Asymptotic Behavior and Stability of Stochastic SIR Model with Variable Diffusion Rates

In this paper, we propose random fluctuation on contact and recovery rates in deterministic SIR model with disease deaths in nonparametric manner and derive a new stochastic SIR model with distributed time delay and general diffusion coefficients. By analysis of the introduced model, we obtain the sufficient conditions for the regularity, existence and uniqueness of a global solution by means of Lyapunov function. Moreover, we also investigate the stochastic asymptotic stability of disease free equilibria and endemic equilibria of this model. Finally, we illustrate our general results by applications.


Introduction
SIR models are the foundation for a large number of compartmental models in mathematical epidemiology which classify the population into three classes: Susceptible, Infected and Removed (see [1]- [19]).Generally, these models admit two types of equilibrium: disease free and endemic equilibrium.If the disease free equilibrium is asymptotically stable, it implies the disease dies out.If the endemic equilibrium is asymptotically stable, it implies the disease persists in the population at the equilibrium level.
In 1976, Hethcote [13] considered the following deterministic SIR model with disease deaths:

S t S t I t K S t I t S t I t I t R t I t R t
In Equation ( 1), ( ) ( ) ( ) , , S t I t R t denote the number of the individuals susceptible to the disease, of infected members and of members who have been removed from the population, respectively.The model ( 1) is based on the following assumptions: i) The population considered has a constant size K, that is, ( ) ( ) ( ) for all t; ii) Births and deaths occur at equal rates µ in K.All the newborns are susceptible.µ is called a daily death removal rate; iii) β is the daily contact rate, i.e., the average number of contacts per infective per day.A contact of an infective is an interaction which results in infection of the other individual if it is susceptible; iv) α is the daily recovery removal rate of the infective.Of course, , , R In [8], Beretta and Takeuchi pointed out that when a susceptible vector is infected by a person, there is a time 0 τ > during which the infectious agents develop in the vector and it is only after that time that the infected vector becomes itself infectious, and proposed the following model where ( ) f s is a non-negative function which is square integrable on [ ] 0, h and satisfies ( ) ( ) and the non-negative constant h is the time delay, In fact, all infectious diseases are subject to randomness in terms of the nature of transmission.Recently, Tornatore et al. [16] investigate the dynamics of system (2) by perturbing the functional contact rates and modified (2) as: where σ is a positive constant and W is a real Wiener process defined on a stochastic basis In order to make the SIR system (2) more realistic, we consider the case of ( ) ( ) ( ) and we perturbed the deterministic system (2) by a white noise and obtained a stochastic counterpart by replacing the rates β by and hence we modify the SIR system (2) as the following model: where , , ,K Besides, there are deaths due to disease the total population size may vary in time so that we always assume the total population size is less than K in the context, where K represents a carrying capacity.Note that if we consider the population size is a constant K and the disease-related death rate 0 γ = , besides we also take ( ) ( ) (where σ is a positive constant), the system (3) becomes the model which has been discussed in [16].In [16], Tornatore proved the stability of disease-free equilibrium under some restricted conditions.However, they didn't consider the dynamics of the endemic equilibrium.It is of great importance from a theoretical point of view to investigate the stability of the endemic equilibrium.
In this paper, we mainly study the stochastic SIR model (5) with distributed delay which has more general diffusion coefficients than model (4)'s.By means of averaged Itô formula and Lyapunov function, we obtain the sufficient conditions for the regularity, existence and uniqueness of a global solution.
Furthermore, we also investigate the stochastic asymptotic stability of disease free equilibria and the dynamics of endemic equilibria which has not been discussed in [16].
The remaining parts of the paper are organized as follows: In Section 2, we will give some basic concepts and conclusions.In Section 3, we employ the averaged Itô formula to obtain the regularity, existence and uniqueness of the global solution of SIR model (5).In Section 4, we derive the sufficient condition to ensure the global stochastic asymptotic stability of disease free equilibrium in SIR model (5), besides we also consider the stochastic asymptotic stability of endemic equilibrium in Section 5. Finally, we illustrate our general results by applications.

Some Preliminary Definition and Lemmas
At first, we recall the notation of regularity of continuous time stochastic processes as introduced in [10].Let ( ) be a fixed closed domain.
For Simplicity, we only consider deterministic domains otherwise irregular with respect to  (or not invariant with respect to  ).Consider the d-dimensional stochastic differential equation of the form with an initial value ( ) Definition 2. The infinitesimal generator  associated with the SDE ( 6) is given by , and , and suppose f and g satisfy the existence and uniqueness conditions for solutions of (6) on each set ( ) for some positive constant c and , that is to say the stochastic process is called regular on  .Regularity on  implies boundedness, uniqueness, continuity and Markov property of the strong solution process X of SDE (6) with ( ) X t ∈ for all 0 t > (a.s.).Definition 3. The equilibrium solution * x of the SDE ( 6) is stochastically stable (stable in probability) if for every t denotes the solution of ( 6) satisfying Definition 4. The equilibrium solution * x of the SDE ( 6) is stochastically asymptotically stable (stable in probability) if it is stochastically stable and ( ) ( ) Definition 5.The equilibrium solution * x of the SDE ( 6) is said to be globally stochastically asymptotically stable (stable in probability) if it is stochastically stable and for every 0 X and every s ( ) 4]) Assume that f and g satisfy locally Lipschitz-continuous and satisfy linear growth condition and they have continuous coefficients with respect to t. 1) Suppose that there exists a positive definite function Then the equilibrium solution * x of ( 6) is stochastically stable.
2) In addition, if V is descresent (that is to say there exists a positive definite ( ) negative definite, then the equilibrium solution * x is stochastically asymptotically stable.
3) If the assumptions of part (2) hold for a radially unbounded function defined everywhere then the equilibrium solution * x is globally stochastically asymptotically stable.and ( )

Existence, Uniqueness and Regularity of Stochastic SIR Model Solution
≥ governed by Equation ( 5) is regular on  ; i.e. we have Moreover, regularity on  implies stochastic SIR model (5) admits a a unique, continuous-time, Markovian global solution process Proof.First we consider the result under the condition of (A).Denote drift term ( ) and the diffusion term e ,e e , , .
Since Equation ( 5) is well-defined on  and n  , and the coefficients .
Define  as infinitesimal generator as in Definition 2, then calculate , , 2 2

S f s I t s s K S f s I t s s K S S S f s I t s s S f s I t s s
, and hence we have , , , , .sup 2 In what follows, to show that ( ) ( ) where c is defined as above.And hence and apply averaged Itô formula, we have .
Using this fact and Equation ( 7), one can estimates , , e inf , , , , e 0 as , 2 2 ∈ +∞ , because of the appearance of the function ( ) for ( ) ) Then it proves the regularity and the global existence of the solution , , S t I t R t ∈ and by means of Lemma 2.1 under the condition of (A), we also derive the uniqueness and continuity of the solution.
Similarly the above discussions, we only need to take the function ( ) , , V S I R In what follows, we distinguish three cases to investigate the solution of these special situations.
1) If 0 S = , then the system (5) will reduce to . By using the similar analysis, we know that the above SDE is regular which implies there exists a unique global solution on 1 D ; 2) If 0 I = , then the system (5) will reduce to an ODE ( ) . By using the theory of ODE, we know that the above ODE is regular which implies there exists a unique global solution on 2 D ; 3) If 0 R = , then the system (5) will become . By using the similar analysis, we know that the above SDE is regular which implies there exists a unique global solution on 3 D .

Global Stochastic Asymptotic Stability of Disease Free Equilibrium
Theorem 4.1.Assume that 5) is globally stochastically stable on  .
Proof.Notice that the assumption , , , V S I R  becomes negative definite on  , and hence it completes the proof of Theorem 4.1 by applying Lemma 2.2. Remark 4.1.As we known, the basic reproduction number 0 R is one of the most important parameters in epidemiology, which reflects the expected number of secondary infections produced when one infected individual entered a fully susceptible population.If 0 1 R < then the outbreak will disappear, on the other hand, if 0 1 R > then the epidemic will spread a population.In this context, the basic reproduction number of the SIR model is 0 K R β α γ µ = + + .

Stochastic Asymptotic Stability of Endemic Equilibrium
, then there exists a unique endemic equilibrium solution ( ) , , S I R for the model (5), where ) ( ) ( ) Theorem 5.1.The endemic equilibrium solution, ( ) , , S I R of the Equation ( 5) is stochastically asymptotically stable on , , 0 i F S I R = and satisfies ( )

, , 2
Proof.It is a fact that the endemic equilibrium solution of system (5) exists if , , 0 , where a and b are defined as Equations (( 13) and ( 14)), c is an arbitrary positive constant.An elementary computation leads to 0 V > for any point ( ) , , S I R ∈ , and we have From the following formulas and the definitions of a , b can help to simplify ( )   , that is, 1 , and hence we can obtain the disease free equilibrium solution ( ) ( ) , , ,0,0 S I R K = of Equation ( 5) is globally stochastically stable on  .In the simulation, the parameters are chosen as follows the force of infection at time t.
R are locally Lipschitz-continuous and satisfy linear growth condition on  , then there exists a unique, bounded and Markovian solution up to random time ( ) τ  (or ( ) n τ  ), where ( ) τ  (or ( ) n τ  represents the random time of the first exit of stochastic process we can also obtain the same results under the condition of (B).Here, we omits the details.This completes the proof of Theorem 3.1. Remark 3.1.Because 0 I S R = = = are undefined in the domain  .

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. H. Xie et al.DOI: 10.4236/am.2017.880801039 Applied Mathematics and hence one can estimates that there exists a positive constant Just note that the infinitesimal generator  satisfies ( )