system (2.2) with for some are called endemic equilibria, at which the disease persists. Our main task is to find some conditions that determine whether the disease dies out (i.e., the fraction goes to zero) or remains endemic (i.e., the fraction remains positive) for system (2.1).
Because no solution paths leave through any boundary, it can be verified that region Γ is positively invariant for system (2.1) and the model is well posed. Our results in this paper will be stated for system (2.1) in Γ.
It is clear that is the interior of Γ.
where and denote the spectral radius and the set of eigenvalues of a matrix, respectively. Since it can be verified that system (2.1) satisfies conditions (A1)-(A5) of Theorem 2 of , we have the following proposition.
Proposition 2.1 For system (2.1), the disease-free equilibrium is locally asymptotically stable if while it is unstable if .
3 Asymptotic stability of the disease-free equilibrium
Then the following lemma immediately follows.
Lemma 3.1 if and only if .
Theorem 3.2 Assume is irreducible. If , then the disease-free equilibrium of system (2.2) is globally asymptotically stable in Γ and there does not exist any endemic equilibrium .
on , where , and . Note that , where , , , and first we claim that there does not exist any endemic equilibrium in Γ. Suppose that . Then we have . Since nonnegative matrix is irreducible, it follows from the Perron-Frobenius theorem (see ) that . This implies that equation has only the trivial solution , where . Hence the claim is true.
Hence, if , then and thus is the only solution of (3.3). Summarizing the statements, we see that if and only if or , which implies that the compact invariant subset of the set where is only the singleton . Thus, from the LaSalle invariance principle , it follows that the disease-free equilibrium is globally asymptotically stable in Γ. □
4 Asymptotical stability of an endemic equilibrium in the deterministic MSIRS model
We discuss the persistence of the deterministic MSIRS model (2.1) in this section, our goal is to find some conditions that determine when the disease remains endemic (i.e., the fraction remains positive) for system (2.1). First, we introduce some specifics of the graph theory that are useful for the proofs of asymptotic stability of an endemic equilibrium in this section and the next section.
The matrix denotes the contact matrix. Associated to B, one can construct a directed graph whose vertex k represents the k th group, . A directed edge exists from vertex k to vertex j if and only if . Throughout the paper, we assume that B is irreducible, which is equivalent to being strongly connected. Biologically, this is the same as assuming that any two groups k and j have a direct or indirect route of transmission. More specifically, individuals in can infect ones in directly or indirectly.
and let denote the directed graph associated with matrix B (and ), and denote the cofactor of the entry of .
We have the following fundamental lemma .
Lemma 4.1 (Kirchhoff’s Matrix-Tree theorem)
The solution space of system (4.2) has dimension 1, with a basis .
- (2)For ,
where is the set of all directed spanning subtrees of that are rooted at vertex k, is the weight of a directed tree T, and denotes the set of directed arcs in a directed tree T.
Let be defined in (2.3). If , it follows from Proposition 2.1 that the disease-free equilibrium is unstable. From a uniform persistence result of [8, 9, 33, 36, 37], we can deduce that the instability of implies the uniform persistence of system (2.1) in , the following proposition for system (2.1) is known in the literature, and its proof is standard (see [8, 9, 33, 36, 37]).
Proposition 4.2 Assume is irreducible. Then the following statement applies. If , then is unstable and system (2.1) is uniformly persistent in Γ.
The uniform persistence of (2.1), together with uniform boundedness of solutions in , implies the existence of an endemic equilibrium of system (2.1) in . Summarizing the statements, we have the following corollary.
Corollary 4.3 Assume is irreducible. If , then system (2.1) has at least one endemic equilibrium.
Moreover, based on the result of the existence of endemic equilibrium, we can prove the following theorem, which is one of the main results of this paper.
Theorem 4.4 Assume that is irreducible. If , then system (2.1) has a unique endemic equilibrium which is asymptotically stable.
Hence is negative-definite in a sufficiently small neighborhood of for . However, it is easy to see is a positive-definite decrescent function, which implies the endemic equilibrium is asymptotically stable. □
5 Stochastic stability of the endemic equilibrium of a multi-group stochastic MSIRS model
If the assumptions of the existence-and-uniqueness theorem are satisfied, then, for any given initial value , (5.3) has a unique global solution denoted by . For the purpose of stability we assume in this section and for all . So (5.3) admits a solution , which is called the trivial solution or the equilibrium position.
- (1)The trivial solution of (5.3) is said to be stochastically stable or stable in probability if for every pair of and , there exists a such that
- (2)The trivial solution is said to be stochastically asymptotically stable if it is stochastically stable and for every , there exists a such that
- (3)The trivial solution is said to be stochastically asymptotically stable in the large if it is stochastically stable and for all
Before presenting the main theorem we put forward a lemma from .
Lemma 5.3 
If there exists a positive-definite decrescent function such that is negative-definite, then the trivial solution of (5.3) is stochastically asymptotically stable.
From the above lemma, we can obtain the stochastically asymptotically stability of equilibrium as follows.
the endemic equilibrium is stochastically asymptotically stable.
where , and is an infinitesimal of higher order of for . Hence is negative-definite in a sufficiently small neighborhood of for . According to Lemma 5.3, we therefore conclude that the zero solution of (5.2) is stochastically asymptotically stable. The proof is complete. □
6 Numerical simulation
By Lemma 4.1, we see that the endemic equilibrium of the deterministic model (2.1) is asymptotically stable. The computer simulations shown in Figure 2 clearly support this result.
where the time increment , and , , and are -distributed independent random variables, which can be generated numerically by pseudo-random number generators.
Values of , when fixing and changing
Values of , when fixing and changing
Analyzing the data in Tables 2 and 3, it shows that the higher the value of the rate of immunity loss is, the higher the value () of the endemic equilibrium is. Thus, it will be of great importance for health management to take some effective measures to diminish the rate the immunity loss. For example, when the antibody concentration of a recovered person decreased, he can be required to undergo vaccination to achieve the protective antibody levels.
This paper presented a mathematical study describing the dynamical behavior of an MSIRS epidemic model. Our purpose was based on analyzing this behavior using both a deterministic model and a stochastic model. This result differs from the previous results obtained in [4, 5] for single-group MSIRS models. We proved that the deterministic model has a unique endemic equilibrium, which is asymptotically stable if the reproduction number is greater than one; this means that the disease will persist at the endemic equilibrium level if it is initially present and the disease die out if . Furthermore, concerning the stochastic model, we obtained sufficient conditions for stochastic asymptotical stability of the endemic equilibrium by using a suitable Lyapunov function and other stochastic analysis techniques. The investigation of this stochastic model revealed that the stochastic stability of depends on the magnitude of the intensity of the noise as well as the parameters involved within the model system.
All authors of this paper were partially supported by Project of Science and Technology of Heilongjiang Province of China No. 12531187 and Natural Science Foundation of Heilongjiang Province of China No. A201410.
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