Dynamics of deterministic and stochastic multi-group MSIRS epidemic models with varying total population size

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1 Wang et al. Advances in Difference Equations 014, 014:70 R E S E A R C H Open Access Dynamics of deterministic and stochastic multi-group MSIRS epidemic models with varying total population size Zhigang Wang 1, Xiaoming Fan 1*, Fuquan Jiang and Qiang Li 3 * Correspondence: fanxm093@163.com 1 School of Mathematical Sciences, Harbin Normal University, Harbin, , P.R. China Full list of author information is available at the end of the article Abstract 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 framewor to a stochastic one and formulate it as a stochastic differential equation. In the deterministic multi-group model, the basic reproduction number R 0 is a threshold that completely determines the persistence or extinction of the disease. By using Lyapunov function techniques, we show that if R 0 > 1, then the disease will prevail, the infective condition persists and the endemic state is asymptotically stable in a feasible region. If R 0 1, 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 R 0,when R 0 > 1, 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. Keywords: multi-group MSIRS model; stochastic perturbation; graph theory; Brownian motion 1 Introduction Many models of the outbrea and spread of disease have been analyzed mathematically and applied to specific diseases, and these models have provided some useful and valid reference data for the characteristics of disease transmission. Based on the results of these theoretical analyses, one can predict the future course of an outbrea and evaluate the strategies used to control an epidemic. In 197, Kermac and McKendric investigated theclassicsirepidemicmodelandproveheexistenceofathreshold1]; thereafter, the number of studies on epidemiological modeling has rapidly increased, and a tremendous variety of models have now been formulated, mathematically analyzed, and applied to various infectious diseases. These models have involved many aspects of infectious diseases, such as stage of infection, age structure, vertical transmission, spatial spread, loss of vaccine and disease-acquired immunity, vaccination, and quarantine 9]. Compartments with the labels S, E, I, and R susceptible, exposed, infectious, and recovered are often used for the epidemiological classes. For some infections, including measles, infants are not born into the susceptible compartment but are immune to the disease for the first few 014 Wang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited.

2 Wang et al. Advances in Difference Equations 014, 014:70 Page of 5 months of life due to protection from maternal antibodies. Thus it is reasonable to add an M class maternally derived immunity at the beginning of the model. In 4], Hethcote discuss the MSEIR model in detail. Threshold theorems involving the basic reproduction number R 0,thecontactnumberσ, and the replacement number R are surveyed for the classic MSEIR epidemic 4]. Based on themseir model in 4], Lou and Ma alsoproposed a class of single-group MSIRS models 5], which are formulated as follows: dm = bn Sδ dm, ds = bs τr δm n d γ I, di = βsi N dr βsi N = γ I d τr, dn =b dn, =1,,...,n. ds, 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 newborn 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 4]. Considering the different contact patterns, a distinct number of sexual partners, or different geography, among other variables, it is more appropriate to divide individual hosts into groups in the modeling of an epidemic disease. Therefore, it is reasonable to propose multi-group models to describe the transmission dynamics of diseases in heterogeneous host populations. In fact, there are already many scholars who focus their studies on the various formsof multi-groupepidemic models 611]. Kuniya investigated the global stability of a multi-group SVIR epidemic model and considered the heterogeneity of the population and the effect of immunity induced by vaccination 10]. Muroya et al. also proved the global stability of an endemic equilibrium of a multi-group SIRS epidemic model using varying population sizes by extending the Lyapunov function techniques, which is one of the main mathematical challenges in analyzing multi-group models 11]. Recently, Li et al. more closely examined these multi-group models 114]. They first proposed a graphtheoretic approach to the method of global Lyapunov functions and used it to establish global stability of the interior equilibrium for more general models 1]. In the present paper, based on system 1.1, we divide the population of size Nt inton distinct groups, and then the n-group n MSIRS epidemic models are formulated by dm = b N S δ d MM, ds di = S τ R δ M n = n β j S ti j t N j d I γ I, dr = γ I td R τ R, dn =b d N, =1,,...,n. β j S ti j t N j d S S,

3 Wang et al. Advances in Difference Equations 014, 014:70 Page 3 of 5 Because the natural births and deaths are not balanced, that is, d, the total population of the model is of an exponentially changing size. Thus, it is more difficult to analyze mathematically because the population size N is an additional variable that is governed by a differential equation. In accordance with Guo et al. 14], we investigate the asymptotic behavior of system 1.. 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 1518]. Beretta et al. proved the stability of the epidemic model using stochastic time delays influenced by the probability under certain conditions 19]. Such stochastic perturbations were first proposed in 19, 0] and later were successfully used in many other papers for many different systems see, e.g.,15]. Yuan et al. 6] and Yu et al. 7] 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 R 0 1, which means the disease will die out; the determined that when R 0 >1,thedisease will prevail, which is measured through the difference between the solution and the endemic equilibrium of the deterministic model using time-averaged data 8]. Imhof and Walcher 9] 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 wor, 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 30]; we also investigated a two-group epidemic model with distributed delays and random perturbation 31]. Because of the similarity between the transmission of human infectious diseases and the transmission of malicious objects in a computer networ, we used the epidemic models to describe the transmission of malicious objects in the cyber world 3]. In the current paper, to examine the influence of white noise on system 1., we also consider a stochastic version of the MSIRS model by perturbing the deterministic system 1. 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 by providing the necessary bacground 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. in Section 3. We derive the asymptotic stability of a unique epidemic state in Section 4 see Theorem 4.4. In Section 5, wederive the stochastic version from the deterministic model 1. and perform an analysis of the asymptotic behavior of the stochastic model by means of the method of Lyapunov

4 Wang et al. Advances in Difference Equations 014, 014:70 Page 4 of 5 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. 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. is global and nonnegative. The parameters in the model 1. are summarized in Table 1. The disease transmission diagram is depicted in Figure 1. We assume that d M = ds = di = dr = d >0, >0, d = q > 0, and that the rest of the parameters be nonnegative for all. It is clear that the population size changes in an exponentially increasing manner. In particular, β j = 0 if there is no transmission of the disease between compartments S and I j. The fractions of the population in the classes of M, S, I, R are M = M, S N = S, I N = I, R N = R, respectively. Note that the number of infectives I N could go to infinity even though the fraction I goes to zero if the population size N grows faster than I.Toavoid Table 1 Summary of notation Notation M S I R N β j d M, ds, di, dr τ δ γ Explanation Passively immune infants in th group Susceptibles in the th group Infectives in the th group Recovered people with immunity in the th group Total population size in the th group Fraction at which new-borns of group have the passive immunity Rate of disease transmission between susceptible individuals in the th group and infectious individuals in the jth group Mortality rates of susceptible, infectious and recovered individuals in the th group, respectively The rate of immunity loss in the th group Rate of the transfer out of the passively immune class in the th group Recovery rate of infectious individuals in the th group Figure 1 The transfer diagram for MSIRS model 1..

5 Wang et al. Advances in Difference Equations 014, 014:70 Page 5 of 5 this ambiguity, we focus on the behavior of the fractions in the epidemiological classes. It is convenient to convert to differential equations for the fractions in the epidemiological classes with simplifications by using the differential equation for N ;wethencalculate dm = dm N q M, ds ds = q S, N di di = q I, N dr = dr N q R. Furthermore, eliminating the differential equation for S by using S =1M I R and using d = q > 0, the ordinary differential equations for the MSIRS model becomes dm =δ M d q I d q R, di = n β j1 M I R I j d γ q I, dr = γ I d q τ R, =1,,...,n..1 Equilibrium solutions M 1, I 1, R 1,...,M n, I n, R n R3n of system.1 obey the following equation: 0=δ M d q I d q R, 0= n β j1 M I R I j d γ q I,. 0=γ I d q τ R, =1,,...,n. Then it is easy to verify that the trivial solution of system. is given by P 0 = M1 0, I0 1, R0 1,...,M0 n, I0 n, R0 n R3n,whereM0 = I0 = R0 =0, {1,...,n}. Itisclearthat S1 0 = = S0 n = 1. In epidemiology it is called the disease-free equilibrium, at which the population remains in the absence of disease. Nontrivial solutions of system. with I >0forsome {1,...,n} are called endemic equilibria, at which the disease persists. Our main tas is to find some conditions that determine whether the disease dies out i.e., the fraction I goes to zero or remains endemic i.e., the fraction I remains positive for system.1. We can chec that the right sides of.1 aresmooth,sothatsolutionofsystem.1 with initial condition M 1 0, I 1 0, R 1 0,...,M n 0, I n 0, R n 0 R 3n has a unique solution and remain nonnegative. Therefore in what follows we consider system.1 inr 3n A suitable bounded region in the nonnegative cone of R 3n for system.1is Ɣ = { M 1, I 1, R 1,...,M n, I n, R n R 3n 0 M, I, R, M I R 1, 1 n }. Because no solution paths leave through any boundary, it can be verified that region Ɣ is positively invariant for system.1 and the model is well posed. Our resultsin this paper will be stated for system.1inɣ. Let Ɣ= { M 1, I 1, R 1,...,M n, I n, R n R 3n 0<M, I, R, M I R <1,1 n }.. It is clear that Ɣ is the interior of Ɣ.

6 Wang et al. Advances in Difference Equations 014, 014:70 Page 6 of 5 Set β 11 β 1n F = β n1 β nn and d 1 γ 1 q d γ q 0 V = diagd γ q = d n γ n q n. Then the next generation matrix is β 11 d 1 γ 1 q 1 FV 1 = β n1 d 1 γ 1 q 1 β 1n d n γ n q n β nn d n γ n q n, and hence the basic reproduction number R 0 is R 0 = ρ FV 1 = max { λ ; λ σ FV 1},.3 where ρ andσ denote the spectral radius and the set of eigenvalues of a matrix, respectively. Since it can be verified that system.1 satisfies conditions A1-A5 of Theorem of 33], we have the following proposition. Proposition.1 For system.1, thedisease-freeequilibriump 0 is locally asymptotically stable if R 0 <1while it is unstable if R 0 >1. 3 Asymptotic stability of the disease-free equilibrium In the study of population systems, extinction and persistence are two of the most important issues. We will discuss the extinction of the deterministic MSIRS model.1 inthis section, that is, we will find some conditions that determine when the disease dies out i.e., the fraction I goes to zero for system.1. First, following 10, 14], we prepare a matrix whose spectral radius has a similar threshold property to that of R 0.Let β 11 d 1 γ 1 q 1 V 1 F = β n1 d n γ n q n β 1n d 1 γ 1 q 1 β nn d n γ n q n. Then the following lemma immediately follows. Lemma 3.1 ρv 1 F 1 if and only if R 0 1. Using Lemma 3.1, we obtain the following theorem, which is one of the main results of this paper. This proof is similar to that of 10, 14].

7 Wang et al. Advances in Difference Equations 014, 014:70 Page 7 of 5 Theorem 3. Assume B =β j is irreducible. If R 0 1, then the disease-free equilibrium P 0 of system. is globally asymptotically stable in Ɣ and there does not exist any endemic equilibrium P. Proof From Lemma 3.1 we have ρv 1 F 1. Following 10, 14], we define the matrixvalued function MM, I, R= β 11 1M 1 I 1 R 1 β 1n 1M 1 I 1 R 1 d 1 γ 1 q 1 d 1 γ 1 q β n1 1M n I n R n d n γ n q n β n1 1M n I n R n d n γ n q n on R 3n,whereM =M 1,...,M n T, I =I 1,...,I n T and R =R 1,...,R n T.NotethatMM 0, I 0, R 0 =V 1 F,whereM 0 =M 0 1,...,M0 n T, I 0 =I 0 1,...,I0 n T, R 0 =R 0 1,...,R0 n T,and first we claim that there does not exist any endemic equilibrium P in Ɣ. Supposethat M, I, R M 0, I 0, R 0. Then we have 0 < MM, I, R<V 1 F. Since nonnegative matrix MM, I, RV 1 F is irreducible, it follows from the Perron-Frobenius theorem see 34] that ρmm, I, R < ρv 1 F 1. This implies that equation MM, I, RI = I has only the trivial solution I =0,whereI =I 1,...,I n T.Hencetheclaimistrue. Next we claim that the disease-free equilibrium P 0 is globally asymptotically stable in Ɣ. From the Perron-Frobenius theorem see Theorem.1.4 of 34] it follows that the nonnegative irreducible matrix V 1 F has a strictly positive left eigenvector l := l 1,...,l n 0 associated with the eigenvalue ρv 1 F. Let us define a Lyapunov function LI= i=1 l i I i d i γ i q i 3.1 on R n, whose derivative along the trajectories of system.1is L I= i=1 1 Mi I i R i n l β iji j i I i d i γ i q i = l MM, I, RE n I l V 1 F E n I = l ρ V 1 F 1 I, 3. where E n and denote the n n identity matrix and the inner product of vectors, respectively. If ρv 1 F 1, then l ρv 1 F 1I 0. Suppose that ρv 1 F <1.Then L I =0ifandonlyifI =0.SupposethatρV 1 F = 1. Then it follows from 3. that L I=0implies l MM, I, R=l I. 3.3 Hence, if M, I, R M 0, I 0, R 0, then l MM, I, R <l V 1 F =ρv 1 F l = l and thus I =0istheonlysolutionof3.3. Summarizing the statements, we see that L I=0if and only if I =0orM, I, R=M 0, I 0, R 0, which implies that the compact invariant subset of the set where L I = 0 is only the singleton {P 0 } Ɣ. Thus, from the LaSalle invariance

8 Wang et al. Advances in Difference Equations 014, 014:70 Page 8 of 5 principle 35], it follows that the disease-free equilibrium P 0 is globally asymptotically stable in Ɣ. 4 Asymptotical stability of an endemic equilibrium in the deterministic MSIRS model We discuss the persistence of the deterministic MSIRS model.1 in this section, ourgoal is to find some conditions that determine when the disease remains endemic i.e.,the fraction I remains positive for system.1.first,weintroducesomespecificsofthegraph theory that are useful for the proofs of asymptotic stability of an endemic equilibrium in this section and the next section. The matrix B =β j denotes the contact matrix. Associated to B, onecanconstructa directed graph L = GB whosevertex represents the th group, =1,...,n. A directed edge exists from vertex to vertex j if and only if β j > 0. Throughout the paper, we assume that B is irreducible, which is equivalent to GB being strongly connected. Biologically, this is the same as assuming that any two groups and j have a direct or indirect route of transmission. More specifically, individuals in I j caninfectonesins directly or indirectly. We define β j = β j S I j = β j 1M I R I j, β j >0,1, j n, 4.1 where M1, I 1, R 1,...,M n, I n, R n is the endemic equilibrium solution of system.1. Now consider the linear system where Bς =0, 4. i 1 β 1i β 1 β n1 β B = 1 i β i β n , β 1n β n i n β ni and let L = GB denote the directed graph associated with matrix B and β j n n, and C j denote the cofactor of the j, entryofb. We have the following fundamental lemma 1]. Lemma 4.1 Kirchhoff s Matrix-Tree theorem Assume β j n n is irreducible and n. Then the following results hold: 1 The solution space of system 4. has dimension 1, with a basis ς 1, ς,...,ς n =C 11, C,...,C nn. For 1 n, C = T T WT= T T r,m ET β rm >0, where T is the set of all directed spanning subtrees of L that are rooted at vertex, WT is the weight of a directed tree T, and ET denotes the set of directed arcs in a directed tree T.

9 Wang et al. Advances in Difference Equations 014, 014:70 Page 9 of 5 Let R 0 be defined in.3. If R 0 > 1, it follows from Proposition.1 that the disease-free equilibrium P 0 is unstable. From a uniform persistence result of 8, 9, 33, 36, 37], we can deduce that the instability of P 0 implies the uniform persistence of system.1 inɣ,the following proposition for system.1 is nown in the literature, and its proof is standard see 8, 9, 33, 36, 37]. Proposition 4. Assume B =β j is irreducible. Then the following statement applies. If R 0 >1,then P 0 is unstable and system.1 is uniformly persistent in Ɣ. The uniform persistence of.1, together with uniform boundedness of solutions in Ɣ, implies the existence of an endemic equilibrium of system.1 in Ɣ. Summarizing the statements, we have the following corollary. Corollary 4.3 Assume B =β j is irreducible. If R 0 >1,then system.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 B =β j is irreducible. If R 0 >1,then system.1 has a unique endemic equilibrium P which is asymptotically stable. Proof When n =1,system.1becomes dm/ =δm d qi d qr, di/ = β1 M I RI βm I RI d γ qi, dr/ = γ I d q τr. 4.3 For this single-group model, Lou and Ma proved that the endemic equilibrium P is globally asymptotically stable, which of course is asymptotically stable. Here, we will consider the case of n. First, using the change the variables of M = M M, Ĩ = I I, R = R R and system.1canbewrittenas d M =δ M d q Ĩ d q R, dĩ = n β j1 M I R Ĩ j n β j M Ĩ R I j Ĩ j d γ q Ĩ, d R = γ Ĩ d q τ R, =1,,...,n. 4.4 Note that B is the Laplacian matrix of the matrix β j n n.becauseβ j n n is irreducible, the matrices β j n n and B are also irreducible. The column sums of the Laplacian matrix B are zero. Therefore, it follows from Lemma 4.1 that the solution space of the linear system Bς = 0 has dimension 1, with a basis ς := ς 1, ς,...,ς n T =C 11, C,...,C nn T,

10 Wang et al. Advances in Difference Equations 014, 014:70 Page 10 of 5 where C denotes the cofactor of the th diagonal entry of B.Forsuchς =ς 1, ς,...,ς n, we define the Lyapunov function WM, I, R:= I β j Ij τ δ M d q τ δ d q γ W M, I, R:= τ δ R γ τ δ d q γ M R τ δ d q γ I ] I β j Ij τ δ M M d q τ δ d q γ τ δ R R γ τ δ d q γ M R M R ] τ δ d q γ = W 1 W, I Ĩ, 4.5 Ĩ Ĩ where W 1 = W = I β j Ij τ δ M M d q τ δ d q γ τ δ R R γ τ δ d q γ M R M R ], τ δ d q γ Ĩ Ĩ. We calculate I W 1 = I β j Ij τ δ δ M d q τ δ d q γ τ δ M Ĩ τ δ d q γ = τ δ M R τ δ d q γ τ δ R Ĩ τ δ d q γ d q γ R γ τ δ d q γ δ M τ R d q γ M R Ĩ τ δ d q γ τ δ d q γ τ δ d q γ ] τ δ M R τ δ d q γ β j Ij τ δ d q δ M d q τ δ d q γ M Ĩ R Ĩ I d q γ γ τ R ], γ τ δ d q γ

11 Wang et al. Advances in Difference Equations 014, 014:70 Page 11 of 5 W = I Ĩ β j 1M I R Ĩj β j M Ĩ R Ij Ĩ j d γ q Ĩ = = = I β j Ij Ĩ β j 1M I R Ĩ Ĩ j β j 1 M I R I j I β j Ij M R Ĩ I Ĩ β j Ĩ M Ĩ R Ĩ j β j 1M I R I Ij β j 1M I R I Ij Ĩ I β j Ij Ĩ β j Ij M R Ĩ I β j I v Ĩ j I Ij β j Ij M R Ĩ β j I Ĩ I Ĩ Ĩ j I Ij ] β j Ĩ M Ĩ R Ĩ j β j Ij Ĩ β j Ĩ M Ĩ R Ĩ j ]. It follows from the arithmetic-geometric mean inequality that I Ĩ j I Ij 1 Ĩ I 1 Ĩj Ij, therefore 1 Ĩ W β j I 1 ] Ĩj Ĩ Ij I ] β j Ij Ĩ β j Ij M R Ĩ β j Ĩ M Ĩ R Ĩ j I = 1 ] Ĩj Ĩ β j I j I β j Ij Ĩ β j Ij M R Ĩ I β j Ĩ M Ĩ R Ĩ j ]. 4.6

12 Wang et al. Advances in Difference Equations 014, 014:70 Page 1 of 5 Note that from 4.1, 4., and Lemma 4.1,wehave β j ς j = β i = β j. 4.7 It follows that Ĩj β j Ij = = ς j Ĩ I Ĩ β j I = β j = Ĩ I β j ς j β j Ĩ I. 4.8 Substituting this into 4.6yields W I β j Ij Ĩ β j Ij M R Ĩ β j Ĩ M Ĩ R Ĩ j ], 4.9 thus, W M, I, R=W 1 W = I β j Ij τ δ d q δ M d q τ δ d q γ d q γ γ τ R ] γ τ δ d q γ Ĩ β j Ĩ M Ĩ R Ĩ j W 0 I I β j Ĩ M Ĩ R Ĩ j, where W 0 := I β j Ij τ δ d q δ M d q τ δ d q γ d q γ γ τ R γ τ δ d q γ Ĩ ]. We denote Y = M, Ĩ, R andy =Y 1,...,Y n ; then 1/. Y = M tĩ R, Y = Y

13 Wang et al. Advances in Difference Equations 014, 014:70 Page 13 of 5 It is clear that n n I β jĩ M Ĩ R Ĩ j is an infinitesimal of higher order of Y for Y 0. Let ω = min {1,...,n} I { I β j Ij β j Ij τ δ d q δ d q τ δ d q γ, d q γ γ τ γ τ δ d q γ, I }. Thus W M, I, R ω Y o Y Hence W M, I, R is negative-definite in a sufficiently small neighborhood of Y = 0 for t 0. However, it is easy to see WM, I, R is a positive-definite decrescent function, which implies the endemic equilibrium P is asymptotically stable. 5 Stochastic stability of the endemic equilibrium of a multi-group stochastic MSIRS model In this section, we consider the stochastic version of the deterministic MSIRS model. Under the assumption that R 0 >1andB =β j n n is irreducible, we now from Section 4 that there exists a unique positive endemic equilibrium P in Ɣ. Furthermore, we assume stochastic perturbations on the M t, I t, R t are of white-noise type, which are directly proportional to deviations M t, I t, and R t fromthevaluesofm, I, R,respectively. Thus, system.1resultsin dm =δ M d q I d q R σ 1 M M db 1, di = n β j1 M I R I j d γ q I σ I I db dr = γ I d q τ R σ 3 R R db 3, =1,,...,n,, 5.1 where B 1 t, B t, and B 3 t are independent standard Brownian motions and σ i >0 represent the intensities of B i t i =1,,3, respectively. Obviously, the stochastic system 5.1 has the same equilibrium points as system.1. Next, let us now proceed to discuss asymptotic stability of system 5.1. In this paper, unless otherwise specified, let, F, {F t } t t0, P be a complete probability space with a filtration {F t } t t0 satisfying the usual conditions i.e. it is increasing and right continuous while F 0 contains all P-null sets. Let B i t be the Brownian motions defined on this probability space. If R 0 > 1, then the stochastic system 5.1 can be centered at its endemic equilibrium P =M1, I 1, R 1,...,M n, I n, R n, by the change of variables u = M M, v = I I, w = R R,

14 Wang et al. Advances in Difference Equations 014, 014:70 Page 14 of 5 we obtain the following system: du dv dw db =δ u d q v d q w σ 1 u 1, = n β j1 M I R v j n β ju v w Ij db v j d γ q v σ v = γ v d q τ w σ 3 w db 3, =1,,...,n., 5. It is clear that the stability of equilibrium of system 5.1 is equivalent to the stability of zero solution of system 5.. Considering the d-dimensional stochastic differential equation 38, 39] dxt=f xt, t g xt, t dbt, t t If the assumptions of the existence-and-uniqueness theorem are satisfied, then, for any given initial value xt 0 =x 0 R d,5.3hasauniqueglobalsolutiondenotedbyxt; t 0, x 0. For the purpose of stability we assume in this section f 0, t=0andg0, t = 0 for all t t 0. So 5.3 admits a solution xt 0, which is called the trivial solution or the equilibrium position. Let κ denote the family of all continuous nondecreasing functions μ : R R such that μ0 = 0 and μr >0ifr >0.For0<h and S h = {x R d, x < h}, denote C,1 S h t 0, ; R the family of all nonnegative functions V x, tons h t 0, which are continuously twice differentiable in x and once in t. Define the differential operator L associated with 5.3by L = t d i=1 f i x, t x i 1 d g T x, tgx, t ]. ij x i x j i, If L acts on a function V C,1 R d t 0, ; R, then LV x, t=v t x, tv t x, tf x, t 1 trace g T x, tv xx x, tgx, t ]. Definition The trivial solution of 5.3 is said to be stochastically stable or stable in probability if foreverypairofε 0, 1 and r >0,thereexistsaδ = δε, r, t 0 >0such that P { xt; t0, x 0 < r for all t t0 } 1ε whenever x 0 < δ. Otherwise, it is said to be stochastically unstable. The trivial solution is said to be stochastically asymptotically stable if it is stochastically stable and for every ε 0, 1,thereexistsaδ 0 = δ 0 ε, t 0 >0such that { } P lim xt; t 0, x 0 =0 1ε t whenever x 0 < δ 0.

15 Wang et al. Advances in Difference Equations 014, 014:70 Page 15 of 5 3 The trivial solution is said to be stochastically asymptotically stable in the large if it is stochastically stable and for all x 0 R d { } P lim xt; t 0, x 0 =0 =1. t Definition 5. A continuous nonnegative function V x, t is said to be decrescent if for some μ κ: V x, t μ xt for all x, t S h t 0,. Before presenting the main theorem we put forward a lemma from 39]. Lemma ] If there exists a positive-definite decrescent function V x, t C,1 S h t 0, ; R such that LV x, t 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. Theorem 5.4 Assume that B =β j is irreducible and R 0 >1.Then, if the following condition is satisfied: σ 1 <δ, σ < β j Ij, σ 3 < d q τ τ δ τ γ τ δ γ, 5.4 the endemic equilibrium P is stochastically asymptotically stable. Proof It is easy to see that we only need to prove the zero solution of 5.1 isstochastically asymptotically stable. Let x t =u t, v t, w t T R 3, = 1,...,n and xt = x 1 t,...,x n t T R 3n.WedefinetheLyapunovfunctionV xt as follows: V x= 1 a u v c w e u w, 5.5 where a >0, >0,c > 0 are real positive constants to be chosen later. Then it can be described as the quadratic form V x= 1 x T Qx, where a e 0 e Q = 0 0 e 0 c e is a symmetric positive-definite matrix. So it is obviously that V x is positive-definite decrescent. For the sae of simplicity, 5.5 may be divided into four functions: V x =

16 Wang et al. Advances in Difference Equations 014, 014:70 Page 16 of 5 V 1 xv xv 3 xv 4 x, where V 1 x= 1 V 3 x= 1 a u, V x= 1 c w, V 4x= 1 Using Itô s formula, we compute LV 1 = = v, e u w. a u δ u d q v d q w 1 a σ1 u a δ 1 σ 1 u a d q u v d q u w. 5.6 Similarly, from Itô s formula, we obtain LV = v β j 1M I R vj β j u v w I j v j d γ q v 1 σ v = β j Ij 1 σ v β j 1M I R v v j 1 M β I R I j j I v β j Ij u w v β j v u v w v j = β j 1M I R I I v j I β j 1M I R I Ij v = I v j Ij β j Ij 1 σ v β j Ij u w v β j v u v w v j β j I v v j I Ij β j I β j Ij u w v v I β j Ij 1 σ β j v u v w v j v

17 Wang et al. Advances in Difference Equations 014, 014:70 Page 17 of 5 = 1 1 β j I v I β j Ij u w v vj β j v u v w v j β j I vj I j I j β j I β j v u v w v j β j I v I β j Ij 1 σ v I v β j Ij u w v β j Ij 1 σ v. 5.7 Let = C I,so = I. It follows from Bς =0andβ j = β j 1 Mj Ij R j I that β j ς j = β j, which implies β j I vj Ij = β j I v I. Hence inequality 5.7becomes LV LV 3 = LV 4 = = = β j Ij u w v β j Ij 1 σ v, c w γ v d q τ w 1 β j v u v w v j c σ3 w c d q τ 1 σ 3 w c γ v w, e u w δ u d q γ v τ w 1 e δ u e τ w e τ δ u w e d q γ u w v 1 e σ 1 u σ 3 w. e σ 1 u σ 3 w

18 Wang et al. Advances in Difference Equations 014, 014:70 Page 18 of 5 Then we compute LV = LV 1 LV LV 3 LV 4 a e δ 1 σ 1 u β j Ij 1 σ = c d q τ e τ 1 ] c e σ3 w c γ v w a d q u v d q u w β j Ij u w v e τ δ u w e d q γ u w v β j v u v w v j a e δ 1 σ 1 u β j Ij 1 σ c d q τ e τ 1 ] c e σ3 w c γ ] β j Ij e d q γ v w a d q ] β j Ij e d q γ u v a d q e τ δ ] u w We can choose a, c, e such that v v β j v u v w v j. i.e., a d q e τ δ =0, a d q c γ a = β j Ij e d q γ =0, β j Ij e d q γ =0, n β ji j τ δ d q τ δ d q γ,

19 Wang et al. Advances in Difference Equations 014, 014:70 Page 19 of 5 then n e = β jij, τ δ d q γ n c = β jij τ δ γ τ δ d q γ, LV a e δ 1 σ 1 u β j Ij 1 σ c d q τ τ γ τ δ γ τ δ τ δ σ 3 w = V 0 β j v u v w v j β j v u v w v j, 5.8 v where V 0 = A u B v C w, A =a e δ 1 σ 1, B = β j Ij 1 σ, C = c d q τ τ γ τ δ γ τ δ τ δ σ 3, and the proofs above show that if the condition 5.4issatisfied,thenA, B, C are positive constants. Let λ = min {A, B, C }, {1,...,n} then λ >0.From5.8, one sees that LV λ x t o x t =λ xt o xt, 5.9 where x t = u tv tw t, xt = n x t 1/ and o xt isaninfinitesimal of higher order of xt for t.hencelv x, t is negative-definite in a sufficiently small neighborhood of x = 0 for t 0. According to Lemma 5.3, we therefore conclude that the zero solution of 5. is stochastically asymptotically stable. The proof is complete.

20 Wang et al. Advances in Difference Equations 014, 014:70 Page 0 of 5 6 Numerical simulation Numerical methods are used to solve the systems.1and5.1 and to depict the behavior of the passively immune, infectious, and recovered with respect to time. We numerically simulate the solution of systems.1and5.1whenn =. In this case, we have and M 0 = β 11 β 1 d1 I q 1γ 1 d1 I q 1γ 1 β 1 β d I q γ d I q γ ] β 11 K 1 β 1 K 1 := β 1 K β K R 0 = ρm 0 = β 11K 1 β K β 11 K 1 β K 4β 1 β 1 K 1 K. The system parameters are given by β 11 =0.5, β 1 =0.35, d 1 =0.0, q 1 =0.03, δ 1 =0.15, τ 1 =0.05, γ 1 =0.1, β 1 =0.45, β =0.65, d =0.03, q =0.05, δ =0.5, τ =0.10, γ =0.5. Hence, we obtain M 1 =0.1763,I 1 =0.405,R 1 =0.886,M =0.1717,I =0.47,and R =0.310.Itiseasytocomputethat R 0 =3.5406>1. Moreover, we choose M 1 0 = 0.35, I 1 0 = 0., R 1 0 = 0.1, M 0 = 0., I 0 = 0.35, R 0 = 0.05 as the initial values, and we eep all these initial values and parameter values unchanged in each example unless otherwise stated. In the absence of noise, we simulate the asymptotic stability of the endemic equilibrium of the deterministic system.1 in Figure. By Lemma 4.1, we see that the endemic equilibrium P of the deterministic model.1 is asymptotically stable. The computer simulations shown in Figure clearly support this result. Next, we show the numerical simulation of the stochastic system 5.1. Given the discretization of system 5.1fort =0, t, t,...,n t,and =1,. M,i1 = M,i δ M,i d q I,i d q R,i t σ 1 M,i M tε 1,i, I,i1 = I,i β 1 1 M,i I,i R,i I 1,i β 1 M,i I,i R,i I,i 6.1 d γ q I,i t σ I,i I tε,i, R,i1 = R,i γ I,i d q τ R,i t σ 3 R,i R tε 3,i, where the time increment t >0,andε 1,i, ε,i,andε 3,i are N0, 1-distributed independent random variables, which can be generated numerically by pseudo-random number generators.

21 Wang et al. Advances in Difference Equations 014, 014:70 Page 1 of 5 Figure Deterministic trajectories of the MSIRS model.1 for the initial conditions M 1 0 = 0.35, I 1 0 = 0., R 1 0 = 0.1, M 0 = 0., I 0 = 0.35, and R 0 = Figure 3 Stochastic trajectories of the MSIRS model 5.1forσ 11 =0.37,σ 1 =0.5,σ 31 =0., σ 1 =0.35,σ =0.5,σ 3 =0.3,and t =10 3. To determine the effect of the noise intensity, we consider three series of different values. Figure 3 corresponds to σ 11 =0.37,σ 1 =0.5,σ 31 =0.,σ 1 =0.35,σ =0.5,and σ 3 = 0.3, and it is easy to verify the noise intensity and system parameters obey condition 5.4. We can therefore conclude, by Theorem 5.4, that the endemic equilibrium P of stochastic model 5.1 is asymptotically stable. The computer simulations shown in Figure 3 agree well with the mathematical result. Figure 4 corresponds to σ 11 =5.5,σ 1 =7.5, σ 31 =5.5,σ 1 =5.3,σ =6.8,andσ 3 =5.7,anhecomparisonofFigures3 left and 4 left suggests that the fluctuations of at least one of the curves increase as the noise level increases. The same situation occurs for the comparison of Figures 3 right and 4 right. Note that condition 5.4 is just a sufficient condition. When this condition is not satisfied, the stochastic system 5.1 may or may not be stable. For example, the intensities of the Brownian motions σ 11 =5.5,σ 1 =7.5,σ 31 =5.5,σ 1 =5.3,σ =6.8,andσ 3 =5.7donot obey the condition 5.4, but we can see from Figure 4 that the endemic equilibrium of the stochastic system 5.1 is still asymptotically stable. If we choose σ 11 =43.5,σ 1 =46.5, σ 31 =51.3,σ 1 =65.3,σ =5.,andσ 3 = 43.3, then the solution of the stochastic sys-

22 Wang et al. Advances in Difference Equations 014, 014:70 Page of 5 Figure 4 Stochastic trajectories of MSIRS model 5.1forσ 11 =5.5,σ 1 =7.5,σ 31 =5.5,σ 1 =5.3, σ =6.8,andσ 3 =5.7. Figure 5 Stochastic trajectories of the MSIRS model 5.1forσ 11 = 43.5, σ 1 =46.5,σ 31 =51.3, σ 1 = 65.3, σ = 5., and σ 3 = tem 5.1 is not asymptotically stable but rather explodes to infinity in a finite time see Figure 5. To better understand the long time behavior of the deterministic and stochastic systems, we show the phase space portraits for the deterministic and MSIRS stochastic endemic model in Figures 6 and 7, respectively. For the stochastic case, Figure 7 corresponds to σ 11 = 0.37, σ 1 =0.5,σ 31 =0.,σ 1 =0.35,σ =0.5,andσ 3 = 0.3, in which the noise intensity and system parameters obey the condition 5.4. Therefore, by Theorem 5.4,theendemic equilibrium P of stochastic model 5.1 is asymptotically stable; we can see from Figure 7 that an oscillation appears under environmental driving forces, which actually affect the deterministic curves shown in Figure 6. These two trajectories in Figure 7 still maintains thesameoveralltrendasthoseoffigure6,anheyreachthesameequilibriumpointsas the deterministic version. The simulations agree with our results. The major difference between the MSIRS model and the MSIR model is that the MSIRS model does not confer permanent immunity to individuals in the model. Thus, let us an-

23 Wang et al. Advances in Difference Equations 014, 014:70 Page 3 of 5 Figure 6 Phase space portrait for the deterministic MSIRS endemic model. Figure 7 Phase space portrait for the MSIRS endemic model with stochastic perturbations. Table Values of P, when fixing τ = 0.1 and changing τ 1 M 1 I1 R 1 M I R τ 1 = τ 1 = τ 1 = τ 1 = Table 3 Values of P, when fixing τ 1 = 0.05 and changing τ M 1 I1 R 1 M I R τ = τ = τ = τ = alyze the impact of the rate τ of passive immune loss on the MSIRS model by assigning different values to it, as provided in Tables and 3 by calculating the equilibrium of system.1, which is, of course, an equilibrium of system 5.1. Analyzing the data in Tables and 3, it shows that the higher the value τ of the rate of immunity loss is, the higher the value I j j =1,...,n of the endemic equilibrium is. Thus, it will be of great importance for health management to tae 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.

24 Wang et al. Advances in Difference Equations 014, 014:70 Page 4 of 5 7 Conclusion 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 R 0 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 R 0 1. Furthermore, concerning the stochastic model, we obtained sufficient conditions for stochastic asymptotical stability of the endemic equilibrium P by using a suitable Lyapunov function and other stochastic analysis techniques. The investigation of this stochastic model revealed that the stochastic stability of P depends on the magnitude of the intensity of the noise as well as the parameters involved within the model system. Competing interests The authors declare that they have no competing interests. Authors contributions All authors completed the paper together and contributed equally to this wor. All authors read and approved the final manuscript. Author details 1 School of Mathematical Sciences, Harbin Normal University, Harbin, , P.R. China. Department of Foundation, Harbin Finance University, Harbin, 15003, P.R. China. 3 College of Sciences, Qiqihar University, Qiqihar, , P.R. China. Acnowledgements All authors of this paper were partially supported by Project of Science and Technology of Heilongjiang Province of China No and Natural Science Foundation of Heilongjiang Province of China No. A Received: 18 September 014 Accepted: 15 October 014 Published: 1 Oct 014 References 1. Kermac, WO, McKendric, AG: Contributions to the mathematical theory of epidemics, part 1. Proc. R. Soc. Lond. Ser. A 115, Mulone, G, Straughan, B, Wang, W: Stability of epidemic models with evolution. Stud. Appl. Math. 118, Wang, X, Wang, W, Zhang, G: Global analysis of predator-prey system with Haw and Dove tactics. Stud. Appl. Math. 14, Hethcote, HW: The mathematics of infectious diseases. SIAM Rev. 4, Lou, J, Ma, Z: Stability of some epidemic models with passive immune. Acta Math. Sin. 3, in Chinese 6. Lajmanovich, A, Yore, JA: A deterministic model for gonorrhea in a nonhomogeneous population. Math. Biosci. 8, Beretta, E, Capasso, V: Global stability results for a multigroup SIR epidemic model. In: Hallam, TG, Gross, LJ, Levin, SA eds. Mathematical Ecology. World Scientific, Teanec Hethcote, HW: An immunization model for a heterogeneous population. Theor. Popul. Biol. 14, Thieme, HR: Local stability in epidemic models for heterogeneous populations. In: Capasso, V, Grosso, E, Paveri-Fontana, SL eds. Mathematics in Biology and Medicine. Lecture Notes in Biomathematics, vol. 57, pp Springer, Berlin Kuniya, T: Global stability of a multi-group SVIR epidemic model. Nonlinear Anal., Real World Appl. 14, Muroya, Y, Enatsu, Y, Kuniya, T: Global stability for a multi-group SIRS epidemic model with varying population sizes. Nonlinear Anal., Real World Appl. 14, Guo, H, Li, MY, Shuai, Z: A graph-theoretic approach to the method of global Lyapunov functions. Proc. Am. Math. Soc. 136, Li, MY, Shuai, Z, Wang, C: Global stability of multi-group epidemic models with distributed delays. J. Math. Anal. Appl. 361, Guo, H, Li, MY, Shuai, Z: Global stability of the endemic equilibrium of multigroup SIR epidemic models. Can. Appl. Math. Q. 14, Dalal, N, Greenhalgh, D, Mao, X: A stochastic model of AIDS and condom use. J. Math. Anal. Appl. 35, Ji, C, Jiang, D, Shi, N: Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation. J. Math. Anal. Appl. 359, Ji, C, Jiang, D, Li, XY: Qualitative analysis of a stochastic ratio-dependent predator-prey system. J. Comput. Appl. Math. 35, Tornatore, E, Buccellato, SM, Vetro, P: Stability of a stochastic SIR system. Physica A 354,

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