The Limit Behavior of a Stochastic Logistic Model with Individual Time-Dependent Rates

where λ and μ are the infection rate of susceptibles and the recovery rate of infectives, respectively.This simplemodel has found applications in a variety of fields, including population biology, metapopulation ecology, chemistry, and physics. Properties such asmean epidemic size [2, 3], mean extinction time [1, 4, 5], and quasi-stationary distributions [6–8] have been extensively studied under different initial conditions. The stochastic logistic model has an interesting limit property that it can be approximated by deterministic differential equations. In particular, based on Theorem 3.1 of [9], the rescaled process n−1Yn converges in probability uniformly on finite time intervals to the solution of an ordinary differential equation. The issue of differential equation approximations for stochastic processes traces back to the pioneer work of Kurtz [9, 10], where deterministic limit of pure jump density-dependent Markov processes was add-ressed via Trotter type approximation theorems. Recently, McVinish and Pollett [11] show that a deterministic limit process can be established for a stochastic logistic model with individual variation, where the coefficients of the transition rates of the Markov chain can vary with the nodes. We refer the interested reader to a comprehensive survey [12] for numerous sufficient conditions for this type of convergence. In this paper, along the above line of study, we investigate a variant of the stochastic logistic model where both individual variation and time-dependent infection and recovery rates are allowed. Specifically, we consider a continuous-time Markov chain Xn = (Xn 1 , . . . , X n


Introduction
The stochastic logistic model, also called the endemic SIS model in the epidemiological context, was first discussed by Weiss and Dishon [1].This model describes the evolution of an infection in a fixed population of size  by a continuoustime Markov chain for the number of infected individuals   .The state space is {0, 1, . . ., }, and the transition rates are given by   →   + 1 at rate  −1   ( −   ) ,   →   − 1 at rate   , (1) where  and  are the infection rate of susceptibles and the recovery rate of infectives, respectively.This simple model has found applications in a variety of fields, including population biology, metapopulation ecology, chemistry, and physics.Properties such as mean epidemic size [2,3], mean extinction time [1,4,5], and quasi-stationary distributions [6][7][8] have been extensively studied under different initial conditions.
The stochastic logistic model has an interesting limit property that it can be approximated by deterministic differential equations.In particular, based on Theorem 3.1 of [9], the rescaled process  −1   converges in probability uniformly on finite time intervals to the solution of an ordinary differential equation.The issue of differential equation approximations for stochastic processes traces back to the pioneer work of Kurtz [9,10], where deterministic limit of pure jump density-dependent Markov processes was add-ressed via Trotter type approximation theorems.Recently, McVinish and Pollett [11] show that a deterministic limit process can be established for a stochastic logistic model with individual variation, where the coefficients of the transition rates of the Markov chain can vary with the nodes.We refer the interested reader to a comprehensive survey [12] for numerous sufficient conditions for this type of convergence.
In this paper, along the above line of study, we investigate a variant of the stochastic logistic model where both individual variation and time-dependent infection and recovery rates are allowed.Specifically, we consider a continuous-time Markov chain   = (  1 , . . .,    ) on the state space {0, 1}  with transition rates given by where  = 1, 2, . . ., ,  , ,  , ∈ R + , and    is the -dimensional vector whose th entry is 1, and other entries are 0. Let    = 0 represent that individual  is susceptible and represent that individual  is infected.Then   = ∑  =1    is equivalent to the stochastic logistic model by setting  , = ,  , = , () = , and   () =  for all , , and .
The above model ( 2) can be viewed as a generalization of that treated in [11] in twofolds.Firstly, the infection and recovery rates are time varying, incorporating realistic scenarios where the infection and recovery capacities may change over time [13].As such, the transition rates of the Markov chain are explicitly time dependent, making the previously obtained sufficient conditions no longer applicable.
Secondly, the introduction of functions   () accommodates needed flexibility for some applications.Indeed, since    ∈ {0, 1}, the linear form   () =    used in [11] implies that no contribution can be made by susceptible individuals.This is only a rough approximation.For one thing, susceptible individuals aware of a disease in their proximity can take measures (such as wearing masks, avoiding public places, and frequent hand washing) to reduce their susceptibility, which in turn affect the epidemic dynamics dramatically [14,15].For another, the epidemic progression strongly depends on the contact patterns between susceptible and infected individuals, especially in the network context.Identifying individual role is an interesting and demanding task [16].In the present framework, each node  applies individual contribution   (0) (and   (1)), indicating the underlying interaction structure/strength among individuals.
The rest of the paper is organized as follows.We state the main result in Section 2 and provide the proof in Section 3.

The Result
In what follows, we assume naturally that   (1) ≥   (0) ≥ 0 for all .We will show that the stochastic logistic model (2) converges weakly to the solution of an integral equation as the population size  → ∞.
Let (, B) be a measurable space with  ⊆ R 2 + and B the Borel -algebra on .Denote by   () the set of bounded, continuous functions on .Let Ω be the set of finite measures on .For  ∈ N and ℎ ∈   (), we define the measure-valued nonrandom process {   ,  ∈ R + } and the measure-valued Markov process {   ,  ∈ R + } by For some set , let (R + , ) be the set of right-continuous functions with left-hand limits mapping R + to .
as  → ∞, where  is the unique solution of with Let  1 () be the space of   -integrable functions on .We have the following corollary.
Corollary 2. Suppose that  is the unique solution to (5).Then there exists a unique function for all Borel set  ∈ , and

Proofs
In this section, Theorem 1 will be proved through a series of lemmas by tightness and uniqueness arguments [18].The idea of proofs is similar to that in [11], and we include the complete proofs here, not only for the convenience of the reader but also to convince the reader that the results do hold in our setting.
From Lemma 3, the sequence   is tight, and then, there exists a weakly convergent subsequence    in (R + , Ω).We still denote it by   in the next lemma for convenience.

Lemma 5. If 𝜌 𝑛 𝑑
→  as  → ∞, then for any ℎ ∈   (), Proof.For each  ∈ N, define a random process   (ℎ) by From Theorem 3.1 in [18, page 28] and the arguments in the proof of Lemma 4, it is sufficient to show the following two conditions: To show (A), note that sup From the argument following (17), there is some constant  > 0 satisfying as  → ∞, for any bounded, uniformly continuous function  : (R + , R) → R. By the definition of   (ℎ) and the assumption     → , we only need to show that (  (ℎ)) is a continuous function of   mapping (R + , Ω) to R, invoking the dominated convergence theorem.Furthermore, it is sufficient to show that   (ℎ) is a continuous function of   mapping (R + , Ω) to (R + , R).In the following, we take one term in   (ℎ) as an example to show the continuity with respect to   .Other terms can be shownanalogously.Proof.Suppose that  and ρ are two solutions to (5), and they are the limits of two weakly convergent subsequences of   , respectively.By the assumption (A2),  0 = ρ0 .We need to show that   = ρ for all  ∈ R + . From where  > 0 is some constant.To see this, note that by the assumptions (A1), (A2), and (38), we have for some constant   > 0. A simple application of Gronwall's lemma yields (  , ρ ) = 0 for all  ∈ R + , which concludes the proof.
Proof of Theorem 1.By Lemma 4 and Lemma 5, the limit of any weakly convergent subsequence of   must satisfy (5).By Lemma 6, we find that the sequence   must itself converge weakly to that unique solution (see the corollary in [18, page 59]).The proof of Theorem 1 is thus completed.
Proof of Corollary 2. For any open set  ⊆ , we obtain from (38) that ∫    (, )   (d, d) ≤ ∫    (, )   (d, d) , (46)where   is taken as a continuous function upwardly converging to the indicator function of .By using the dominated convergence theorem, we know that   () ≤   ().A regularity property (see, e.g., [17, page 18, Lemma 1.34]) implies that   () ≤   () for all  ∈ .This means that   is absolutely continuous with respect to   for any  ≥ 0. An application of the Radon-Nykodym theorem yields the existence of   such that (8) holds with 0 ≤   ≤ 1.Now the result follows straightforwardly from Theorem 1.