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

1.   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, higher-order 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 $SIR$ model ^[11], $SIS$ model ^[12], $SIRS$ model ^[13], $SEIR$ model ^[14], $SI_1\cdots I_kR$ ^[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 ^[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 ${\beta}SI$ ^[26]. The standard incidence is ${\beta}SI/N$, 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 ${\beta}SI/(1+aS)$ or ${\beta}SI/(1+aI)$ ^[23]. Other incidence forms, such as the nonlinear incidence rate ${\beta}SI^{p}/(1+{\alpha}S^{q})$ and Beddington-DeAngelis incidence $\beta SI/(1+aS+bI)$, have been discussed in ^[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 ${\beta}SI/(1+aS)(1+bI)$, 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 $N$, ${\beta}_{i}$ 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 ${\delta}_{i}$ respectively, and ${\alpha}_{i}$ 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:

In fact, disease transmission is quite sensitive to disturbances caused by external environmental factors, such as temperature, light, rainstorms and human intervention. These stochastic factors could have a significant impact on almost all parameters of the model in multiple ways ^{[36,37,38,39,40]}. The transmission rate $\beta$ oscillates around an average value as a result of the environment's ongoing oscillations brought on by the impact of white noise ${\beta}+{\sigma}\dot{B}(t)$, where $B(t)$ represents the standard Brownian motions and ${\sigma} > 0$ is the intensity of environmental fluctuations. Then, we obtain a stochastic epidemic model as follows:

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 ${\sigma}_{i}$ and parameters $a_{i}$ and $b_{i}$ on the dynamics of the system. The paper ends with a short discussion and conclusion.

2.   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

We denote

then, regarding the solutions of system (1.1), ${\Gamma}$ 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:

$({\rm{i}})$ disease-free equilibrium $E_{0} = \left(\frac{N}{{m}}, 0, 0\right)$;

$({\rm{ii}})$ boundary equilibrium $E_{1} = \left(\bar{S_{1}}, \bar{I_{1}}, 0\right)$, where $0 < \bar{S}_{1} < \frac{N}{m}, \bar{I_{1}} > 0$;

$({\rm{iii}})$ boundary equilibrium $E_{2} = \left(\bar{S_{2}}, 0, \bar{I_{2}}\right)$, where $0 < \bar{S}_{2} < \frac{N}{m}, \bar{I_{2}} > 0$;

$({\rm{iv}})$ endemic equilibrium $E_{3} = \left(S^{*}, I_{1}^{*}, I_{2}^{*}\right)$, where $S^{*}, I_{1}^{*}, I_{2}^{*} > 0$.

By (2.1), when $I_{2} = 0$ and $I_{1}\neq 0$, we obtain the boundary equilibrium

and $\bar{I_{1}}$ is the positive root of

where

when $\mathcal{R}_{1} > 1$, we have

If $\Delta = Y^{2}-4XZ = 0$, there exists a unique $E_{1}$ equilibrium with $\bar{I_{1}} = -\frac{Y}{2X}$; if $\Delta = Y^{2}-4XZ > 0$, then system (1.1) has two $E_{1}$ equilibria with $\bar{I_{1}} = \frac{-Y\pm\sqrt{Y^{2}-4XZ}}{2X}$.

Similarly, by (2.1), when $I_{1} = 0$ and $I_{2}\neq 0$, we obtain the boundary equilibrium

if $\mathcal{R}_{2} > 1$ then there may exist one or two-type equilibria $E_{2}$ as well.

By equilibrium equation (2.1), when $\mathcal{R}_{i} > 1, i = 1, 2$, the endemic equilibrium $E_{3} = \left(S^{*}, I_{1}^{*}, I_{2}^{*}\right)$ exists and satisfies

where the relationship between $I_{1}^{*}$ and $I_{2}^{*}$ satisfies

Theorem 1. 1). If $\mathcal{R}_{1} < 1$ and $\mathcal{R}_{2} < 1$, then the disease-free equilibrium $E_{0}$ is locally asymptotically stable.

2). If $\mathcal{R}_{1} > 1$ and equilibrium $E_{1}$ exists, then the boundary equilibrium $E_{1}$ is locally asymptotically stable.

3). If $\mathcal{R}_{2} > 1$ and equilibrium $E_{2}$ exists, then the boundary equilibrium $E_{2}$ is locally asymptotically stable.

4). If $\mathcal{R}_{1} > 1$, $\mathcal{R}_{2} > 1$ and equilibrium $E_{3}$ exists, then the endemic equilibrium $E_{3}$ is locally asymptotically stable.

The proof is given in Appendix A.

3.   Dynamics of stochastic system

3.1.   Existence and uniqueness of the positive solution

Lemma 1 ([^[43]]). For any initial value $\big(S(0), I_{1}(0), I_{2}(0)\big)\in \mathbb{R}^{3}_{+}$, there exists a unique solution $\big(S(t), I_{1}(t), I_{2}(t)\big)$ to system (1.2) on $t\geq0$, and the solution will remain in $\mathbb{R}^{3}_{+}$ with probability 1, i.e., $\big(S(t), I_{1}(t), I_{2}(t)\big)\in \mathbb{R}^{3}_{+}$ for all $t\geq0$ a.s.

Proof. The proof of Lemma 1 is similar to that in Theorem 2.1 of ^[44]; we therefore omit it here.

Lemma 2 ([^[29]]). ${\Gamma}$ is an almost positive invariant set of system (1.2), that is, if $\big(S(0), I_{1}(0), I_{2}(0)\big)\in{\Gamma}$, then $\mathbb{P}\big(S(t), I_{1}(t), I_{2}(t)\in{\Gamma}\big) = 1$ for all $t\geq0$.

Define the stochastic basic reproduction numbers

3.2.   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 $\big(S(t), I_{1}(t), I_{2}(t)\big)$ be a solution of system (1.2) with initial value $\big(S(0), I_{1}(0), I_{2}(0)\big)\in \mathbb{R}^{3}_{+}$. Then

Theorem 2. Suppose that one of the following two assumptions is satisfied:

$({{\rm{H}}_1})$ ${\sigma}_{i} > \hat{{\sigma}}_{i}: = \max\left\{ \sqrt{\frac{{\beta}_{i}(N a_{i}+m)}{N}}, \frac{{\beta}_{i}}{\sqrt{2(m+{\alpha}_{i}+{\delta}_{i})}}\right\}, \quad i = 1, 2$;

$({{\rm{H}}_2})$ ${\sigma}_{i}\leq\sqrt{\frac{{\beta}_{i}(Na_{i}+m)}{N}}$ and $\mathcal{R}^{s}_{i} < 1, \quad i = 1, 2.$

Then the solution $\big(S(t), I_{1}(t), I_{2}(t)\big)$ of system (1.2) with any initial value $\big(S(0), I_{1}(0), I_{2}(0)\big)\in {\Gamma}$ satisfies that

Proof. By using $\mathrm{It\hat{o}}$'s formula, we have

Case 1: Under assumption $({{\rm{H}}_1})$, integrating both sides of (3.1), we have

where

dividing both sides of (3.2) by $t$, we have

by Lemma 3, we have

since ${\sigma}_{i} > \frac{{\beta}_{i}}{\sqrt{2(m+{\alpha}_{i}+{\delta}_{i})}}$ for $i = 1, 2$, taking the limit superior on both sides of (3.4) leads to

Thus, $\lim\limits_{t\rightarrow +{\infty}}I_{i}(t) = 0$ a.s.

Case 2: Under assumption $({{\rm{H}}_2})$, similar to the calculation in Case 1, we have

where the function $\Upsilon(x)$ is defined as

Take note that $\Upsilon(x)$ increases monotonically for $x\in\left[0, \frac{{\beta}_{i}}{{\sigma}_{i}^{2}}\right]$ and $x < \frac{N}{a_iN+m}$; thus, when ${\sigma}_{i}\leq\sqrt{\frac{{\beta}_{i}(Na_{i}+m)}{N}}$, we have

Taking the limit superior of both sides of (3.5) leads to

which implies that $\lim\limits_{t{\rightarrow }+{\infty}}I_{i}(t) = 0, i = 1, 2.$

We suppose that $0 < I_{i}(t) < {\varepsilon}_{i} (i = 1, 2)$ for all $t\geq0$; by the first equation of system (1.2), we have

Because ${\varepsilon}_{1}\rightarrow0$ and ${\varepsilon}_{2}\rightarrow0$, if we divide (3.6) by the limit inferior on both sides, we have

Combining this with $\limsup\limits_{t\rightarrow +{\infty}}S(t)\leq\frac{N}{m},$ it is easy to see that

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 $\big(S(t), I_{1}(t), I_{2}(t)\big)$ and that $\big(S(0), I_{1}(0), I_{2}(0)\big)\in {\Gamma}$ 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 $\mathcal{R}^{s}_{1} > 1$, $\mathcal{R}^{s}_{2} < 1$ and the disturbance intensity satisfies that ${\sigma}_{2}\leq\sqrt{{\beta}_{2}(a_{2}+\frac{m}{N})}$. Additionally, $I_{1}$ satisfies

(ii) The disease $I_{1}$ becomes extinct and the disease $I_{2}$ becomes permanent in the mean if $\mathcal{R}^{s}_{1} < 1$, $\mathcal{R}^{s}_{2} > 1$ and the disturbance intensity satisfies that ${\sigma}_{1}\leq\sqrt{{\beta}_{1}(a_{1}+\frac{m}{N})}$. Additionally, $I_{2}$ satisfies

(iii) If $\mathcal{R}^{s}_{i} > 1$, then the two infectious diseases $I_{i}$ are permanent in mean; moreover, $I_{i}$ satisfies

where

4.   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.

To simulate the behavior of the stochastic system (1.2), we made use of Milstein's method ^[45,46]. Following ^[23,47], with the exception of $\sigma_i$, the other parameter values of system (1.1) and system (1.2) were derived as given in Table 1. Then, we chose initial values as $\big(S(0), I_{1}(0), I_{2}(0)\big) = (15, 10, 5)$.

4.1.   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 ${\alpha}_{1}$ is changed to 0.7, ${\sigma}_{1} = {\sigma}_{2} = 1$, $\mathcal{R}_{1} = 1.091 > 1$, $\mathcal{R}_{1}^{s} = 1.091-0.379 = 0.712 < 1$, $\mathcal{R}_{2} = 0.9375 < 1$ and $\mathcal{R}_{2}^{s} = 0.9375-0.3125 = 0.625 < 1$. 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)).

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 ${\alpha}_{2}$ is changed to 0.7, ${\delta}_{2}$ is 0.2, $a_{2} = 1.3$, ${\sigma}_{1} = {\sigma}_{2} = 1$, $\mathcal{R}_{1} = 0.909 < 1, \mathcal{R}_{1}^{s} = 0.909-0.416 = 0.496 < 1, \mathcal{R}_{2} = 1.071 > 1$ and $\mathcal{R}_{2}^{s} = 1.071-0.255 = 0.816 < 1$. 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 ${\delta}_{2} = 0.1, {\sigma}_{1} = 1$ and ${\sigma}_{2} = 0.1$, it follows that $\mathcal{R}_{1} = 1.091 > 1, \mathcal{R}_{1}^{s} = 1.091-0.413 = 0.678 < 1, \mathcal{R}_{2} = 1.041 > 1$ and $\mathcal{R}_{2}^{s} = 1.041-0.001 = 1.04 > 1$. 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 ${\delta}_{2} = 0.1, {\sigma}_{1} = 0.1$ and ${\sigma}_{2} = 1$, $\mathcal{R}_{1} = 1.091 > 1, \mathcal{R}_{1}^{s} = 1.091-0.0041 = 1.0869 > 1, \mathcal{R}_{2} = 1.041 > 1$ and $\mathcal{R}_{2}^{s} = 1.041-0.195 = 0.846 < 1$. 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 ${\delta}_{2} = 0.1, {\sigma}_{1} = 1$ and ${\sigma}_{2} = 1$, $\mathcal{R}_{1} = 1.091 > 1$, $\mathcal{R}_{1}^{s} = 1.091-0.413 = 0.678 < 1$, $\mathcal{R}_{2} = 1.041 > 1$ and $\mathcal{R}_{2}^{s} = 1.041-0.195 = 0.846 < 1$. 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)).

4.2.   The impact of environmental noise

By $({\rm H}_1)$ in Theorem 3.1, we can see that when the strengths of the perturbations are large, $\mathcal{R}^{s}_{i}$ loses its meaning and the diseases go to extinction. We chose different perturbation strengths for when ${\sigma}_{i} = 0, 0.3, 0.9$ to observe the trend of the disease. When ${\sigma}_{i}$ is larger, the infectious disease $I_{i}$ goes to extinction (see Figure 3). These simulations support our results for $({\rm H}_1)$ in Theorem 3.1 well.

4.3.   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 ${\sigma}_{i} = 0.3, i = 1, 2$. 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 $\mathcal{R}_{i}$ and $\mathcal{R}_{i}^{s}$. Additionally, we found that the infected population $I_i(t)$ also decreases when $b_{i}$ increases, but $b_i$ cannot lead to extinction.

5.   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 $\mathcal{R}_{i} < 1$; $E_{1}$ is locally asymptotically stable when $\mathcal{R}_{1} > 1$ and $\mathcal{R}_{2} < 1$; $E_{2}$ is locally asymptotically stable when $\mathcal{R}_{2} > 1$ and $\mathcal{R}_{1} < 1$; and $E_{3}$ is locally asymptotically stable when the basic reproduction number $\mathcal{R}_{i} > 1$. Subsequently, we have given the stochastic basic reproduction number $\mathcal{R}_{i}^{*}$ of the stochastic system and proven the stochastic extinction and persistence of the system. Finally, numerical simulations show that appropriate stochastic perturbations ${\sigma}_{i}$ 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.

Acknowledgments

Suli Liu was supported by the National Natural Science Foundation of China (Grant number 12301627), the Science and Technology Research Projects of the Education Office of Jilin Province, China (JJKH20211033KJ) and the Technology Development Program of Jilin Province, China (20210508024RQ).

Conflict of interest

The authors declare that there is no conflict of interest.

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 $\mathcal{R}_{1} < 1$ and $\mathcal{R}_{2} < 1$, then $\lambda_1$, $\lambda_2$ and $\lambda_3 < 0$. 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 $\mathcal{R}_{2} < 1$ and $\bar{S_{1}} < \frac{N}{m}$ is used. What follows is the characteristic equation:

where

By the Routh-Hurwitz condition, if $\mathcal{R}_{1} > 1$ and $\mathcal{R}_{2} < 1$, 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

Then

where $p > 0$ when $\mathcal{R}_{1} > 1$ and $\mathcal{R}_{2} > 1$, and we have

When both $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ 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 $(\text{i})$. It is easy to see from the theorem that $\lim\limits_{t\rightarrow +{\infty}}I_{2}(t) = 0$. Since $\mathcal{R}^{s}_{1} > 1$, there exists a ${\varepsilon}$ small enough such that $0 < I_{2}(t) < {\varepsilon}$ 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

when $\langle f(t)\rangle = \frac{1}{t}\int^{t}_{0}f(\theta)d\theta$ is defined for an integrable function $f$ on $[0, +\infty)$, one may get

By using $\mathrm{It\hat{o}'s}$ formula, it follows that

On the sides of (A3), dividing by $t > 0$ and integrating from $0$ to $t$ gives

where $Q_{2}(t) = \int^{t}_{0}{\sigma}_{1}S({\tau})\mathrm{d}B_{1}({\tau})$. It is possible to rewrite the inequality (A4) as

where ${\Delta} = \frac{{\beta}_{1}(m+{\delta}_{1})}{m}+b_{1}(m+a_{1}N)(m+{\alpha}_{1}+{\delta}_{1})$.

By Lemma 3, we get that $\lim\limits_{t\rightarrow +{\infty}}\frac{Q_{2}(t)}{t} = 0$. We can observe that $I_{1}(t)\leq\frac{N}{m}$; thus, we have that $\lim\limits_{t\rightarrow +{\infty}}\frac{I_{1}(t)}{t} = 0$ and $\lim\limits_{t\rightarrow +{\infty}}\frac{\ln I_{1}(t)}{t} = 0$ as $I_{1}(t)\geq 1$ and $\lim\limits_{t\rightarrow +{\infty}}{\Theta}(t) = 0$.

When the limit inferior of both sides of (A5) are taken into account, we have

Allowing for ${\varepsilon}\rightarrow 0$, we have

Due to the fact that the methods of proving parts $(\text{ii})$ and $(\text{i})$ are similar, this step will not be repeated. Part $(\text{iii})$. Take note that

Define

Consequently, $V(t)$ is bounded. We have

On the sides of (A8), dividing by $t > 0$ and integrating from $0$ to $t$ gives

where

It is possible to rewrite the inequality (A9) as

By Lemma 3, we have that $\lim\limits_{t\rightarrow +{\infty}}\frac{Q_{i}(t)}{t} = 0$ for $i = 1, 2$. And we can see that $\lim\limits_{t\rightarrow +{\infty}}{\Theta}(t) = 0$ and $\lim\limits_{t\rightarrow +{\infty}}\frac{V(t)}{t} = 0$.

Taking the limit inferior of both sides of (A10) yields