Threshold dynamics of stochastic models with time delays: A case study for Yunnan, China

1.   Introduction

At present, in order to better study the transmission mechanism of infectious diseases, many researchers introduced noises into the deterministic models and studied the effect of noise on the dynamics of the established stochastic epidemic models. Stochastic models could be a more appropriate way of modeling epidemics in many circumstances [11,24]. In particular, Liu et al. [11] established a deterministic model with non-linear incidence rate, and studied the global stability of the model by the basic reproduction number of the model. Then, based on the deterministic model, the stochastic model is established, the dynamics of the stochastic model is proved theoretically, and the properties of these two models are compared. The paper [24] shows that the stochastic models are able to consider randomness of infectious contacts occurring in the latent or infectious periods. The combination of stochastic model and deterministic counterpart can make people understand the epidemic trend of infectious diseases more comprehensively, and make the established theory and prevention strategy more reliable. Many realistic stochastic epidemic models can be derived based on their corresponding deterministic counterparts. Britton [3] gave an excellent survey on stochastic differential equation (SDE) epidemic models which presented the exact and asymptotic properties of a simple stochastic epidemic model, and illustrated by studying effects of vaccination and inference procedures for important parameters such as the basic reproduction number and the critical vaccination coverage. Allen [1] provided a great introduction to the methods of derivation for various types of stochastic models including SDE epidemic models. There are different possible approaches including random effects in the models, both from biological and mathematical perspective [9]. The general stochastic differential equation SIRS model introduced in this paper adopts the modeling approach from Mao et al. [13], which has been pursued in [11,24], and assume that the parameters involved in the model always fluctuate around some average value due to continuous fluctuation in the environment.

On the other hand, the spread of infectious diseases is not only related to the current state, but also to the historical state (see [25,23]). Therefore, it is more practical to use stochastic differential equation model with time delay to describe the spread of infectious diseases. Recently, many scholars have studied the stochastic infectious disease model with time delay. In particular, Liu et al. [10] proposed and studied the stochastic SEIR epidemic model with infinite distribution delay, and obtained the sufficient conditions for the global asymptotic stability of the endemic equilibrium. Another part of the work is to introduce the disturbance of some parameter variables in the system, which can make the system have a disease-free equilibrium point and give the stability conditions of the disease-free equilibrium point. However, when the parameters of the delay infectious disease model are disturbed randomly, the stochastic model generally does not have disease-free equilibrium and endemic equilibrium. Tornatore et al. [19] proposed a stochastic model with latency and time delay for mosquito transmission, and proposed a strategy for controlling disease transmission. After that, Fan et al. [5] considered the persistence and extinction of a class of stochastic SIR epidemic model with generalized nonlinear incidence and transient immunity and time delay, and obtained the threshold of the model. On the basis of [19,5], Berrhazi et al. [2] considered the stochastic SIR epidemic model with Beddington-DeAngelis incidence and delayed immune loss under Lévy noise disturbance, and studied its persistence, extinction and threshold problems. Considering the effect of vaccination and the incubation period of some diseases (such as tuberculosis, AIDS, measles) after infection, Liu et al. [12] discussed the stochastic SVEIR epidemic model with distribution delay. By constructing appropriate stochastic Lyapunov function, the existence, uniqueness and ergodicity of the positive distribution of the system were obtained.

Motivated by above mentioned papers, we introduce time delays and stochastic into the deterministic differential equations. This paper is organized as follows: In Section 2.1, we give the basic theory and related lemmas. In Section 2.2, we construct a class of stochastic differential equations models with time delays, and give the thresholds and dynamics of the models by using the theorems and lemmas of Section 2.1. In Section 3, we do a case study of the models in Section 2.2. In addition, we carry out some numerical simulations aiming to HIV/AIDS and Gonorrhea transmission in Yunnan by using the models. In Section 4, we summarize the whole paper.

2.   Modeling and theoretical analysis

2.1.   Basic theory and related lemmas

Firstly, we introduce some lemmas and notations, which will be used in the following parts. Generally speaking, the $d$-dimensional stochastic differential equation can be expressed as follows:

where $f(t,X(t))$ is a function in $\mathbb{R}^d$ defined in $[t_0 , + \infty ] \times \mathbb{R}^d$ and $g(t,X(t))$ is a $d \times m$ matrix, $f,g$ are locally Lipschitz with respect to the second variable. $B(t)$ is an $m$-dimensional standard Brownian motion defined on the above probability space. The differential operator $L$ of system (2.1.1) is defined by

If $L$ acts on a function $V \in \mathbb{C}^{2,1} (\mathbb{R}^d \times [t_0 , + \infty ];\mathbb{R}_ + )$, then

where $V_t (x,t) = \frac{{\partial V}} {{\partial t}},V_x (x,t) = (\frac{{\partial V}} {{\partial x_1 }}, \cdots ,\frac{{\partial V}} {{\partial x_d }}),V_{xx} = (\frac{{\partial ^2 V}} {{\partial x_1 \partial x_j }})_{d \times d} .$ In view of Itô's formula, if $x(t) \in \mathbb{R}^d$, then ${\text{d}}V(x,t) = LV(x,t){\text{d}}t + V_x (x,t)g(x,t){\text{d}}B(t).$

Lemma 2.1. [27] Let $A(t)$ and $U(t)$ be two continuous adapted increasing processes on $t \geqslant 0$ with $A(0) = U(0) = 0$ a.s. Suppose that $M(t)$ is a real-valued continuous local martingale with $M(0) = 0$ a.s. and $X(0)$ a nonnegative $F_0$-measurable random variable with $E\left( {X(0)} \right) < \infty$. Define $X(t) = X(0) + A(t) - U(t) + M(t)$ for all $t \geqslant 0$. If $X(t)$ is nonnegative, then $\lim _{t \to \infty } A(t) < \infty$ implies that $\lim _{t \to \infty } U(t) < \infty ,\lim _{t \to \infty } X(t) < \infty$ and $- \infty < \lim _{t \to \infty } M(t) < \infty$ with probability one.

Lemma 2.2. [27] Let $M(t),\;t \geqslant 0$, be a local martingale vanishing at time 0 and define

where $\left[ {M,M} \right](t)$ is Meyers angle bracket process. Then $\mathop {\lim }\limits_{t \to \infty } \frac{{M(t)}}{t} = 0$ a.s. provided $\mathop {\lim }\limits_{t \to \infty } \rho _M (t) < \infty$ a.s.

Lemma 2.3. [4] Set $f \in \mathbb{C}\left[ {[0,\infty ] \times \Omega ,(0,\infty )} \right]$, and $\langle {x(t)} \rangle = \frac{1}{t}\int_0^t {x(s){\mathit{\text{d}}}s}$. Suppose that there exist positive constants $\lambda _0 ,\lambda$ such that

for all $t \geqslant 0$, where $F \in \mathbb{C}\left[ {[0,\infty ] \times \Omega ,(0,\infty )} \right]$ and $\lim _{t \to \infty } \frac{{F(t)}}{t} = 0\;{\mathit{\text{a}}}{\mathit{\text{.s}}}.$ Then

In real world, a group of people will often be infected with a variety of diseases, especially two of them have the same or similar transmission routes. We discuss a class of models with two infectious diseases as follows.

2.2.   Modeling and analysis for double diseases

In the paper [14], Meng and Zhao formulated a kind of epidemic model with two classes of epidemics as follows

where $S$ denotes the number of the population susceptible to the diseases, $I_1$ and $I_2$ are the total population of the infective in terms of two diseases at time $t$, respectively. The recruitment to the susceptible population is to be considered as a constant $A$, $\beta _1$ and $\beta _2$ are the contact rates, $d$ is natural mortality rate, $\alpha _1$ and $\alpha _2$ are the rates of diseases-related death, $r_1$ and $r_2$ are the treatment cure rates of two diseases, respectively. $a_i$ is the parameter that measure the inhibitory effect for $I_i$. The incidence rate $\frac{{\beta _i SI_i }} {{1 + a_i I_i }}(i = 1,2)$ of susceptible individuals through their contacts with infectious. The two thresholds of the model are $\mathcal R_1 = \frac{{\beta _1 A}} {{d(d + \alpha _1 + r_1 )}}$ and $\mathcal R_2 = \frac{{\beta _2 A}} {{d(d + \alpha _2 + r_2 )}}$, and the four equilibria are $E_0 = \left( {\frac{A} {d},0,0} \right)$, $E_1 = \left( {S_1^* ,I_1^* ,0} \right)$, $E_2 = \left( {S_2^* ,0,I_2^* } \right)$ and $E^* = \left( {S^* ,I_1^{**} ,I_2^{**} } \right)$, respectively. In addition, the dynamic behavior for the model (2.2.1) is described in the following theorem.

Theorem 2.4. [14] For system (2.2.1), the following conclusions are true.

(i) If $\mathcal R_1 < 1$ and $\mathcal R_2 < 1$, then both diseases go extinct and system (2.2.1) has a unique stable diseases-extinction equilibrium $E_0$.

(ii) If $\mathcal R_1 > 1$ and $\mathcal R_2 < 1$, then the disease $I_2$ goes extinct and system (2.2.1) has a unique stable equilibrium $E_1$.

(iii) If $\mathcal R_1 < 1$ and $\mathcal R_2 > 1$, then the disease $I_1$ goes extinct and system (2.2.1) has a unique stable equilibrium $E_2$.

(iv) If $\mathcal R_1 > 1$ and $\mathcal R_2 > 1$, then $E^*$ is a unique stable equilibrium, which implies both diseases of system (2.2.1) are permanent.

Next, we study the stochastic model with time delays corresponding to model (2.2.1).

where $B_i(t)$ represents a standard Brownian motion with $B_i(0) = 0$ and $\sigma _i^2 > 0\;(i = 1,2,3)$ denotes the intensity of the white noise, ${\tau _i}$ denotes that $S$ has become the susceptible population of $I_i$ before ${\tau _i}$. On the basis of model (2.2.2), we define two thresholds: $\mathcal R_1^* = \frac{1} {{d + \alpha _1 + r_1 }} \cdot \left( {\frac{{\beta _1 A}} {d} - \frac{{\sigma _1^2 }} {2}} \right)$ and $\mathcal R_2^* = \frac{1} {{d + \alpha _2 + r_2 }} \cdot \left( {\frac{{\beta _2 A}} {d} - \frac{{\sigma _2^2 }} {2}} \right)$, respectively.

Lemma 2.5. For the solution $(S(t),I_1 (t),I_2 (t))$ of model (2.2.2) with any initial value $(S(0),I_1 (0),I_1 (0)) \in \mathbb{R}_ + ^3$, we have

Moreover,

Proof. From (2.2.2), we get

This equation has the solution

where

is a continuous local martingale with $M(0) = 0$ a.s. Define

with $X(0) = (S(0) + I_1(0) + I_2(0)),\;A(t) = \frac{A}{d}(1 - e^{ -d t} )$ and $U(t) = (S(0) + I_1(0) + I_2(0))(1 - e^{ - dt} )$ for all $t \ge 0.$ Due to the stochastic comparison theorem, $S(t) + I_1(t) + I_2(t) \le X(t)$ a.s. It is easy to check that $A(t)$ and $U(t)$ are continuous adapted increasing processes for $t \ge 0$ with $A(0) = U(t) = 0$. By using Lemma 2.1, we have $\lim _{t \to \infty } X(t) < \infty$ a.s. This completes the proof of (2.2.3).

For convenience, we denote

Computing $\left[ {{M_1},{M_1}} \right](t) = \sigma _1^2 \int_0^t {I_1^2 (s){\text{d}}s}$, by Lemma 2.2 and (2.2.3), we obtain

Then, by Lemma 2.2, $\lim _{t \to \infty } \frac{1} {t}\int_0^t {\sigma _1 I_1(s){\text{d}}B_1 (s)} = 0.$ The left can be proved similarly. This completes the proof.

Lemma 2.6. For the solution $(S(t),I_1 (t),I_2 (t))$ of model (2.2.2) with any initial value $(S(0),I_1 (0),I_1 (0)) \in \mathbb{R}_ + ^3$, we have

Proof. We set

By Lemma 2.5, we have

From (2.2.6), since

by (2.2.12), we obtain $\lim _{t \to \infty } \langle {M(t)} \rangle = 0.$ From (2.2.6), we get

By (2.2.6), (2.2.13) and (2.2.14), we obtain

This completes the proof.

Lemma 2.7. For any given initial value $(S(0),I_1 (0),I_1 (0)) \in \mathbb{R}_ + ^3$, there exists a unique positive solution $(S(t),I_1(t),I_2(t))$ to system (2.2.2) on $t \ge 0$ and the solution will remain in $\mathbb{R}_ + ^3$ with probability one, that is to say, $(S(t),I_1(t),I_2(t)) \in \mathbb{R}_ + ^3$ for all $t \ge 0$ almost surely.

Proof. Since the coefficients of system (2.2.2) are locally Lipschitz continuous, for any given initial value $(S(0),I_1 (0),I_1 (0)) \in \mathbb{R}_ + ^3$, there exists a unique local solution $(S(t),I_1(t),I_2(t))$ on $t \in [0,\tau _e )$, where $\tau _e$ denotes the explosion time (see [3]). To verify that this solution is global, we only need to prove $\tau _e = + \infty$ a.s.

Let $k_0 > 0$ be enough large such that each component of $(S(0),I_1 (0),I_1 (0))$ is no large than $k_0$. For each integer $k > k_0$, define the stopping time

where throughout this paper we set $\inf \emptyset = + \infty$. Obviously, $\tau _k$ is increasing as $k \to \infty$. Set $\tau _\infty = \lim _{k \to \infty } \tau _k$, then we can get $\tau _\infty \le \tau _e$ a.s.

Define a $\mathbb{C}^2$-function $V:\mathbb{R}_ + ^3 \to \mathbb{R}_ +$ by

By Itô's formula, we get

where

For any $k \geqslant k_0$, there exists $T > 0$ such that $\tau _k \in (0,T \wedge \tau _k ]$. By the generalized Itô's formula, for any $t \in (0,T \wedge \tau _k ]$, we have

where $E$ is the expectation of the function. Let $k \to \infty ,$ then $t \to \infty ,$ it follows that $\lim _{k \to \infty } P(\tau _k \leqslant T) = 0$, therefore $P(\tau _\infty \leqslant T) = 0.$ Since $T > 0$ is arbitrary, it results in

This completes the proof.

Theorem 2.8. Let $(S(t),I_1 (t),I_2 (t))$ be the positive solution of model (2.2.2) with initial value $(S(0),I_1(0),I_2(0)) \in \mathbb{R}_+^3$. Then as $\mathcal R_i^* < 1$, two infectious diseases of model (2.2.2) go extinct almost surely, i.e. $\lim _{t \to \infty } I_i (t) = 0,\;i = 1,2$. Moreover, $\lim _{t \to \infty } S(t) = \frac{A}{d}$ a.s.

Proof. Define a function $V = \ln I_i (t)$. By Itô's formula, we obtain

Integrating (2.2.15) from 0 to $t$ gives

where

Dividing both sides of (2.2.16) by $t$, we have

By integrating the model (2.2.2) and dividing the two sides by $t$, we conclude that

Calculating the sum of (2.2.18), we can assert that

where

According to (2.2.19), we have

Putting (2.2.20) into (2.2.17), we obtain

By Lemma 2.5 and Lemma 2.6, we get

That is, when $\mathcal R_i^* = \frac{1} {{d + \alpha _i + r_i }} \cdot \left( {\frac{{\beta _i A}} {d} - \frac{{\sigma _i^2 }} {2}} \right) < 1$, there obviously holds $\mathop {\lim \sup }\limits_{t \to \infty } \frac{{\ln I_i (t)}} {t} < 0$, which implies $\mathop {\lim }\limits_{t \to \infty } I_i (t) = 0,\;\mathop {\lim }\limits_{t \to \infty } S(t) = \frac{A} {d} = S_0$. This completes the proof.

Theorem 2.9. Let $(S(t),I_1 (t),I_2 (t))$ be any positive solution of model (2.2.2) with initial value $(S(0),I_1 (0),I_2 (0)) \in \mathbb{R}_ + ^3$, then we have the following results.

(i) If $\mathcal R_1^* > 1$ and $\mathcal R_2^* < 1$, then the disease $I_2$ goes extinct and the disease $I_1$ is permanent on average. Moreover, $I_1$ satisfies

(ii) If $\mathcal R_2^* > 1$ and $\mathcal R_1^* < 1$, then the disease $I_1$ goes extinct and the disease $I_2$ is permanent on average. Moreover, $I_2$ satisfies

(iii) If $\mathcal R_1^* > 1$ and $\mathcal R_2^* > 1$, then the two infections diseases $I_1$ and $I_2$ are permanent on average. Moreover, $I_1$ and $I_2$ satisfy

where

Proof. Firstly, we prove (ⅰ). From model (2.2.2), we obtain

Integrating (2.2.23) and dividing the two sides by $t$, we may assert that $\mathop {\lim }\limits_{t \to \infty } I_2 (t) = 0$ as $\mathcal R_2^* < 1$. From Theorem 2.8, we obtain

where

By (2.2.24), we deduce

From model (2.2.2), we also obtain

where

Define function $V = \ln I_1 (t)$. By using Itô's formula and combining with (2.2.17), we obtain

Putting (2.2.25) and (2.2.26) into (2.2.27), we obtain

According to Lemma 2.3, Lemma 2.5 and Lemma 2.6, when $\mathcal R_1^* > 1$ and $\mathcal R_2^* < 1$, we have

This completes the proof of (ⅰ).

(ii) The proof of (ⅱ) is similar to that of (ⅰ).

(iii) From model (2.2.2), we get

Similar to (2.2.23)-(2.2.25), integrating (2.2.29) yields

From model (2.2.2), we also get

where

Define the function $V = \ln ( a_2 I_1 (t) + a_1 I_2 (t) )$. Similar to (ⅰ), using Itô's formula, we get

Substituting (2.2.30) and (2.2.31) into (2.2.32), we get

Then we obtain

where

By Lemma 2.3, Lemma 2.5 and Lemma 2.6, we take the limit on both sides of (2.2.34) to get

This completes the proof.

From Theorem 2.8 and Theorem 2.9, we can claim that the thresholds $\mathcal R_i^*(i = 1,2)$ of the model (2.3.2) can describe the persistence and extinction of two diseases. In other words, if $\mathcal R_1^* <1$ and $\mathcal R_2^* <1$, then the two infections diseases $I_1$ and $I_2$ of model (2.3.2) go extinct; if $\mathcal R_1^* >1$ and $\mathcal R_2^* <1$, then the disease $I_2$ goes extinct and the disease $I_1$ is permanent on average; if $\mathcal R_2^* >1$ and $\mathcal R_1^* <1$, then the disease $I_1$ goes extinct and the disease $I_2$ is permanent on average; if $\mathcal R_1^* >1$ and $\mathcal R_2^* >1$, then the two infections diseases $I_1$ and $I_2$ are permanent on average.

3.   A case study for Yunnan in China

In this section, basing on the model (2.2.1) and (2.2.2), we do a case study for HIV/AIDS and Gonorrhea in Yunnan Province, China. The transmission routes of these two kinds of diseases are close to those of the main infected population. The main transmission modes are as follows: sexual transmission, blood transmission, mother to child transmission. According to the report data [26]: The cumulative number HIV positives reported at the end of September 2018 was 850,000, including 260,000 recorded deaths in China, and the estimated number living with HIV/AIDS was 36.9 million around the world. At present, there are many papers about the spread of HIV/AIDS. The basic mathematical models of HIV/AIDS in-host has been developed to describe interactions between immune system and viruses [21]. In [7] and [16], a class of HIV/AIDS model with time delay and a class of HIV/AIDS model with age structure are studied respectively. In [15], a cell-to-cell transmission model of HIV/AIDS is studied. There are few literatures about Gonorrhea, the way of transmission is similar to AIDS, and its harm is not as serious as AIDS. Gonorrhea is a purulent inflammatory disease of genitourinary system caused by Neisseria gonorrhoeae. It is also because the transmission route and AIDS are similar, so it is necessary to study the co-infection model of these two infectious diseases. Yunnan is located in southwest of China, bordering the countries of Myanmar, Laos and Vietnam. According to the sixth national census in 2011 [17], there are 45,596,000 people in Yunnan. From the cumulative number of HIV/AIDS infections in Yunnan Province in 2007 [26], combining with the number of newly increased infections and deaths of HIV/AIDS from 2007 to 2016 [18], the cumulative number of HIV/AIDS infections in Yunnan Province from 2007 to 2016 is obtained (see Table 1). In addition, the number of Gonorrhea infections increased annually from 2007 to 2016 in Yunnan Province [18] is shown in Table 1.

Using Eviews 7.0, we will test the stationarity of the data for the number of people infected with HIV/AIDS and Gonorrhea from 2007 to 2016 in Yunnan Province, respectively. The autocorrelation and partial correlation coefficients of the test results show that the data series is stable and the statistics are good. Next, the parameters of the model (2.2.2) are further determined for fitting. The specific idea is that part of the parameters are based on the known literatures and some biological values, and then, based on the data in Table 1 and the parameters obtained, the least square method is used to fit the model to estimate the remaining parameters. For the natural mortality of people in Yunnan Province, we choose $d = {\scriptstyle{}^{1}\!\!\diagup\!\!{}_{67}\;} = {\text{0}}{\text{.0149}}$ where 67 is the average life of people in Yunnan [6]. For HIV/AIDS and Gonorrhea patients' death rate $\alpha _i (i = 1,2)$ in Yunnan, we know that $1 - e^{ - \alpha _i t}$ is the death probability of HIV/AIDS and Gonorrhea patients. We can take $\alpha _1 = 0.7114$ (see [7]) and detailed parameters values are shown in Table 2. We take 2007 as the initial time $t = 0$, according to the cumulative number of HIV/AIDS infections in Yunnan Province in 2007 [26], so we take $I_1 (0) = 57325$, and the other two initial values $S(0)$ and $I_2 (0)$ are obtained by estimation, as shown in Table 2. The number of susceptible persons, the population infected with HIV/AIDS and the population infected with Gonorrhea are obtained by numerical fitting using the parameters of Table 2, as shown in Fig. 1, including the comparison of fitting data and statistical data.

When the parameters in Table 2 are substituted into the two thresholds of the model (2.2.1), they are both greater than unity, that is, both diseases are persistent. To illustrate the significance of model (2.2.2) in disease control, we first describe the dependence of each parameter in thresholds $\mathcal R_i^* (i = 1,2)$ of model (2.2.2), namely the partial rank correlation coefficients (PRCCs). As shown in Fig. 2. Considering the objective conditions of medical equipment and human life span, combining with the results of PRCC, we can adopt four ways to control the two diseases: (1) improving the recovery rate of diseases $r_i$; (2) reducing the infections rate of infectious diseases $\beta _i$; (3) reducing the number of imports to susceptible persons $A$ and (4) using big noises $\sigma _i (i = 1,2)$.

According to the results of PRCC, we take the following values: ${\text{A = 400,}}$ $r_1 {\text{ = 0}}{\text{.75,}}$ $r_2 {\text{ = 0}}{\text{.2,$ $d = 0}}{\text{.0149,}}$ $\beta _1{\text{ = 0}}{\text{.00007,}}$ $\beta _2 {\text{ = 0}}{\text{.00002,}}$ $\alpha _1 {\text{ = 0}}{\text{.7114,}}$ $\alpha _2 {\text{ = 0}}{\text{.1,}}$ $a_1 {\text{ = 0}}{\text{.0001,}}$ $a_2 {\text{ = 0}}{\text{.0001,}}$ $\sigma _1 {\text{ = 0}}{\text{.95,}}$ $\sigma _2 {\text{ = 0}}{\text{.9,}}$ $\sigma _3 {\text{ = 0}}{\text{.2,}}$ $\tau _1 = \tau _2 = 0.5$. It follows that $\mathcal R_1 = 1.2729 > 1 > \mathcal R_1^* = 0.9672,\:\mathcal R_2 = 1.7050 > 1 > \mathcal R_2^* = 0.4189$. Under these conditions, the two diseases of the deterministic model (2.2.1) will be persistent, whereas the two diseases described by the model (2.2.2) will be extinct (see Fig. 3 and Fig. 4).

4.   Discussions

The fluctuation of natural environment will bring variability to biological system [20]. And environmental changes have a vital impact on the development of epidemics. Variability of temperature and rainfall may cause significant fluctuations in the dynamics of pathogenic fungi [22,8]. In terms of human disease, the nature of epidemic spread and growth is inherently random due to the unpredictability of person-to-person contacts [11,24]. Therefore, the variability and randomness of the environment are introduced into the epidemic model [20]. In general, the threshold of the model is a very important quantity for theoretical analysis of differential equation models describing infectious diseases. That is to say, the relationship between the threshold and 1 is used to analyze whether the disease is extinct or not. Therefore, in this paper, we discuss two classes of differential equations models. Specifically in each class of models, considering the introduction of randomness and time-delays, the change of infection rate, the existence of immunity loss and so on, then we theoretically analyze the thresholds changes in each class. We obtain the sufficient conditions for the extinction and persistence of diseases. In addition, we do a case study of (2.2.1) and (2.2.2), and we also carry out some numerical simulations aiming to HIV/AIDS and Gonorrhea transmission in Yunnan by using the models.

Through the case study, the results of numerical simulation and theoretical analysis are consistent, i.e., when $\mathcal R_i^* > 1\;(i = 1,2)$, the diseases will be persistent; when $\mathcal R_i^* < 1\;(i = 1,2)$, the diseases will be extinct. By comparing the thresholds of model (2.2.1) and model (2.2.2), and combining with the numerical simulations results, its are found that there are always $\mathcal R_i > \mathcal R_i^* \;(i = 1,2)$. Therefore, we are mostly concerned about the situations $\mathcal R_i > 1 > \mathcal R_i^* \;(i = 1,2)$, which are the significances of stochastic differential equations in controlling infectious diseases. Finally, we discuss some biological implications and focus on the impact of some key model parameters, the strategies of controlling HIV/AIDS and Gonorrhea are given, i.e., controlling the spread of infectious diseases by improving the recovery rate of diseases $r_i$, reducing the infections rate of infectious diseases $\beta _i$, declining the number of imports to susceptible people $A$ and using big noises $\sigma _i (i = 1,2)$.

Another possible and important extension for future work is that there is a class of stochastic model and its corresponding deterministic model, and the threshold of the stochastic model is larger than that of the deterministic model. That is, when the threshold of the stochastic model is greater than 1 and then greater than the threshold of the deterministic model, the disease of the deterministic model will be extinct, but the disease of the stochastic model will be persistent. This is the risk of introducing random noises into the deterministic models for the infectious diseases models, but for the population models, this can maintain the growth of the population, so it is very meaningful to study this problem. All the aforementioned possible extensions are interesting, biologically important but yet mathematically challenging, and we have to leave them for future research projects.

Acknowledgments

We are grateful to reviewers for their valuable comments and suggestions, which greatly improved the presentation of this paper.