Stochastic SIR Epidemiological Model With Two Levels of Severity

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Introduction
The mathematical analysis of the dynamics of disease spread within a population has consistently piqued the interest of many researchers.These investigations primarily revolve around mathematical modeling, with the compartmental SIR model remaining the most classical and fundamental in epidemiology due to its robustness and simplicity.Most models are compartmental models which involve dividing the population into disjointed classes, each containing individuals with the same clinical state regarding the disease.In 1927, some researchers (Kermack & McKendrick, 1927) pioneered the study of the compartmental SIR model, followed by other authors (Kaddar et al., 2011;Korobeinikov & Wake, 2002) who delved into models such as SIS, SEIR and SIRS.
With the emergence of the Covid-19 disease in 2019, researchers have become interested in two-compartment models of infected individuals.Indeed, according to researchers (C.Waechter, 2021), there are two states of desease: symptomatic infected individuals who exhibit signs of the disease, and asymptomatic infected individuals.The number of asymptomatic infection cases at a given moment refers to the number of individuals affected by a disease who do not show any symptoms at that specific time.These situations can elude detection because infected individuals do not exhibit any apparent signs of the disease, but they still have the ability to transmit the infection to others.
We can mention the work of some researchers (Liu et al., 2020), who modeled this phenomenon and used data from three countries to make predictions.The autors (Coulibaly & N´zi, 2021) have also worked on the same model, with a jump perturbation.
In these models, the borders remain closed, which means that there is no recruitment.It is true that authorities have implemented various measures to close the borders in order to slow down the spread, but some borders have remained open with entry controls.This allows the entry of healthy and susceptible individuals into the population.
It is important to emphasize that several authors have examined models involving recruitment within the population of susceptible individuals (Chen & Li, 2022;Lahrouz et al., 2011;Settati et al., 2021).However, in the models examined by these authors, there is only one compartment for infected individuals, who are individuals displaying symptoms of infection.This article is devoted to a qualitative study of an SIaIsR model that incorporates a recruitment process within the population.
The rest of the paper is organized as follows: section 2 is dedicated to the mathematical formulation of the model.we provide some preliminary results in Section 3, while our main results are presented in section 4. In the deterministic model, we establish the stability of the free equilibrium state.In the stochastic model, we justify the existence and positivity of a solution to the system that almost surely resides in a specific domain of R 4 .Subsequently, we show that, under appropriate conditions, the solution of the stochastic system converges almost surely exponentially to the desease-free equilibrium.
Finally, under assumptions regarding the intensity of the white noise, we demonstrate that the free equilibrium state is globally stochastically stable.In section 5, we perform some numerical simulations to compare the dynamic behaviors of deterministic system and stochastic system.

Deterministic Model
The model that we present is governed by the following system: where • S (t) is the number of susceptible individuals at time t, i.e people who are not infected yet but might become infectious individuals in the future.
• I a (t) is the number of asymptomatic infectious individuals at time t, i.e people who have contracted the disease but have not yet developed it at time t.
• I s (t) is the number of symptomatic infectious individuals with mild symptoms at time t.
• R(t) is the number of recovered individuals at time t.
• λ represents the recruitment rate, corresponds to the number of individuals joining the population per unit of time.
• µ is the death rate.
• τ is the transmission rate.
• γ 1 and γ 2 are the recovered rate respectively of asymptomatic infectious individuals and symptomatic infectious individuals.
All the parameters are positive.
An equilibrium state (S , I a , I s , R) of the system (1) satisfies the following equations: A disease-free equilibrium (DFE) is an equilibrium state where I a = I s = 0 in system (2) .Its stability implies the disappearance of the disease.System (1) admits only one disease-free equilibrium E 0 = ( λ µ , 0, 0, 0).An endemic equilibrium is an equilibrium state where the compartments of infected individuals (I a and I s ) are non-zero.Its stability implies the persistence of the disease.
If the recovery rates of both groups of infected individuals are identical, then we have a classical SIR model.
If the recovery rates are different, then there is no endemic equilibrium state with the simultaneous presence of both types of infected individuals (I * a 0 and I * s 0).Indeed, the existence of such a state, denoted as (S * , I * a , I * s , R * ), would lead to S * = µ+γ 1 τ = µ+γ 2 τ .That is contradictory.However, we can have a single endemic equilibrium where only one of the compartments of infected individuals is non-zero.
The basic reproduction number R 0 representing how many secondary infectious result from the introduction of one infected individual into a population of susceptible.Using the (Van den Driessche & Watmough, 2002) method, we obtain: We consider different recovery rates and study the stability of the disease-free equilibrium.

Stochastic Model
The system ( 1) is obviously a deterministic model that abstracts from any randomness in the parameter values.This assumption often deviates from reality.
Indeed, random fluctuations can influence the parameters or variables of the model.Stochastic perturbations can be introduced to account for the inherent uncertainty in epidemiological processes.According to the literature, these perturbations come in two types.Some authors consider perturbations of the numbers of individuals in compartments through independent Brownian motions.The works of (Cai et al., 2017;Zhang & Wang, 2014;Ikram et al., 2022) can be mentioned in this context.On the other hand, others consider a perturbation of the contact rate, denoted as τ.The works of (Lahrouz et al., 2011;N´zi & Kanga, 2016;N´zi & Tano, 2017) can be cited in this regard.The rate of contact between healthy individuals and infected individuals is subject to random phenomena.The reception of the measures taken by the authorities by the populations often leads to a disruption of the contact rate.To account for this aspect, we have formulated a stochastic version of the model by adding white noise to the contact rate.
Moreover, to account for the stochastic nature of the contact rate, we have added white noise σ dB(t) dt to it.Where (B(t)) t≥0 is standard Brownian motion.
We have taken this type of perturbation into account in our model.We get the following system: Its easy to prove that domain ∆ is invariant.In effect, the total population in system (1) verifies the equation Theorem 2.1 (A.M.Lyapunov:1992) If there exists a continuously differentiable function" V : R n → R such that: Therefore, the equilibrium state x e is globally asymptotically stable.
Theorem 2.2 The unidimensional Itô's formula (Seidler, 1991) (I.Karatzas and S.E Shere) x) be a real function that is twice differentiable in x and once differentiable in t, and let X be an Itô process.Then we have: Theorem 2.3 Comparison Theorem (Ikeda:1976) Let σ, b 1 , and b 2 be three continuous functions defined on [0 : +∞[×R with values in R, such that b 1 (t, x) ≤ b 2 (t, x) for all t ≥ 0 and for all x ∈ R. We consider the following stochastic differential equations: If (X 1 t ) t≥0 and (X 2 t ) t≥0 are respective solutions of equations ( 6) and (7) such that X 1 0 ≤ X 2 0 , then P − p.s X 1 t ≤ X 2 t for all t ≥ 0.
Proof Let S = λ µ − S .Then system (1) becomes as follows Let and C positive constants such that, The last term in ( 10) is negative and using Young inequality we have: where is the constant in (8).Those inequalities injincting in (10), we obtain: where Using the fact that R 0 < 1 and (9) it esay to verify that K 2 < 0 and K 3 < 0.
According to the theorem, the free equilibrium state is globally asymptotically stable.

Stochastic Model
Let (Ω, F , P) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions.We consider the system (4) and the domain ∆ (see ( 5)).
The coefficients of the system (4) are locally Lipschitz continuous, so for any given initial value (S (0), I a (0), I s (0), R(0)) there is a unique local solution on [0, τ e ] where τ e is the explosion time.
By using the comparison theorem, In particular at the point T ∧ τ this inequality remains true.Now taking the expectation of both parts of the above inequality and using the fact that t 0 (σ(I a (u) is a mean zero process, we deduce that for all T ≥ 0 Furthermore, in view of ( 11), we have V(X(T ∧ τ ) ≥ 0 thus, By continuity there is some component of X(τ ) equal to , therefore V(X(τ )) ≥ −ln µ λ .So, we have By combining ( 13) and ( 14) we obtain for all T ≥ 0 By letting goes to zero, we derive that for all T ≥ 0, P(τ ≤ T ) = 0. Hence P(τ = ∞) = 1.As τ e ≥ τ, we have τ e = τ = ∞ a.s .

Proof
Let (S (0), I a (0), I s (0), R(0)) ∈ ∆.Consider function where, Since the last term is negative and S (t) ≤ λ µ , ∀t ≥ 0 we have, Therefore, (S (t)) 2 is bounded, then by the strong law of large number for local martingales we have From ( 15) and ( 16) we have, This completes the proof.

Simulation and Discussions
To illustrate the various theoretical results presented above, the systems ( 1) and ( 4 Figures 1 and 2 illustrate the dynamical behavior of the S IaI sR model described by the deterministic system (1), when R 0 < 1. Table 1.Estimated parameters of figure 1   We also observe the global stability of the Disease-Free Equilibrium (DFE).In this case, the recovery rates are higher than those presented in Figure 1, leading to a rapid decrease of symptomatic and asymptomatic infections towards zero.
Figures 3 and 4 illustrate the dynamical behavior of the S IaI sR model described by the deterministic system (5), when R 0 > 1.In both cases, we observe an instability of the Disease-Free Equilibrium (DFE).

Conclusion
One primary objective of mathematical epidemiology is to comprehend how to control or eradicate diseases.Mathematical models are extensively employed in the investigation of ecological and epidemiological phenomena.One of the key challenges in studying epidemic behavior is the analysis of the model's steady states and their stability.
In this paper, we studied an epidemiological model with 2 compartments of infected individuals in both deterministic and stochastic cases.This model is suitable for studying Covid-19, which involves both symptomatic and asymptomatic infected individuals.
In the deterministic case, if the basic reproduction number is less than 1, then the disease-free equilibrium state is globally asymptotically stable, indicating disease eradication.However, if the basic reproduction number is greater than 1, the disease persists, as illustrated in Figures 3 and 4. In the stochastic case, we demonstrated that a small perturbation in the contact rate ensures global asymptotic stability of the disease-free equilibrium state.At two levels, our model extends the work of Liu et al. (Liu et al., 2020).This involves incorporating the recruitment of susceptible populations and introducing a stochastic version of the model.Our findings stem from a qualitative study (stability of disease-free equilibrium).Disease control conditions have been identified based on the basic reproduction number and the intensity of the perturbation.However, such a result does not hold with high noise intensity, which could be a subject for future investigation.
) were simulated for various sets of parameters.Figures 1 to 4 illustrate the deterministic model (1), and Figures 5 and 6 depict the stochastic model (4).

Figure 2 .
Figure 2. Deterministic trajectories of S IaI sR epidemics model

Figure 3 .
Figure 3. Deterministic trajectories of S IaI sR epidemics model The recovery rates are distinct, and we observe stability of a single endemic equilibrium (I * a 0 and I * s = 0).

Figure 4 .
Figure 4. Deterministic trajectories of S IaI sR epidemics model

Figure 5 .
Figure 5. Stochastic trajectories of S IaI sR epidemics model

Table 2 .
Estimated parameters of figure3 and 4

Table 3 .
Estimated parameters of figure5 and 6