key: cord-0580567-bbetwrx0
authors: Easlick, Terry; Sun, Wei
title: A Unified Stochastic SIR Model
date: 2022-01-10
journal: nan
DOI: nan
sha: 4e01909ed3706e04178e9de4c9fc02807ccbfd42
doc_id: 580567
cord_uid: bbetwrx0

We propose a unified stochastic SIR model. The model is structural enough to allow for time-dependency, nonlinearity, demography and environmental disturbances. We present concise results on the existence and uniqueness of positive global solutions and investigate the extinction and persistence of the model. Examples and simulations are provided to illustrate the main results.

Epidemiological compartment models have garnered much attention from researchers in an attempt to better understand and control the spread of infectious diseases. Mathematical analysis of such models aids decision-making regarding public health policy changes-especially in the event of an epidemic (e.g., . One such compartmental model introduced by Kermack and McKendrick [9] in 1927 divides a population into three compartments-susceptible, infected, and recovered (SIR). The classical SIR model is as follows: where β is the transmission rate and γ the recovery rate. Additionally, demography may be introduced to include birth rate Λ and mortality rate µ as:

The basic SIR models (1.1) and (1.2) have many variations including the SIRD, SIRS, SIRV, SEIR, MSIR, etc. (cf. e.g., [1] , [2] and [11] ). Further, these deterministic models have been put into different stochastic frameworks, which makes the situation more realistic (cf. e.g., [3] , [6] , [8] and [12] ). The existing models are often analyzed with a focus on specific diseases or parameters. Such studies have been successful in achieving new results; however, often it is the case that structural variability is lacking in these models. To overcome the drawbacks inherent in traditional approaches, we propose and investigate in this paper the unified stochastic SIR model:

Hereafter, R + denotes the set of all positive real numbers, the drift functions b i (t, x, y, z) and the diffusion functions σ ij (t, x, y, z) are measurable on [0, ∞) × R 3 + , i = 1, 2, 3, j = 1, 2, . . . , n, and (B (j) t ) t≥0 , j = 1, 2, . . . , n, are independent standard one-dimensional Brownian motions. We will show that the unified model (1.3) is structural in design that allows variability without sacrificing key results on the extinction and persistence of diseases. Namely, the model allows for time-dependency, nonlinearity (of drift and diffusion) and demography. Environmental disturbances can have profound effects on transmission, recovery, mortality and population growth. The above model encapsulates the stochastic perturbations driven by white noises (B (j) t ) t≥0 with intensities σ ij (t, X t , Y t , Z t ). An important structural feature we emphasize is time-dependency. Time-dependency can capture the progression of a disease insofar as mutations/transmissibility (e.g., Delta and Omicron variants of COVID-19, vaccination programs).

In the following sections, we establish results on the existence and uniqueness of positive global solutions, extinction and persistence of diseases, and provide illustrative examples and simulations. Section 2 is concerned with the model (1.3) for population proportions whereas Section 3 covers the model (1.3) for population numbers. Both approaches are commonly found in studies of SIR models; hence, the importance of investigation for a unifying model. At the time of writing this paper we are unaware of existing work on the unified SIR model and aim to add such a model to the existing literature; additionally, we will close the paper with proposed future work.

In this section, we let X t , Y t and Z t denote respectively the proportions of susceptible, infected and recovered populations at time t. Define

We make the following assumptions.

1. For any T ∈ (0, ∞) and N ∈ N, there exists K T,N > 0 such that

Hereafter | · | denotes the Euclidean norm of R 3 .

2. For any (x, y, z) ∈ ∆, σ ij (t, x, y, z) = 0, j = 1, 2, . . . , n.

First, we prove the result on the existence and uniqueness of solution. 

We will show below that τ = ∞ a.s.. Assume the contrary that there exists T > 0 such that

By Itô's formula, we obtain that for t < τ ,

Define

x, y, z) z ,

Then, we get

which contradicts with (2.1). Therefore, τ = ∞ a.s..

One can check that many existing models are special cases of our model, e.g., the semi-parametric SIR model introduced in [5] and the stochastic SIR model with vaccination considered in [4] .

Now we consider the extinction and persistence of diseases. Namely, we investigate whether a disease will extinct with an exponential rate or will be persistent in mean.

If there exist positive constants λ 0 and λ such that

Proof. (i) By Itô's formula, we have that (ii) By (2.5) and (2.8), we obtain that for any η ∈ (0, λ) there exists T η > 0 such that

Therefore, (2.6) holds by (2.9) and [7, Lemma 5.1].

(iii) Obviously, condition (2.7) implies condition (2.5). Hence, the assertion is a direct consequence of assertion (ii).

If we take the following assumption of our model:

In fact, we have α ≤ α * and hence condition (2.10) implies that

Throughout this paper, we denote by L ∞ + [0, ∞) the set of all bounded, non-negative, measur-

We consider the system

t .

Suppose that ξ ≥ 1. 

Additionally, a key feature of the system (2.11) to note is that the transmission function is in the form of power function which differs from the often seen bilinear form.

(2.12)

We have dXt dt + dYt dt + dZt dt = 0. Hence, by Theorem 2.1, equation (2.12) has a unique strong solution taking values in ∆.

Suppose that

Set λ 0 = γ 2 and λ = γ 2 − γ 1 − 2σ 2 . Then, condition (2.7) is satisfied. Therefore, by Theorem 2.3(iii), we obtain that the disease is persistent and

In this section, we let X t , Y t and Z t denote respectively the numbers of susceptible, infected and recovered individuals at time t. First, we make the following assumption.

1. For any T ∈ (0, ∞) and N ∈ N, there exists K T,N > 0 such that

By this assumption, we have the existence and uniqueness of local strong solution to equation (1.3) on [0, τ ), where τ is the explosion time. Next, we make the following assumption.

2. There is an invariant subset Γ ⊂ R 3 + of the system (1.3), i.e., (X t , Y t , Z t ) ∈ Γ for any (X 0 , Y 0 , Z 0 ) ∈ Γ. Moreover, for any T ∈ (0, ∞),

Now we present the result on the existence and uniqueness of solution. Proof. By Assumption 1, we have the existence and uniqueness of local strong solution to equation (1.3) on [0, τ ), where τ is the explosion time. We will show below that τ = ∞ a.s.. Define

We have that τ ∞ ≤ τ so it suffices to show τ ∞ = ∞ a.s.. Hence assume the contrary that there exist ε > 0 and T > 0 such that P(τ ∞ < T ) > ε, which implies that

By Itô's formula, we obtain that for t ≤ τ N ,

Define

x, y, z) z ,

Then, we get

However, by (3.1), we get

We have arrived at a contradiction. Therefore, τ = ∞ a.s..

Similar to Theorem 2.3, we can prove the following result on the extinction and persistence of diseases.

x, y, z)| y < ∞. (ii) If there exist positive constants λ 0 and λ such that

(iii) If there exist positive constants λ 0 and λ such that

Let Λ, µ, β, γ, ε, σ ∈ L ∞ + [0, ∞). We consider the system

Suppose that µ > 0.

which implies that

is an invariant set of the system (3.3). Hence, equation (3.3) has a unique strong solution taking values in Γ by Theorem 3.1.

Define

Then, we have

Thus, by Theorem 3.2(i), we obtain that if

then the disease extincts with exponential rate

then the disease extincts with exponential rate

This result generalizes the result given in [7, Theorem 2.1].

which implies that

Then, by (3.3)-(3.5), we get lim inf

Therefore, by Theorem 3.2(ii), we obtain that the disease is persistent and

This result generalizes the result given in [7, Theorem 3.1].

We now present simulations corresponding to Examples 2.5 and 3.3. Simulations are completed using the Euler-Maruyama scheme with a time step ∆t = 0.001 which run from time t = 0 to t = 1000. We include both the stochastic and deterministic results to demonstrate the effect of noise on such systems. Time t is not given with a specific calendar time unit but we may consider that t represents days, weeks or years, etc.

(i) We assume that the system (2.11) in Example 2.5(a) has initial values (X 0 , Y 0 , Z 0 ) = (0.8, 0.19, 0.01) and set the parameters as follows: In Figure 1 below, it is illustrated that the extinction of the disease occurs at an exponential rate.

In accordance with Example 2.5(a), the disease will extinct with exponential rate −α ≥ γ − β = 0.05. (ii) We assume that the system (2.12) in Example 2.5(b) has initial values (X 0 , Y 0 , Z 0 ) = (0.57, 0.32, 0.11) and parameters: We achieve results which illustrate persistence of the disease, as is displayed in Figure 2 below. (iii) The remaining simulations are concerned with Example 3.3, system (3.3). We assume that the total population is 95 million and set the initial values using proportional values for convenience. The initial condition is set to (X 0 , Y 0 , Z 0 ) = 298 95 , 300 95 , 5 95 . In the following simulations, the parameters will change to demonstrate their effects on a system with unchanging initial condition. The first two simulations illustrate extinction of the disease and the final simulation will illustrate persistence of the disease. We initially set the parameters as follows: It is important to note that since the initial condition is unchanging this forces two parameters, namely Λ(t) and µ(t), to remain unchanged for these simulation purposes. Moreover, we have that

as the invariant set for the system (3.3). That is, this system has a unique strong solution taking values in Γ per Theorem 3.1. Given these parameters and following Example 3.3, we havẽ

≤ 0.5983 < 1 and σ 2 < 0.1296 < 0.1418 = µβ Λ .

As demonstarted below in Figure 3 , the disease will go extinct with exponential rate −α ≥ (µ + γ + ε) 1 −R 0 ≥ 2.45. We now make only the alteration of a single parameter in the system (3.3) . Assume that σ(t) has the following form: 2(µ + γ + ε) .

Thus, we have the scenario in which the disease goes extinct with exponential rate −α ≥ (µ + γ + ε) − β 2 2σ 2 ≥ 5.87.

Moreover, if we compare Figure 4 to the above Figure 3 we notice the disease appears to go extinct at a faster rate which is as expected given the above results. This modification yieldsR 0 = βΛ µ(µ + γ + ε) − σ 2 Λ 2 2µ 2 (µ + γ + ε) ≥ 1.73 > 1.

In Figure 5 below, we see such a modification yields disease persistence as opposed to disease extinction achieved in the previous two simulations for the system (3.3). 

In this paper, we propose and investigate the unified stochastic SIR model given by the system (1.3). We have presented two forms of the model-one for population proportions and the other for population numbers. For both forms of the model, we have given results on the extinction and persistence of diseases; moreover, we have shown that these results still hold with time-dependent, nonlinear parameters and multiple noise sources. Notably, we give examples and simulations that agree with the theoretical results and illustrate the impact that noise has on a given SIR model system.

In a further investigation of this work, we plan to introduce jumps and periodicity into the system (1.3) and obtain results on the extinction and persistence of diseases. This investigation will begin immediately with the concluding of this paper.

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