Dynamic analysis of a stochastic epidemic model incorporating the double epidemic hypothesis and Crowley-Martin incidence term

: The host population in epidemiology may actually be at risk of more than two infectious diseases with stochastic complicated interaction, e.g., HIV and HBV. In this paper, we propose a class of stochastic epidemic model that applies the double epidemic hypothesis and Crowley-Martin incidence rate in order to explore how stochastic disturbances a ﬀ ect the spread of diseases. While disregarding stochastic disturbances, we examine the dynamic features of the system in which the local stability of equilibria are totally determined by the basic reproduction numbers. We focus particularly on the threshold dynamics of the corresponding stochastic system, and we obtain the extinction and permanency conditions for a pair of infectious diseases. We ﬁnd that the threshold dynamics of the deterministic and stochastic systems vary signiﬁcantly: (i) disease outbreaks can be controlled by appropriate stochastic disturbances; (ii) diseases die out when the intensity of environmental perturbations is higher. The e ﬀ ects of certain important parameters on deterministic and stochastic disease transmission were obtained through numerical simulations. Our observations indicate that controlling epidemics should improve the e ﬀ ectiveness of prevention measures for susceptible individuals while improving the e ﬀ ectiveness of treatment for infected individuals.


Introduction
Smallpox, cholera, AIDS, COVID-19 and other infectious disease epidemics have wreaked immense havoc on the economy and way of life of the populace.Many mathematical models have been developed by researchers to explore the dynamical behavior of infectious diseases and thus control their transmission and gain a deeper understanding of these diseases [1][2][3][4][5][6]; among which, higherorder networks are widely used the spreading dynamics [7][8][9].Compartmental models, which were originally established by Kermack and McKendrick [10], constitute a class of representative infectious disease models that includes the S IR model [11], S IS model [12], S IRS model [13], S EIR model [14], S I 1 • • • I k R [15] model and other variations [16][17][18][19].
Most epidemic models only concentrate on the transmission of a unique infectious disease; however, the host population may actually be at risk of more than two infectious diseases with complicated interaction, and they could occur as parallel, competitive or stimulative.A large percentage of people at risk for HIV infection is also at risk for HBV infection due to shared mechanisms of transmission [20].Casalegno et al. [21] discovered that during the first half of fall 2009, in France, rhinovirus interference slowed the influenza pandemic and affected the transmission of the H1N1 virus.During the COVID-19 pandemic, it was discovered in [22] that the SARS-COV-2 Delta (B.1.617.2) variant had replaced the Alpha (B.1.1.7)variation on a significant scale, which is related to the Delta version's earlier invasion and superior transmissibility.In this paper, we only focus on two epidemics spreading parallelly, and we assume that an epidemic caused by one virus prevents the occurrence of the other.For related works, we recommend the references [23][24][25] and the references therein.
The rate at which new infections emerge, known as the disease incidence, is a crucial variable in mathematical models of infectious disease dynamics.The incidence rate has different forms which are commonly used as follows.It is assumed that the exposure rate is proportionate to the whole population and that the mass-action (bilinear) incidence is βS I [26].The standard incidence is βS I/N, and it requires the assumption that the number of people exposed to a sick person per unit time is constant [27].If the exposure rate is saturation of the susceptible S or infective I, the incidence will be the saturation incidence βS I/(1 + aS ) or βS I/(1 + aI) [23].Other incidence forms, such as the nonlinear incidence rate βS I p /(1 + αS q ) and Beddington-DeAngelis incidence βS I/(1 + aS + bI), have been discussed in [28,29].
A particular Crowley-Martin functional response function was proposed in 1975 [30], and it is widely used in prey-predator models [31], eco-epidemic models [32] and epidemic models [33][34][35].In infectious disease models, the Crowley-Martin incidence is represented by βS I/(1 + aS )(1 + bI), which takes into account the interaction between susceptible and infected populations, where a measures the preventive effect of susceptible individuals and b measures the treatment effect with respect to infected individuals.
For these reasons, this paper presents a deterministic epidemic model with the double epidemic hypothesis and Crowley-Martin nonlinear incidence term.We divided the population into three compartments: the susceptible population S , the infected population I 1 infected with virus D 1 and the infected population I 2 infected with virus D 2 .In addition, susceptible individuals enter at a rate of constant N, β i is the rate of transmission from a susceptible person to an infected person, the natural and causal mortality rates of the population are m and δ i respectively, and α i is the rate of infected people transitioning to the susceptible class.The flowchart of disease transmission and progression is as shown in Figure 1; we formulate the following dynamical model: (1.1) In fact, disease transmission is quite sensitive to disturbances caused by external environmental
The transmission rate β oscillates around an average value as a result of the environment's ongoing oscillations brought on by the impact of white noise β + σ Ḃ(t), where B(t) represents the standard Brownian motions and σ > 0 is the intensity of environmental fluctuations.Then, we obtain a stochastic epidemic model as follows: dB 2 (t). ( The following describes how this manuscript is structured.In Section 2, we discuss the dynamics of deterministic systems, especially for the asymptotic stability of equilibria.In Section 3, we establish the extinction and persistence conditions of the corresponding stochastic system.In Section 4, through a number of numerical simulations, we explore the effects of the perturbation strength σ i and parameters a i and b i on the dynamics of the system.The paper ends with a short discussion and conclusion.

Dynamics of deterministic system
Prior to investigating the stochastic system, it is also essential to ascertain the dynamical behaviors of the deterministic system.For the deterministic system (1.1) or the stochastic system (1.2), we obtain This implies that lim sup t→∞ We denote then, regarding the solutions of system (1.1),Γ is a positively invariant set.
Utilizing the next generation matrix method [41,42], we can obtain the basic reproduction number: The equilibrium equation is listed as follows: System (1.1) has four possible equilibria: where S * , I * 1 , I * 2 > 0. By (2.1), when I 2 = 0 and I 1 0, we obtain the boundary equilibrium and Ī1 is the positive root of . Similarly, by (2.1), when I 1 = 0 and I 2 0, we obtain the boundary equilibrium where the relationship between I * 1 and I * 2 satisfies .
2).If R 1 > 1 and equilibrium E 1 exists, then the boundary equilibrium E 1 is locally asymptotically stable.
The proof is given in Appendix A.

Extinction and persistence of stochastic system
We focus on disease extinction and persistence in this subsection since stochastic systems have distinct extinction and persistence conditions compared to deterministic systems.First, the following lemma is presented to demonstrate the extinction and persistence of diseases.Lemma 3 ( [42,45]).Let S (t), I 1 (t), I 2 (t) be a solution of system (1.2) with initial value S (0), I 1 (0), I 2 (0) ∈ R 3 + .Then Theorem 2. Suppose that one of the following two assumptions is satisfied: Then the solution S (t), I 1 (t), I 2 (t) of system (1.2) with any initial value S (0), Proof.By using Itô's formula, we have Case 1: Under assumption (H 1 ), integrating both sides of (3.1), we have where dividing both sides of (3.2) by t, we have by Lemma 3, we have 2(m+α i +δ i ) for i = 1, 2, taking the limit superior on both sides of (3.4) leads to lim sup Thus, lim t→+∞ I i (t) = 0 a.s.Case 2: Under assumption (H 2 ), similar to the calculation in Case 1, we have where the function Υ(x) is defined as Take note that Υ(x) increases monotonically for x ∈ 0, β i σ 2 i and x < N a i N+m ; thus, when Taking the limit superior of both sides of (3.5) leads to lim sup which implies that lim t→+∞ I i (t) = 0, i = 1, 2. We suppose that 0 < I i (t) < ε i (i = 1, 2) for all t ≥ 0; by the first equation of system (1.2), we have We will present the persistence result of system (1.2) in the following theorem, whose proof is given in Appendix B.
Theorem 3. If we assume that the solution to system (1.2) is S (t), I 1 (t), I 2 (t) and that S (0), I 1 (0), I 2 (0) ∈ Γ is the initial value, then we get the following: (i) The disease I 2 becomes extinct and the disease I 1 becomes permanent in the mean if R s 1 > 1,R s 2 < 1 and the disturbance intensity satisfies that σ 2 ≤ β 2 (a 2 + m N ).Additionally, I 1 satisfies (ii) The disease I 1 becomes extinct and the disease I 2 becomes permanent in the mean if (iii) If R s i > 1, then the two infectious diseases I i are permanent in mean; moreover, I i satisfies where

Simulations
In this section of this paper, we will continue with our investigation of the deterministic system and the stochastic system by using the numerical method.Before looking at how changes in the environment influence the spread of diseases and the effect of the parameters a i and b i on the dynamics of the disease, we first compare the extinction conditions for the same parameter values for the stochastic system and the deterministic system.

Extinction conditions of two diseases in deterministic and stochastic systems
In order to investigate the dynamical differences that exist between systems (1.1) and (1.2), we give five examples for numerical simulations.
Example 1.When I 2 is facing extinction in a deterministic system, the stochastic perturbation could change I 1 from prevalence to extinction.When α 1 is changed to 0.7, According to our previous analysis results, disease I 1 is prevalent and disease I 2 is subject to extinction in the deterministic system (1.1); however, disease I 1 and I 2 are both extinct in the stochastic system (1.2) (see Figure 2(a)).
The initial value of all solutions is (15,10,5).The time unit is day.
Example 2. When I 1 is subject to extinction in a deterministic system, the stochastic perturbation could change I 2 from a prevalence to extinction condition.When α 2 is changed to 0.7, δ 2 is 0.2, According to our previous analysis results, disease I 1 goes to extinction and disease I 2 is persistent in the deterministic system (1.1); also, disease I 1 and I 2 both go to extinction in the stochastic system (1.2) (see Figure 2(b)).
Example 3. When I 1 and I 2 are extinct in a deterministic system, the stochastic perturbation could change I 1 from a prevalence to extinction condition.When δ 2 = 0.1, σ 1 = 1 and σ According to our previous analysis results, both disease I 1 and disease I 2 are persistent in the deterministic system (1.1); disease I 1 goes to extinction and I 2 is persistent in the stochastic system (1.2) (see Figure 2(c)).
Example 4. When I 1 and I 2 are extinct in a deterministic system, the stochastic perturbation could change I 2 from a prevalence to extinction condition.When δ 2 = 0.1, σ 1 = 0.1 and According to our previous analysis results, both disease I 1 and disease I 2 are persistent in the deterministic system; also, disease I 1 is persistent and I 2 goes to extinction in the stochastic system (1.2) (see Figure 2(d)).
Example 5. When I 1 and I 2 are extinct in a deterministic system, the stochastic perturbation could change I 1 and I 2 from a prevalence to extinction condition.When δ 2 = 0.1, σ 1 = 1 and According to our previous analysis results, both disease I 1 and disease I 2 are persistent in the deterministic system (1.1).In the stochastic system (1.2), both disease I 1 and disease I 2 go extinct (see Figure 2(e)).

The impact of environmental noise
By (H 1 ) in Theorem 2, we can see that when the strengths of the perturbations are large, R s i loses its meaning and the diseases go to extinction.We chose different perturbation strengths for when σ i = 0, 0.3, 0.9 to observe the trend of the disease.When σ i is larger, the infectious disease I i goes to extinction (see Figure 3).These simulations support our results for (H 1 ) in Theorem 2 well.

The impact of preventive effect a i and treatment effect b i
It should be noted that a i and b i of the Crowley-Martin incidence are key parameters.In this subsection, we discuss the effects of parameters a i and b i on the population and trend of infections by presenting some numerical simulations.
First, we study the influence of preventive effects a i on the population of infective individuals in the deterministic system (1.1).For the case that the parameters in Table 1 are fixed in Table 1, we chose five different sets of values for a i .For the deterministic system (1.1), it can be shown that the bigger the value of a i , the quicker the extinction of disease I i (see Figure 4).Second, we wanted to investigate the influence of the parameter a i on the population of infective individuals in the stochastic system (1.2).We choose the perturbation intensity as σ i = 0.3, i = 1, 2. At last, we observed the effect of a i in the stochastic system (1.2) (see Figure 5).Similarly, we wanted to study the influence of treatment effects b i on the population of infected individuals in deterministic and stochastic systems (see Figures 6 and 7).From the above numerical simulations, we conclude that a larger a i leads to a lower infected prevalence I i (t), and it may result in the extinction in deterministic and stochastic systems.This is because the parameter a i affects R i and R s i .Additionally, we found that the infected population I i (t) also decreases when b i increases, but b i cannot lead to extinction.

Discussion and conclusions
In this paper, we have proposed and studied a class of stochastic double disease models with Crowley-Martin incidence.We discussed the existence conditions and stability of the equilibrium points.E 0 is locally asymptotically stable when the basic reproduction number R i < 1; E 1 is locally asymptotically
stable when R 1 > 1 and R 2 < 1; E 2 is locally asymptotically stable when R 2 > 1 and R 1 < 1; and E 3 is locally asymptotically stable when the basic reproduction number R i > 1. Subsequently, we have given the stochastic basic reproduction number R * i of the stochastic system and proven the stochastic extinction and persistence of the system.Finally, numerical simulations show that appropriate stochastic perturbations σ i can control the spread of the disease, but larger stochastic perturbations can cause the disease to go extinct; the protection effect a i can cause the disease to go extinct; the treatment effect b i can reduce the number of infected individuals, but it cannot cause the disease to go extinct.Therefore, when treatment is given to infected individuals, protective measures for susceptible individuals are more necessary to completely eliminate the virus.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
Appendix A : Proof of Theorem 1 Proof.The system (1.1) has the following Jacobian matrix where , , Case 1.The evaluation of the Jacobian matrix at E 0 is represented by , which has the following eigenvalues: If R 1 < 1 and R 2 < 1, then λ 1 , λ 2 and λ 3 < 0. The disease-free equilibrium E 0 is locally asymptotically stable.
Case 2. The evaluation of the Jacobian matrix at E 1 is represented by , where Using the equilibrium equation (2.1) we obtain one of the three eigenvalues of matrices J(E 1 )'s is represented by where R 2 < 1 and S 1 < N m is used.What follows is the characteristic equation: where Electronic Research Archive Volume 31, Issue 10, 6134-6159.
By the Routh-Hurwitz condition, if R 1 > 1 and R 2 < 1, the boundary equilibrium E 1 is locally asymptotically stable.The proof process for Cases 2 and 3 are similar, so they are omitted.
Case 4. The evaluation of the Jacobian matrix at E 3 is represented by what follows is the characteristic equation: where Electronic Research Archive Volume 31, Issue 10, 6134-6159.
:=p + q, where p > 0 when R 1 > 1 and R 2 > 1, and we have When both R 1 and R 2 are bigger than 1, the Routh-Hurwitz criteria show that the endemic equilibrium E 3 is locally asymptotically stable.

Appendix B : Proof of Theorem 3
Proof.Part (i).It is easy to see from the theorem that lim t→+∞ I 2 (t) = 0. Since R s 1 > 1, there exists a ε small enough such that 0 < I 2 (t) < ε for all t large enough; we have On the sides of the system (1.2), dividing by t > 0 and integrating from 0 to t gives where ∆ = β 1 (m+δ 1 ) m + b 1 (m + a 1 N)(m + α 1 + δ 1 ).By Lemma 3, we get that lim t→+∞ Q 2 (t) t = 0. We can observe that I 1 (t) ≤ N m ; thus, we have that lim Due to the fact that the methods of proving parts (ii) and (i) are similar, this step will not be repeated.Part (iii).Take note that Consequently, V(t) is bounded.We have

Figure 3 .
Figure 3.The evolution of a single path of I 1 and I 2 for the stochastic system (1.2) when σ i changes.And all other parameters are taken as in Table 1.(a)-(c) Influences of three different values of parameter σ i influence the value of I i when diseases I 1 , I 2 are persistent initially; (d)-(f) influence of three different values of parameter σ i on the value of I i when I 1 is persistent and I 2 goes to extinction initially; (g)-(i) influences of three different values of parameter σ i on the values of I i when I 2 is persistent and I 1 goes to extinction initially.The initial value of all solutions is (15,10,5).The time unit is day.

2 Figure 4 . 2 Figure 5 . 9 Figure 6 . 9 Figure 7 .
Figure 4.The impact of the preventive effect a i in the deterministic system (1.1).The preventive effect a i reduces the number of infected individuals and keeps the susceptible individuals from becoming infected: (a) Influences of five different values of parameter a 1 on the values of I 1 .(b) Influences of five different values of parameter a 2 on the values of I 2 .The initial value of all solutions is (15,10,5).The time unit is day.

( 1 + 2 i=1 1 2 i b i 1 2 i b i 1 2 i=1 1 ( 1 + 2 i
a i S )(1 + b i I i ) − 1 + a i N m (m + α i + δ i ) − 1 1 + a i S )(1 + b i I i )) 2 dt + + a i N m σ i S (1 + a i S )(1 + b i I i ) dB i (t) + + a i N m β i S I i (1 + a i S )(1 + b i I i ) − (m + α i + δ i )I i dt + + a i N m σ i S I i (1 + a i S )(1 + b i I i ) dB i (t) 1 + a i S )(1 + b i I i )) 2 dt + b i I i ) − (m + α i + δ i )I i dt + + β 2 )S − 2 i (m + α i + δ i )(1 + b i I i ) 1 + a i N m + σ i S dB i (t).(A8) 1 then there may exist one or two-type equilibria E 2 as well.By equilibrium equation (2.1), when R i > 1, i = 1, 2, the endemic equilibrium E 3 = S * , I *

Table 1 .
Parameter values of system (1.1) and system (1.2) in numerical simulations.