Mathematical analysis of stochastic epidemic model of MERS-corona & application of ergodic theory

Abstract

The “Middle East Respiratory” (MERS-Cov) is among the world’s dangerous diseases that still exist. Presently it is a threat to Arab countries, but it is a horrible prediction that it may propagate like COVID-19. In this article, a stochastic version of the epidemic model, MERS-Cov, is presented. Initially, a mathematical form is given to the dynamics of the disease while incorporating some unpredictable factors. The study of the underlying model shows the existence of positive global solution. Formulating appropriate Lyapunov functionals, the paper will also explore parametric conditions which will lead to the extinction of the disease from a community. Moreover, to reveal that the infection will persist, ergodic stationary distribution will be carried out. It will also be shown that a threshold quantity exists, which will determine some essential parameters for exploring other dynamical aspects of the main model. With the addition of some examples, the underlying stochastic model of MERS-Cov will be studied graphically for more illustration.

1. Introduction

It has been studied that there exist several strains of corona virus in different regions of the world. Among these, MERS-Cov is an emerging one, especially in Arab countries. In the Kingdom of Saudi Arabia and South Korea, the disease was inspected in 2012 and May–June 2015, respectively [36]. According to [14], the MERS-Cov, which spread in 2015, is the 2nd largest outbreak. Initially, the outbreak in South Korea resolved in more than 150 infections, and 36 patients died. The government immediately isolated at least 16 693 people showing suspicious symptoms [18]. The nature of coronavirus strains shows that if this special kind, MERS-Cov, is ignored and not given ample attention by researchers, pharmacists, and scientists, a chaotic situation can occur like COVID-19. Researchers and pharmacists are struggling to invent a vaccine for MERS-Cov, but till now, no particular treatment has been suggested, nor vaccine has been manufactured [24]. Apparently, the first aid in the treatment of MERS-Cov patients is to give supportive care and symptomatic treatment. Furthermore, patients should be isolated from crowed properly, and carers should use safeguard equipment to avoid diseases. Researchers, scientists, and pharmacists always work for humanity and try to develop standard operating procedures and manufacture antidotes for new emerging viral infections. Mathematical modeling is also playing an important role in the thick of these widespread efforts. The purpose of mathematical modeling is to show when a disease is emerge, when it will increase, when it reaches its peak, and, if possible, when it starts to decrease. Also, it helps to explore the facts that influence the disease. Moreover, mathematical modeling shows sensitivity analysis and describes the impact of optimal control strategies (If they exist) for controlling the disease. Recently many authors studied the dynamics of MERS-Cov in a mathematical framework. The researchers in  [6] analyzed a compartmental model. Epidemiological data on the propagation of viral disease that outbreak in 2013 in the Kingdom of Saudi Arabia were studied in [6]. The model incorporated community and hospital compartments and distinguished index transmission among humans.

Afterwards, in [5] the properties and various dynamics of MERS-Cov and SARS were studied through a statistical approach. The researcher’s endeavor in [13] provides some insight into non-social outbreaks in South Korea via data fitting and taking a closed form of solution. To formulate different ways for selective detection and differentiability to diagnose a case of MERS-Cov with upper respiratory symptoms among imported cases. Depending upon the known incubation interval, the author of [13] adopted probabilistic modeling and Bayesian approaches, which helped in the differential diagnosis. Later, new cases and transmission dynamics were estimated by authors in [23]. A compartmental model of the discrete version was formulated to explore spatial heterogeneity in [26]. Studying the mathematical review of MERS-Cov, another work can be found in  [6]. When Chowell et al. studied the epidemic of the MERS-Cov model incorporating index as well as secondary cases, it was found in [8] that the factors which influence the transmission of MERS-Cov depend on the history of incubation time upon the outbreak of a new viral disease and its initial growth. Malik et al. [21] studied MERS-Cov using deterministic modeling and accommodating infectious individuals. The formulation of their model depends upon dividing humans into two mutually exclusive sets tagged as visitors and residents. The findings showed that effective contact rates and deaths induced by the epidemics are among the influential parameters that help determine the spread of MERS-Cov. Another proposal for integrative maximum likelihood for assessing the MERS spread scenario and incidence of sporadic infection was given by poletto et al. in [25]. Using ordinary differential equations, the epidemic of MERS-Cov from May to July 2015 has also been modeled by authors in [37]. For some interesting scientific works in relation to endemic equilibria, MERS, Covid-19, Zika virus, modelling perceptive-based information stability of various differential equations of higher order, stochastic models, stochastically perturbed SIR and SEIR epidemic models, etc., we refer the readers to look at the sources [3], [4], [7], [9], [12], [16], [19], [20], [27], [28], [29], [30], [31], [32], [33], [34], [38], [39] and their references.

Remark 1

The complicated dynamics of infectious diseases are subject to many biologically important questions, which are assisted by appropriate mathematical models. The parameters of determinist MERS-Cov models are supposed to be absolute constants, with the caveat that environmental noise will always affect them. Among the essential elements of the environment, environmental noise has the greatest impact on MERS-Cov models. Due to frequent environmental fluctuations, the parameters used in MERS-Cov oscillate around average levels. Due to the uncertain interferences of ecological factors, the parameters utilized in deterministic models are not absolute constants. Therefore, despite the fact that traditional deterministic mathematical models can forecast a system’s dynamical characteristics with accuracy, they are constrained. To take into account the impact of an altering environment, several authors have created epidemic models with some parameters altered. The assumption of Gaussian noise in the SDE setting is appropriate for this model because physical noise can be estimated adequately by white noise in such epidemiological problems. Assume that the memory span of the noise affecting mathematical models is finite. In that case, finding an approximating system confused by the Gaussian white noise is frequently accomplished by varying the time scale.

1.1. Hypothesis

• •$\Delta$ and $\Lambda$ are the recruitment ratio of the human and camel population, respectively.
• •Human susceptible become infected after interaction with Human infected individuals.
• •Human susceptible become infected after interaction with infected camels.
• •After successful treatment, the infected individuals go to recovery class.
• •susceptible camel become infected after interaction with infectious camels.

1.2. Formulation of stochastic model

Here, a mathematical modal for the transmission of the MERS-Cov virus and the nature of transmission of individuals among different compartments will be discussed. The total population is divided into three categories, susceptible, infected, and recovered individuals. The susceptible and infected individuals of the human population are represented by $S_h,I_h$, respectively, while the camel population is represented by $S_c,I_c$. $R_h$ represents the recovered individuals. Thus the deterministic version of MERS-Cov is governed by the following set of differential equations;

$$\frac{d{S}_h}{dt}=\Delta -{\beta }_1{S}_h{I}_h-{\beta }_2{S}_h{I}_c-{\mu }_h{S}_h,$$

$$\frac{d{I}_h}{dt}={\beta }_1{S}_h{I}_h+{\beta }_2{S}_h{I}_c-({\mu }_1+\varphi +{\mu }_h)I_h,$$

$$\frac{d{R}_h}{dt}=\varphi I_h-{\mu }_h{R}_h,$$

$$\frac{d{S}_c}{dt}=\Lambda -{\beta }_3{S}_c{I}_c-{\mu }_c{S}_c,$$

$$\frac{d{I}_c}{dt}={\beta }_3{S}_c{I}_c-({\mu }_c+{\mu }_2)I_c,$$ (1)

with initial condition

$$S_h\left(0\right)=S_h^0,I_h\left(0\right)=I_h^0,R_h\left(0\right)=R_h^0,S_c\left(0\right)=S_c^0,I_c\left(0\right)=I_c^0.$$

The dynamics of an epidemic are inexorably influenced by environmental noise, which is essential in the eco-system. In reality epidemic modeling depends on environmental noise. The parameters used in constituting governing equations of an epidemic are not assumed to be absolute constants rather, these parameters fluctuate around some average values. Incorporating random noise in a model turns it into a more complex but realistic one compared to its deterministic counterpart. Recently several authors investigated epidemic models, which took into account environmental noises [2], [11]. Yang et al. considered saturated incidence rates and added random noise into $SIR$ and $SEIR$ models and explored their underlying model’s dynamics depend upon threshold quantity, usually denoted by $R_0$. From a biological and mathematical point of view, various possible approaches exist that result in many effects on the population system by adding random noise in governing equations of the model. Usually, researchers adopt the famous four approaches  [15]. Time Markov chain approach is the first, and parameter perturbation is the second. The third one is to take environmental noise, which is proportional to the used variables, and the last approach is too robust to the positive equilibria of primitive deterministic models.

After discussing the literature review about MERS-Cov and the inclusion of random noise in a model, we present a stochastic version of the MERS-Cov model. The approach that will be followed for our new stochastic model will be stochastic perturbation, which is supposed to be a white noise type proportional to the compartments $S_h$, $I_h$, $S_c$, and $I_c$. In this paper, we will consider a stochastic counterpart of the model (1) by the third approach. That is, the stochastic perturbation is assumed to be of a white noise type, which is directly proportional to $S_h\left(t\right),I_h\left(t\right),S_c\left(t\right)$, and $I_c\left(t\right)$, and influenced by the ${\dot{S}}_h\left(t\right),{\dot{I}}_h\left(t\right),{\dot{S}}_c\left(t\right)$, and ${\dot{I}}_c\left(t\right)$, in model (1). In this way, a reasonable stochastic analog of the system (1) is given by

$$\frac{d{S}_h}{dt}=\Delta -{\beta }_1{S}_h{I}_h-{\beta }_2{S}_h{I}_c-{\mu }_h{S}_h+{\xi }_1{S}_{h}d{B}_1\left(t\right),$$

$$\frac{d{I}_h}{dt}={\beta }_1{S}_h{I}_h+{\beta }_2{S}_h{I}_c-({\mu }_h+{\mu }_1+\varphi )I_h+{\xi }_2{I}_{h}d{B}_2\left(t\right),$$

$$\frac{d{S}_c}{dt}=\Lambda -{\beta }_3{S}_c{I}_c-{\mu }_c{S}_c+{\xi }_3{S}_{c}d{B}_3\left(t\right),$$

$$\frac{d{I}_c}{dt}={\beta }_3{S}_c{I}_c-({\mu }_c+{\mu }_2)I_c+{\xi }_4{I}_{c}d{B}_4\left(t\right).$$ (2)

We will use some notations to represent different rates and ratios in the main model (1) that are given as; $\Delta$ shows birth rate in human population, while ${\beta }_1$ similarly shows the contact rate between susceptible and infected human individuals, the symbol ${\mu }_h$ is used to denote natural death rate in susceptible human population, $\varphi$ represent the recovery rate of human population, ${\mu }_1$ represents disease related death rate of infected human individuals, $\Lambda$ represent new birth rate in camels population, ${\mu }_c$ represent natural death rate in camels, ${\beta }_3$ is the contact ratio between the susceptible and infected camels, more important when the susceptible human individual touch the infected camels in any expect like, blood, wound, saliva or sex the individuals get infection and the rate of that infection is represented by ${\beta }_2$. The feasible region for the system (2) is $\Gamma =\{(S_h,I_h,S_c,I_c)\in R_{+}^4,S_h\ge 0,I_h\ge 0,S_c\ge 0,I_c\ge 0,S_h+I_h\frac{\Delta }{{\mu }_h},S_c+I_c\le \frac{\Lambda }{{\mu }_c}\}$ and it is not tedious to see that it is positive invariant set for the system (2).

2. Supportive materials

Definition 1 [22] —

“A stochastic process is a collection of random variables $\left\{\mathit{X(t)}\right\}$. For any fixed $\mathit{t}=0,1,\dots ,\mathit{T},\mathit{X(t)}$ is a random variable on $\left(\Omega ,{\mathcal{F}}_T\right)$. A stochastic process is called adapted to filtration $\mathbb{F}$ if for all $t=0,1,\dots ,\mathit{T}$, $\mathit{X(t)}$ is a random variable on ${\mathcal{F}}_t$, that is, if $\mathit{X(t)}$ is ${\mathcal{F}}_t-$ measurable”.

Before we move towards the main part of the paper, we here suppose that the mathematical expression, $\left({\mathbb{R}}_{+}^l,\mathcal{B}\left({\mathbb{R}}_{+}^l\right),\left\{{\mathcal{F}}_{t\ge 0}\right\},\mathbb{P}\right)$ will denote complete probability space, while $\left\{{\mathcal{F}}_{t\ge 0}\right\}$ is filtration taken with the space. It will satisfy the condition of right continuity and increasing behavior as well as contains all $\mathbb{P}-$null sets. In addition to this, we also assume the following stochastic differential equation of dimension $l$.

$$dY\left(t\right)={\Phi }_1\left(Y\left(t\right)\right)dt+{\Phi }_2\left(Y\left(t\right)\right)dB,Y\left(0\right)=Y_0\in {\mathbb{R}}^l,t\ge t_0,$$ (3)

where a standard $\mathit{l}$-dimensional Brownian motion is represented by $\mathit{B(t)}$, and is defined on a complete probability space $({\mathbb{R}}_{+}^3,\mathcal{B}\left({\mathbb{R}}_{+}^3\right),\left\{{\mathcal{F}}_{t\ge 0}\right\},\mathbb{P})$. Denote by $\mathit{V}\in C^{2,1}({\mathbb{R}}^3\times [t_o,\infty ];{\mathbb{R}}_{+})$ the family of all non-negative functions $\mathit{V\; (Y,t)}$ defined on ${\mathbb{R}}^3\times [t_o,\infty ]$ such that they are continuously twice differentiable in $\mathit{Y}$ and once in $\mathit{t}$, then the differential operator $\mathit{L}$ of (3) is defined by [22],

$$“L=\frac{\partial }{\partial t}+\sum _{i=1}^3{{\Phi }_1}_i\left(Y\right)\frac{\partial }{\partial Y_i}+\frac{1}{2}\sum _{i,j=1}^3{\left[g^T(Y,t)g(Y,t)\right]}_{ij}\frac{{\partial }^2}{\partial Y_i\partial Y_j}.$$

If $\mathit{L}$ acts on a function $\mathit{V}\in C^{2,1}({\mathbb{R}}^3\times [t_o,\infty ];{\mathbb{R}}_{+})$, then

$$LV(Y,t)=V_t(Y,t)+V_Y(Y,t)f(Y,t)+\frac{1}{2}trace\left[g^T(Y,t)V_{YY}g(Y,t)\right],$$

where $V_t=\partial V/\partial t,V_Y=(\partial V/\partial y_1,\dots ,\partial V/\partial y_3)$ and $V_{YY}={({\partial }^{2}V/\partial y_i\partial y_j)}_{3\times 3}$.

In view of $It\hat{o}$’s formula, if $\mathit{Y}\left(t\right)\in {\mathbb{R}}^l$, then

$$dV(Y,t)=LV(Y\left(t\right),t)+V_Y(Y\left(t\right),t)g(X\left(t\right),t)dB\left(t\right).$$ (4)

A homogeneous Markov Process in the $l$-dimensional Euclidean space is represented by $X\left(t\right)$, which is given as;

$$dX\left(t\right)=b\left(X\right)dt+\sum _{r=1}^k{g}_{r}d{B}_r\left(t\right).$$

Here the relative diffusion matrix is given as:

$$\Lambda =\left({\lambda }_{ij}\left(x\right)\right),{\lambda }_{ij}\left(x\right)=\sum _{r=1}^k{g}_r^i\left(x\right)g_r^j\left(x\right).$$

Lemma 1 [17] —

“The SDE (3) has a unique stationary distribution if there is a bounded open subset $O$ of ${\mathbb{R}}_{+}^3$ with a regular (i.e. smooth) boundary such that its closure $\overline{O}\subset {\mathbb{R}}_{+}^3$ and

• (1) ${inf}_{x\in O}{\lambda }_{min}\left(diag(S,E,I)\right)\sigma {\sigma }^{T}diag(S,E,I)>0$ , where $\sigma ={({\sigma }_1,{\sigma }_2,{\sigma }_3)}^T;$
• (2) ${sup}_{x\left(0\right)\in K-O}E\left({\tau }_O\right)<\infty$ for every compact subset K of ${\mathbb{R}}_{+}^3$ such that $O\subset K$ , where ${\tau }_O=inf\{t\ge 0(S\left(t\right),E\left(t\right),I\left(t\right))\in O\}.”$

3. Positivity and global existence of the solution

Initially it will be shown that the solution of the underlying model (2) is positive and global. Further to prove that no explosion in a finite time occur in the model, we need to prove that for any given initial value of the independent variable, all coefficients appearing in the model (2) need to fulfill the linear growth and local Lipschitz conditions [22]. It can be seen that the coefficients of system (2) do not satisfy linear growth condition. However, the coefficients satisfy local Lipschitz continuous function definition, which suggests that solution of the model (2) can explode in finite interval of time. Using the method of Lyapunov analysis given in [10], it will be shown that solution of model (2) is positive and global in nature.

Theorem 1

For $t\in \left(0\infty \right)$ and with some initial subsidiary condition, $(S_h\left(0\right),I_h\left(0\right),S_c\left(0\right),I_c\left(0\right))$ the model (2) possess unique solution and will entirely lie in the space ${\mathbb{R}}_{+}^4$ with probability 1.

Proof

It is not tedious to show that the coefficients of model (2) satisfy the definition of locally Lipschitz continuous function, for any given condition

$(S_h\left(t\right),I_h\left(t\right),S_c\left(t\right),I_c\left(t\right))\in {\mathbb{R}}_{+}^4$. Moreover, there exist a unique solution

$(S_h\left(t\right),I_h\left(t\right),S_c\left(t\right),I_c\left(t\right))$ for $0\le t<{\tau }_e$, here, $“{\tau }_e”$ represents explosion time [22]. In order to achieve global solution, it need to prove that $“{\tau }_e”$ approaches to infinity almost surly. To do this let us assume a non-negative value of ${\kappa }_0$, which is sufficiently large such that $(S_h\left(0\right),I_h\left(0\right),S_c\left(0\right),I_c\left(0\right))\in [\frac{1}{{\kappa }_0},{\kappa }_0]$. To proceed further we define stopping time as follow:

$${\tau }_{\kappa }=inf\left\{t\in [0,\tau ):(S_h\left(t\right),I_1\left(t\right),S_c\left(t\right),I_2\left(t\right))\notin \left(\frac{1}{\kappa },\kappa \right)\right\}.$$

From the definition of stopping time we can easily infer that ${\tau }_{\kappa }$ increases to infinity. We set ${\tau }_{\infty }={lim}_{\kappa \to \infty }{\tau }_{\kappa }$, where ${\tau }_{\infty }\le {\tau }_e$ a.s. To complete our proof we need to reach the fact that ${\tau }_{\infty }=\infty$ a.s. On contrary we suppose that there exist constants $T>0$ and $\varepsilon \in (0,1)$ such that $P\{{\tau }_{\infty }-T\le 0\}>\varepsilon$. Hence, there is an integer ${\kappa }_0\le {\kappa }_1$ such that

$$\varepsilon \le P\{{\tau }_{\infty }-T\le 0\}forall{\kappa }_1\le \kappa .$$ (5)

Now we define a function $V_1:{\mathbb{R}}_{+}^4\to {\mathbb{R}}_{+}^4$:

$$V(S_h,I_1,S_c,I_2)=(S_h-1-log{S}_h)+(I_1-1-log{I}_1)+(S_c-1-log{S}_c)+(I_2-1-log{I}_2).$$

Now before we use $It\hat{o}$’s formula; for connivance we independently evaluate all terms of $It\hat{o}$’s formula;

$$V_{x}f=\left[\begin{array}{cccc}\hfill 1-\frac{1}{S_h}\hfill & \hfill 1-\frac{1}{I_1}\hfill & \hfill 1-\frac{1}{S_c}\hfill & \hfill 1-\frac{1}{I_2}\hfill \end{array}\right]\left[\begin{array}{c}\hfill f_1\hfill \\ \hfill f_2\hfill \\ \hfill f_3\hfill \\ \hfill f_4\hfill \end{array}\right].$$

$$V_{x}f=\left(1-\frac{1}{S_h}\right)f_1+\left(1-\frac{1}{I_1}\right)f_2+\left(1-\frac{1}{S_c}\right)f_3+\left(1-\frac{1}{I_2}\right)f_4=\left(\sum _{k=1}^4{f}_k\right)-\left(\frac{f_1}{S_h}+\frac{f_2}{I_1}+\frac{f_3}{S_c}+\frac{f_4}{I_2}\right)=(\Delta -{\mu }_h{S}_h-({\mu }_h+{\mu }_1)I_h-\varphi I_h+\Lambda -{\mu }_c{S}_c-({\mu }_c+{\mu }_2)I_c)-\left(\frac{\Delta }{S}-{\beta }_1{I}_h-{\beta }_2{I}_c-{\mu }_h+{\beta }_1{S}_h++\frac{{\beta }_2{S}_h{I}_c}{I_h}-({\mu }_h+{\mu }_1)-\varphi +\frac{\Lambda }{S_c}-{\beta }_3{I}_c-{\mu }_c+{\beta }_3{S}_c-({\mu }_c+{\mu }_2)\right).=-\frac{\Delta }{S}-\frac{{\beta }_2{S}_h{I}_c}{I_h}-\frac{\Lambda }{S_c}-{\mu }_h{S}_h-({\mu }_h+{\mu }_1)I_h-\varphi I_h-{\mu }_c{S}_c-({\mu }_c+{\mu }_2)I_c-{\beta }_1{S}_h-{\beta }_3{S}_c+{\beta }_1{I}_h++{\beta }_2{I}_c+{\beta }_3{I}_c+{\mu }_c+{\mu }_2+({\mu }_h+{\mu }_1)+\varphi +{\mu }_h+\Delta +\Lambda +{\mu }_c.$$ (6)

$$g^T{\mathcal{V}}_{xx}g=\left[\begin{array}{cccc}\hfill {\xi }_1{S}_h\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\xi }_2{I}_h\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_3{S}_c\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_4{S}_c\hfill \end{array}\right]\left[\begin{array}{cccc}\hfill \frac{1}{S_h^2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{I_h^2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{S_c^2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{I_c^2}\hfill \end{array}\right]\left[\begin{array}{cccc}\hfill {\xi }_1{S}_h\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\xi }_2{I}_h\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_3{S}_c\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_4{S}_c\hfill \end{array}\right].$$

$$g^T{\mathcal{V}}_{xx}g=\left[\begin{array}{cccc}\hfill \frac{{\xi }_1^2{S}_h^2}{S_h^2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{{\xi }_2^2{I}_h^2}{I_h^2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{{\xi }_3^2{S}_c^2}{S_c^2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{{\xi }_4^2{I}_c^2}{I_c^2}\hfill \end{array}\right].$$

$$\frac{1}{2}trace\left(g^T{\mathcal{V}}_{xx}g\right)=\frac{{\xi }_1^2}{2}+\frac{{\xi }_2^2}{2}+\frac{{\xi }_3^2}{2}+\frac{{\xi }_4^2}{2}.$$ (7)

$$V_{x}g=\left[\begin{array}{cccc}\hfill 1-\frac{1}{S_h}\hfill & \hfill 1-\frac{1}{I_h}\hfill & \hfill 1-\frac{1}{S_c}\hfill & \hfill 1-\frac{1}{I_c}\hfill \end{array}\right]\left[\begin{array}{c}\hfill g_{1}d{B}_1\hfill \\ \hfill g_{2}d{B}_2\hfill \\ \hfill g_{3}d{B}_3\hfill \\ \hfill g_{4}d{B}_4\hfill \end{array}\right].$$

$$V_{x}gdB=\left(1-\frac{1}{S_h}\right)g_{1}d{B}_1+\left(1-\frac{1}{I_h}\right)g_{2}d{B}_2+\left(1-\frac{1}{S_c}\right)g_{3}d{B}_3+\left(1-\frac{1}{I_c}\right)g_{4}d{B}_4,=\left(\frac{S_h-1}{S_h}\right){\xi }_1{S}_{h}d{B}_1+\left(\frac{I_h-1}{I_h}\right){\xi }_2{I}_{h}d{B}_2+\left(\frac{S_c-1}{S_c}\right){\xi }_3{S}_{c}d{B}_3+\left(\frac{I_c-1}{I_c}\right){\xi }_4{I}_{c}d{B}_4,=(S_h-1){\xi }_{1}d{B}_1+(I_h-1){\xi }_{2}d{B}_2+(S_c-1){\xi }_{3}d{B}_3+(I_c-1){\xi }_{4}d{B}_4.$$ (8)

Substituting (6),(7),(8) in (4), we get

$$dV=LV+(S_h-1){\xi }_{1}d{B}_1+(I_h-1){\xi }_{2}d{B}_2+(S_c-1){\xi }_{3}d{B}_3+(I_c-1){\xi }_{4}d{B}_4.$$ (9)

$$LV=-\frac{\Delta }{S}-\frac{{\beta }_2{S}_h{I}_c}{I_h}-\frac{\Lambda }{S_c}-{\mu }_h{S}_h-({\mu }_h+{\mu }_1)I_h-\varphi I_h-{\mu }_c{S}_c-({\mu }_c+{\mu }_2)I_c-{\beta }_1{S}_h-{\beta }_3{S}_c+{\beta }_1{I}_h++{\beta }_2{I}_c+{\beta }_3{I}_c+{\mu }_c+{\mu }_2+({\mu }_h+{\mu }_1)+\varphi +{\mu }_h+\Delta +\Lambda +{\mu }_c+\frac{{\xi }_1^2}{2}+\frac{{\xi }_2^2}{2}+\frac{{\xi }_3^2}{2}+\frac{{\xi }_4^2}{2},\le {\beta }_1{I}_h+{\beta }_2{I}_c+{\beta }_3{I}_c+{\mu }_c+{\mu }_2+({\mu }_h+{\mu }_1)+\varphi +{\mu }_h+\Delta +\Lambda +{\mu }_c+\frac{{\xi }_1^2}{2}+\frac{{\xi }_2^2}{2}+\frac{{\xi }_3^2}{2}+\frac{{\xi }_4^2}{2},\le \frac{{\beta }_1\Delta }{{\mu }_h}+\frac{{\beta }_2\Lambda }{{\mu }_c}+\frac{{\beta }_3\Lambda }{{\mu }_c}+2{\mu }_c+{\mu }_2+2{\mu }_h+{\mu }_1+\varphi +\Delta +\Lambda +\frac{{\xi }_1^2}{2}+\frac{{\xi }_2^2}{2}+\frac{{\xi }_3^2}{2}+\frac{{\xi }_4^2}{2},\le \Lambda +2{\mu }_c+{\mu }_2+2{\mu }_h+{\mu }_1+\varphi +\Delta +\frac{{\beta }_1\Delta }{{\mu }_h}+\frac{{\beta }_2\Lambda }{{\mu }_c}+\frac{{\beta }_3\Lambda }{{\mu }_c}+\frac{{\xi }_1^2}{2}+\frac{{\xi }_2^2}{2}+\frac{{\xi }_3^2}{2}+\frac{{\xi }_4^2}{2},=\mathcal{C}.$$

where $\mathcal{C}=\Lambda +2{\mu }_c+{\mu }_2+2{\mu }_h+{\mu }_1+\varphi +\Delta +\frac{{\beta }_1\Delta }{{\mu }_h}+\frac{{\beta }_2\Lambda }{{\mu }_c}+\frac{{\beta }_3\Lambda }{{\mu }_c}+\frac{{\xi }_1^2}{2}+\frac{{\xi }_2^2}{2}+\frac{{\xi }_3^2}{2}+\frac{{\xi }_4^2}{2}$.

Integrating both sides of Eq. (9) from 0 to ${\tau }_{\kappa }\wedge T$, we get the following

$${\int }_0^{{\tau }_{\kappa }\wedge T}dV\left(S_h\left(r\right),I_h\left(r\right),S_c\left(r\right),I_c\left(r\right)\right)\le {\int }_0^{{\tau }_{\kappa }\wedge T}Cdr+{\int }_0^{{\tau }_{\kappa }\wedge T}(S_h-1){\sigma }_{1}d{B}_1,+{\int }_0^{{\tau }_{\kappa }\wedge T}(I_h-1){\sigma }_{2}d{B}_2,+{\int }_0^{{\tau }_{\kappa }\wedge T}(S_c-1){\sigma }_{3}d{B}_3,+{\int }_0^{{\tau }_{\kappa }\wedge T}(I_c-1){\sigma }_{4}d{B}_4.$$

here ${\tau }_{\kappa }\wedge T=min\{{\tau }_{\kappa },T\}$. Now the expectations of the preceding inequality leads to

$$EV\left(S_h({\tau }_{\kappa }\wedge T),I_h({\tau }_{\kappa }\wedge T),S_c({\tau }_{\kappa }\wedge T),I_c({\tau }_{\kappa }\wedge T)\right)\le V(S_h\left(0\right),I_h\left(0\right),S_c\left(0\right),I_c\left(0\right))+hT.$$

Set ${\Im }_{\kappa }=\{{\tau }_{\kappa }\le T\}$ for $\kappa \ge {\kappa }_1$, and from (5), we have $\rho \left({\Im }_{\kappa }\right)\ge \varepsilon$. Note that for every $\omega \in {\Im }_{\kappa }$, there is at least one term among these three $S({\tau }_{\kappa }\wedge T),U_1({\tau }_{\kappa }\wedge T),U_2({\tau }_{\kappa }\wedge T)$  and  $U_2({\tau }_{\kappa }\wedge T)$ which is equal to either $\kappa$ or $\frac{1}{\kappa }$, hence

$$V(S_h({\tau }_{\kappa }\wedge T),I_h({\tau }_{\kappa }\wedge T),S_c({\tau }_{\kappa }\wedge T),I_c({\tau }_{\kappa }\wedge T))\ge \left((\kappa -1-ln\kappa )\wedge (\frac{1}{\kappa }-1-ln\frac{1}{\kappa })\right).$$

Thus, implies that,

$$V(S_h\left(0\right),I_h\left(0\right),S_c\left(0\right),I_c\left(0\right))+hT\ge E\left[I_{{\Im }_{\kappa }\left(\omega \right)}V(S_h\left({\tau }_{\kappa }\right),I_h\left({\tau }_{\kappa }\right),S_c\left({\tau }_{\kappa }\right),I_c\left({\tau }_{\kappa }\right))\right]\ge \varepsilon \left((\kappa -1-ln\kappa )\wedge (\frac{1}{\kappa }-1-ln\frac{1}{\kappa })\right).$$

Here $I_{{\Im }_{\kappa }\left(\omega \right)}$ shows indicator function of ${\Im }_{\kappa }\left(\omega \right)$. Aftermath, if $\kappa$ is allowed to reach $\infty$, then we can easily infer that

$$\infty >V(S_h\left(0\right),I_h\left(0\right),S_c\left(0\right),I_c\left(0\right))+hT=\infty a.s.$$

Hence, an impossible situation has created. Therefore, it need to admit that ${\tau }_{\infty }=\infty$. Eventually, at finite time with probability 1, the underlying model (2) explode.

4. Moment exponential stability

Among different aspects of a system of stochastic differential equations, moment exponential stability is an important one [1]. Therefore, in this part of the paper some results will be provided for exploring the moment exponential stability of the equilibrium of the model (2) via Lyapunov function (see [1]).

Theorem 2

Assume that a function $V(t,x)\in C^{1,2}\text{(}\mathbb{R}\times {\mathbb{R}}^n$ ) exist, which obeys the following relation,

$${\mathcal{K}}_1\le V(t,x)\le {\mathcal{K}}_2{\left|x\right|}^p,$$

$$LV(t,x)\le -{\mathcal{K}}_3{\left|x\right|}^p,{\mathcal{K}}_2>0,p>0.$$

Then for the model (2) the equilibrium point is $pth$ moment exponentially stable. Moreover, if $p=2$ , then exponential stability in mean square occur and the equilibrium $X=0$ is globally asymptotically stable.

Remark 2

Assuming conjugate numbers $p,q>0$ and $\frac{1}{p}+\frac{1}{q}=1$, as well as using Young’s inequality, i.e., $xy\le \frac{x^p}{p}+\frac{y^q}{q}$ for $x,y>0$, one can obtain the inequality given below:

Lemma 2 [1] —

“Let $p\ge 2$ and $\phi ,x,y>0$ . Then

$$x^{p-1}y\le \frac{(p-1)\phi }{p}{x}^p+\frac{1}{p{\phi }^{p-1}}{y}^p,$$

$$x^{p-2}{y}^2\le \frac{(p-2)\phi }{p}{x}^p+\frac{1}{p{\phi }^{\frac{p-2}{2}}}{y}^p.”$$ (10)

Theorem 3

Assume that $p\ge 2$ , then the disease-free equilibrium $P^0$ of the model (2) is $pth$ - moment exponentially stable in $\Gamma$ .

Proof

Consider that $p\ge 2$ and $(S_h\left(0\right),I_h\left(0\right),S_c\left(0\right),I_c\left(0\right))\in \Gamma$, then by Theorem 1 the solution of the underlying model (2) entirely lies in $\Gamma$. Before we move forward we her consider a Lyapunov function as follows:

$$g^T{\mathcal{G}}_{xx}g=\left[\begin{array}{cccc}\hfill {\xi }_1{S}_h\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\xi }_2{I}_h\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_3{S}_c\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_4{S}_c\hfill \end{array}\right]\times \left[\begin{array}{cccc}\hfill a_{1}p(p-1){\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^{p-2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill a_{2}p{I_h}^{p-2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill a_{3}p(p-1){\left(\frac{\Lambda }{{\mu }_c}-S_c\right)}^{p-2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill a_{4}p{I_c}^{p-2}\hfill \end{array}\right]\times \left[\begin{array}{cccc}\hfill {\xi }_1{S}_h\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\xi }_2{I}_h\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_3{S}_c\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_4{S}_c\hfill \end{array}\right].$$

$$g^T{\mathcal{G}}_{xx}g=\left[\begin{array}{cccc}\hfill {\xi }_1^2{S}_h^2{a}_{1}p(p-1){\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^{p-2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\xi }_2^2{I}_h^2{a}_{2}p{I_h}^{p-2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_3^2{S}_c^2{a}_{3}p(p-1){\left(\frac{\Lambda }{{\mu }_c}-S_c\right)}^{p-2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_4^2{I}_c^2{a}_{4}p{I_c}^{p-2}\hfill \end{array}\right].$$

$$\frac{1}{2}trac{g}^T{\mathcal{G}}_{xx}g=\frac{{\xi }_1^2{a}_{1}p(p-1)S_h^2}{2}{\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^{p-2}+\frac{{\xi }_2^2{a}_{2}p}{2}{I_h}^p+\frac{{\xi }_3^2{a}_{3}p(p-1)S_c^2}{2}{\left(\frac{\Lambda }{{\mu }_c}-S_c\right)}^{p-2}+\frac{{\xi }_4^2{a}_{4}p}{2}{I_c}^p.$$

Next,

$$L\mathcal{G}=-a_{1}p{\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^p-a_{1}p{\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^{p-1}{\beta }_1{S}_h{I}_h-a_{1}p{\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^{p-1}{\beta }_2{S}_h{I}_c+{\beta }_1{a}_2{I_h}^p{S}_h+\frac{{\xi }_4^2{a}_{4}p}{2}{I_c}^p+{\beta }_2{a}_2{I_h}^{p-1}{S}_h{I}_c-({\mu }_h+{\mu }_1+\varphi )a_2{I_h}^p-a_{3}p{\left(\frac{\Lambda }{{\mu }_c}-S_c\right)}^p-a_{3}p{\left(\frac{\Lambda }{{\mu }_c}-S_c\right)}^{p-1}{\beta }_3{S}_c{I}_c+a_4{I_c}^p{\beta }_3{S}_c-({\mu }_c+{\mu }_2)a_4{I_c}^p+\frac{{\xi }_1^2{a}_{1}p(p-1)S_h^2}{2}{\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^{p-2}+\frac{{\xi }_2^2{a}_{2}p}{2}{I_h}^p+\frac{{\xi }_3^2{a}_{3}p(p-1)S_c^2}{2}{\left(\frac{\Lambda }{{\mu }_c}-S_c\right)}^{p-2}.$$

Using the inequality (10), we can get the following:

$${\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^{p-1}{I}_h\le \frac{(p-1)\phi }{p}{\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^p+\frac{1}{p{\phi }^{p-1}}{I_h}^p,$$

$${\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^{p-1}{I}_c\le \frac{(p-1)\phi }{p}{\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^p+\frac{1}{p{\phi }^{p-1}}{I_c}^p,$$

$${I_h}^{p-1}{I}_c\le \frac{(p-1)\phi }{p}{I_h}^p+\frac{1}{p{\phi }^{p-1}}{I_c}^p,$$

$${\left(\frac{\Lambda }{{\mu }_c}-S_c\right)}^{p-1}{I}_c\le \frac{(p-1)\phi }{p}{\left(\frac{\Delta }{{\mu }_h}-S_c\right)}^p+\frac{1}{p{\phi }^{p-1}}{I_c}^p.$$

Therefore,

$${\mathcal{G}}_{x}f\le -a_{1}p{\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^p-a_{1}p{\beta }_1{S}_h\left(\frac{(p-1)\phi }{p}{\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^p+\frac{1}{p{\phi }^{p-1}}{I_h}^p\right)+{\beta }_1{a}_2{I_h}^p{S}_h+\frac{{\xi }_2^2{a}_{2}p}{2}{I_h}^p+-a_{1}p{\beta }_2{S}_h\left(\frac{(p-1)\phi }{p}{\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^p+\frac{1}{p{\phi }^{p-1}}{I_c}^p\right)+{\beta }_2{a}_2{S}_h\left(\frac{(p-1)\phi }{p}{I_h}^p+\frac{1}{p{\phi }^{p-1}}{I_c}^p\right)+-({\mu }_h+{\mu }_1+\varphi )a_2{I_h}^p-a_{3}p{\left(\frac{\Lambda }{{\mu }_c}-S_c\right)}^p-a_{3}p{\beta }_3{S}_c\left(\frac{(p-1)\phi }{p}{\left(\frac{\Delta }{{\mu }_h}-S_c\right)}^p+\frac{1}{p{\phi }^{p-1}}{I_c}^p\right)++a_4{I_c}^p{\beta }_3{S}_c-({\mu }_c+{\mu }_2)a_4{I_c}^p+\frac{{\xi }_1^2{a}_{1}p(p-1)S_h^2}{2}{\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^{p-2}+\frac{{\xi }_3^2{a}_{3}p(p-1)S_c^2}{2}{\left(\frac{\Lambda }{{\mu }_c}-S_c\right)}^{p-2},\le -a_{1}p{\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^p-\frac{\Delta a_{1}p{\beta }_1(p-1)\phi }{p{\mu }_h}{\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^p-\frac{a_{1}p{\beta }_2\Delta (p-1)\phi }{{\mu }_{h}p}{\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^p-\frac{\Delta a_{1}p{\beta }_1}{{\mu }_{h}p{\phi }^{p-1}}{I_h}^p++\frac{{\beta }_1{a}_2\Delta }{{\mu }_h}{I_h}^p+\frac{{\beta }_2{a}_2\Delta (p-1)\phi }{{\mu }_{h}p}{I_h}^p-({\mu }_h+{\mu }_1+\varphi )a_2{I_h}^p-\frac{a_{3}p{\beta }_3\Lambda (p-1)\phi }{{\mu }_{c}p}{\left(\frac{\Delta }{{\mu }_h}-S_c\right)}^p+-a_{3}p{\left(\frac{\Lambda }{{\mu }_c}-S_c\right)}^p+\frac{{\beta }_2{a}_2\Delta }{{\mu }_{h}p{\phi }^{p-1}}{I_c}^p-\frac{a_{1}p{\beta }_2\Delta }{{\mu }_{h}p{\phi }^{p-1}}{I_c}^p-\frac{a_{3}p{\beta }_3\Lambda }{{\mu }_{c}p{\phi }^{p-1}}{I_c}^p+\frac{a_4{\beta }_3\Lambda }{{\mu }_c}{I_c}^p-({\mu }_c+{\mu }_2)a_4{I_c}^p++\frac{{\xi }_1^2{a}_{1}p(p-1)S_h^2}{2}{\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^{p-2}+\frac{{\xi }_2^2{a}_{2}p}{2}{I_h}^p+\frac{{\xi }_3^2{a}_{3}p(p-1)S_c^2}{2}{\left(\frac{\Lambda }{{\mu }_c}-S_c\right)}^{p-2}+\frac{{\xi }_4^2{a}_{4}p}{2}{I_c}^p,\le -\left[a_{1}p+\frac{\Delta a_{1}p{\beta }_1(p-1)\phi }{p{\mu }_h}+\frac{a_{1}p{\beta }_2\Delta (p-1)\phi }{{\mu }_{h}p}-\frac{{\xi }_3^2{a}_{3}p(p-1){\Delta }^2}{2{\mu }_c^2}\right]{\left(\frac{\Delta }{{\mu }_h}-S_h\right)}^p+-\left[\frac{\Delta a_{1}p{\beta }_1}{{\mu }_{h}p{\phi }^{p-1}}-\frac{{\beta }_1{a}_2\Delta }{{\mu }_h}-\frac{{\beta }_2{a}_2\Delta (p-1)\phi }{{\mu }_{h}p}+({\mu }_h+{\mu }_1+\varphi )a_2-\frac{{\xi }_2^2{a}_{2}p}{2}\right]{I_h}^p+-\left[\frac{a_{3}p{\beta }_3\Lambda (p-1)\phi }{{\mu }_{c}p}+a_{3}p-\frac{{\xi }_3^2{a}_{3}p(p-1){\Lambda }^2}{2{\mu }_c^2}\right]{\left(\frac{\Lambda }{{\mu }_c}-S_c\right)}^p+-\left[\frac{a_{1}p{\beta }_2\Delta }{{\mu }_{h}p{\phi }^{p-1}}+\frac{a_{3}p{\beta }_3\Lambda }{{\mu }_{c}p{\phi }^{p-1}}+({\mu }_c+{\mu }_2)a_4-\frac{{\beta }_2{a}_2\Delta }{{\mu }_{h}p{\phi }^{p-1}}-\frac{a_4{\beta }_3\Lambda }{{\mu }_c}-\frac{{\xi }_4^2{a}_{4}p}{2}\right]{I_c}^p.$$

Here $\phi$ is given sufficiently small value and the constants $a_i,i=1,\dots ,4$ are also chosen in such a way that the coefficients of, ${\left(\frac{\delta }{{\mu }_h}-S_h\right)}^p$, ${I_h}^P$, ${\left(\frac{\Lambda }{{\mu }_c}-S_c\right)}^p$ and, ${I_c}^p$ be negative. Hence, applying Theorem 2 we obtained the required proof.

5. Stationary distribution

Studying epidemic models, mostly it is of great interest to explore that when a disease will remain in the society. To capture such circumstances in determinist modeling, we show that the disease present equilibrium point of the model under consideration is a global attractor (globally asymptotic stable). But it is an unsolved issue to explore the counterpart idea in stochastic modeling. Therefore, the system (2) do not exhibit disease present equilibrium point. Now to study a similar approach in stochastic modeling we will use the theory presented by Khasminskii [17]. It will be shown that an ergodic distribution exist, which will show that the disease will persist in population.

Theorem 4

If $2{\mu }_h>{\xi }_1^2$ , $2{\mu }_h>{\xi }_2^2$ , $2{\mu }_c>{\xi }_3^2$ , and $2{\mu }_c>{\xi }_4^2$ , hold, then the SDE (2) is ergodic.

Proof

The proof of this theorem will be segregated into two sections the first section contains the satisfying the first condition of Lemma 1 while the second section contains the second condition of the Lemma 1.

Condition $i$ in Lemma 1

First, we verify 1st condition of Lemma 1. Therefore, we suppose sufficiently large value of $\mathcal{N}$ and set

$H=\left\{(S_h,I_h,S_c,I_c)\in {\mathbb{R}}_{+}^4:\frac{1}{\mathcal{N}}<(S_h,I_h,S_c,I_c)<\mathcal{N}\right\}\subset \hat{H}\subset {\mathbb{R}}_{+}^4$ here the set, $\hat{H}$ shows the collection of all adherent points of $H$. Now it needs to verify the 1st condition of Lemma 1. The associated diffusion coefficient matrix obtains the following form;

$$A=\left[\begin{array}{cccc}\hfill {\xi }_1{S}_h\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\xi }_2{I}_h\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_3{S}_c\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_4{I}_c\hfill \end{array}\right].$$

It is known that each symmetric matrix over field of real numbers is positive definite and it is due to existence of non zero column matrix $v$ such that $v^{T}Av>0$, i.e. 

$$A=\left[\begin{array}{cccc}\hfill v_1\hfill & \hfill v_2\hfill & \hfill v_3\hfill & \hfill v_4\hfill \end{array}\right]\left[\begin{array}{cccc}\hfill {\xi }_1{S}_h\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\xi }_2{I}_h\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_3{S}_c\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_4{I}_c\hfill \end{array}\right]\left[\begin{array}{c}\hfill v_1\hfill \\ \hfill v_2\hfill \\ \hfill v_3\hfill \\ \hfill v_4\hfill \end{array}\right],$$

$$vA{v}^T=\left[\begin{array}{cccc}\hfill v_1{\xi }_1{S}_h\hfill & \hfill v_2{\xi }_2{I}_h\hfill & \hfill v_3{\xi }_3{S}_c\hfill & \hfill v_4{\xi }_4{I}_c\hfill \end{array}\right]\left[\begin{array}{c}\hfill v_1\hfill \\ \hfill v_2\hfill \\ \hfill v_3\hfill \\ \hfill v_4\hfill \end{array}\right],$$

$$vA{v}^T=v_1^2{\xi }_1{S}_h+v_2^2{\xi }_2{I}_h+v_3^2{\xi }_3{S}_c+v_4^2{\xi }_4{I}_c>0.$$

Here it will be shown that the value of ${\lambda }_{min}\left(A^{T}A\right)$ lies below zero and if value of ${\lambda }_{min}\left(A^{T}A\right)$ become zero, then a vector will exist, $\kappa =({\kappa }_1,{\kappa }_2,{\kappa }_3,{\kappa }_4)\in {\mathbb{R}}^4$ in such a way that $\left|\kappa \right|\ne 0$ as well as $A\kappa =0$.

This results in seeing the fact that ${\kappa }^T{A}^{T}A\kappa =0$ for ${\xi }_i>0$, $i=1,2,3,4$ and from positive uniform definiteness of the obtained matrix, $A^{T}A$ w.r.t $x$ belongs to $\overline{H}$, one can obtain $\kappa =0$, which results in a contradiction. Thus, it need to take that ${\lambda }_{min}\left(A^{T}A\right)\ge 0$. Also for a continuous function of $x\in \overline{H}$, we infer

$$in{f}_{x\in H}{\lambda }_{min}\left(A^{T}A\right)\ge \underset{x\in H}{min}{\lambda }_{min}\left(A^{T}A\right)>0.$$

Hence, 1st condition of Lemma 1 is varied. Afterwards, we will check 2nd condition of Lemma 1 in the sequel.

2nd Condition of Lemma 1

Consider a function $V:{\mathbb{R}}_{+}^4\to {\mathbb{R}}_{+}^4$

$$\mathcal{V}={(S_h+I_h)}^2-log(S_h+I_h)+{(S_c+I_c)}^2-log(S_c+I_c)$$

$$\mathcal{A}=2(S_h+I_h)-\frac{1}{(S_h+I_h)},\mathcal{B}=2(S_c+I_c)-\frac{1}{(S_c+I_c)}$$

$${\mathcal{V}}_{x}f=\left(\begin{array}{cccc}\hfill \mathcal{A}\hfill & \hfill \mathcal{A}\hfill & \hfill \mathcal{B}\hfill & \hfill \mathcal{B}\hfill \end{array}\right)\left(\begin{array}{c}\hfill f_1\hfill \\ \hfill f_2\hfill \\ \hfill f_3\hfill \\ \hfill f_4\hfill \end{array}\right)$$

$${\mathcal{V}}_{x}f=(2(S_h+I_h)-\frac{1}{(S_h+I_h)})(f_1+f_2)+2(S_c+I_c)-\frac{1}{(S_c+I_c)}(f_3+f_4),=\left(2(S_h+I_h)-\frac{1}{(S_h+I_h)}\right)(\Delta -{\beta }_1{S}_h{I}_h-{\beta }_2{S}_h{I}_c-{\mu }_h{S}_h+{\beta }_1{S}_h{I}_h-({\mu }_h+{\mu }_1+\varphi )I_h+{\beta }_2{S}_h{I}_c)++\left(2(S_c+I_c)-\frac{1}{(S_c+I_c)}\right)\times (\Lambda -{\beta }_3{S}_c{I}_c-{\mu }_c{S}_c+{\beta }_3{S}_c{I}_c-({\mu }_c+{\mu }_2)I_c),=\left(2(S_h+I_h)-\frac{1}{(S_h+I_h)}),(\Delta -{\mu }_h{S}_h-({\mu }_h+{\mu }_1+\varphi )I_h\right)+2(S_c+I_c)-\frac{1}{(S_c+I_c)}\left(\Lambda -{\mu }_c{S}_c-({\mu }_c+{\mu }_2)I_c\right),=2(S_h+I_h)(\Delta -{\mu }_h{S}_h-({\mu }_h+{\mu }_1+\varphi )I_h)-\frac{1}{(S_h+I_h)}(\Delta -{\mu }_h{S}_h-({\mu }_h+{\mu }_1+\varphi )I_h)+2(S_c+I_c)(\Lambda -{\mu }_c{S}_c-({\mu }_c+{\mu }_2)I_c)-\frac{1}{(S_c+I_c)}(\Lambda -{\mu }_c{S}_c-({\mu }_c+{\mu }_2)I_c),=2S_h(\Delta -{\mu }_h{S}_h-({\mu }_h+{\mu }_1+\varphi )I_h)+2I_h(\Delta -{\mu }_h{S}_h-({\mu }_h+{\mu }_1+\varphi )I_h)-\frac{\Delta }{(S_h+I_h)}+\frac{({\mu }_h+{\mu }_1+\varphi )I_h}{(S_h+I_h)}+\frac{{\mu }_h{S}_h}{(S_h+I_h)}+2S_c(\Lambda -{\mu }_c{S}_c-({\mu }_c+{\mu }_2)I_c)+2I_c(\Lambda -{\mu }_c{S}_c-({\mu }_c+{\mu }_2)I_c)-\frac{\Lambda }{(S_c+I_c)}+\frac{{\mu }_c{S}_c}{(S_c+I_c)}++\frac{({\mu }_c+{\mu }_2{I}_c)}{(S_c+I_c)},=2\Delta S_h-2{\mu }_h{S}_h^2-2({\mu }_h+{\mu }_1+\varphi )I_h{S}_h+2\Delta I_h-2{\mu }_h{I}_h{S}_h-2({\mu }_h+{\mu }_1+\varphi )I_h^2-\frac{\Delta }{(S_h+I_h)}+\frac{{\mu }_h(S_h+I_h)}{(S_h+I_h)}++\frac{({\mu }_1+\varphi )I_h}{(S_h+I_h)}+2\Lambda S_c-2{\mu }_c{S}_c^2-2({\mu }_c+{\mu }_2)I_c{S}_c+2\Lambda I_c-2{\mu }_c{I}_c{S}_c-2({\mu }_c+{\mu }_2)I_c^2-\frac{\Lambda }{(S_c+I_c)}+\frac{{\mu }_c(S_c+I_c)}{(S_c+I_c)}+\frac{{\mu }_2{I}_c}{(S_c+I_c)},=2+\frac{1}{{(S_h^2+I_h^2)}^2},{\mathcal{C}}_2=2+\frac{1}{{(S_c^2+I_c^2)}^2}.$$

Next,

$$g^T{\mathcal{V}}_{xx}g=\left[\begin{array}{cccc}\hfill {\xi }_1{S}_h\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\xi }_2{I}_h\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_3{S}_c\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_4{S}_c\hfill \end{array}\right]\times \left[\begin{array}{cccc}\hfill {\mathcal{C}}_1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathcal{C}}_1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\mathcal{C}}_2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\mathcal{C}}_2\hfill \end{array}\right]\times \left[\begin{array}{cccc}\hfill {\xi }_1{S}_h\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\xi }_2{I}_h\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_3{S}_c\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_4{S}_c\hfill \end{array}\right],$$

$$g^T{\mathcal{V}}_{xx}g=\left[\begin{array}{cccc}\hfill {\xi }_1^2{S}_h^2{\mathcal{C}}_1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\xi }_2^2{I}_h^2{\mathcal{C}}_1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_3^2{S}_c^2{\mathcal{C}}_2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\xi }_4^2{I}_c^2{\mathcal{C}}_2\hfill \end{array}\right],$$

$$\frac{1}{2}trace\left(g^T{\mathcal{V}}_{xx}g\right)=\frac{{\xi }_1^2{S}_h^2{\mathcal{C}}_1}{2}+\frac{{\xi }_2^2{I}_h^2{\mathcal{C}}_1}{2}+\frac{{\xi }_3^2{S}_c^2{\mathcal{C}}_2}{2}+\frac{{\xi }_4^2{I}_c^2{\mathcal{C}}_2}{2},=\frac{{\xi }_1^2{S}_h^2}{2}\left(2+\frac{1}{{(S_h^2+I_h^2)}^2}\right)+\frac{{\xi }_2^2{I}_h^2}{2}\left(2+\frac{1}{{(S_h^2+I_h^2)}^2}\right)+\frac{{\xi }_3^2{S}_c^2}{2}\left(2+\frac{1}{{(S_c^2+I_c^2)}^2}\right)++\frac{{\xi }_4^2{I}_c^2}{2}\left(2+\frac{1}{{(S_c^2+I_c^2)}^2}\right),={\xi }_1^2{S}_h^2+\frac{{\xi }_1^2{S}_h^2}{2{(S_h^2+I_h^2)}^2}+{\xi }_2^2{I}_h^2+\frac{{\xi }_2^2{I}_h^2}{2{(S_h^2+I_h^2)}^2}+{\xi }_3^2{S}_c^2+\frac{{\xi }_3^2{S}_c^2}{2{(S_c^2+I_c^2)}^2}+{\xi }_4^2{I}_c^2+\frac{{\xi }_4^2{I}_c^2}{2{(S_c^2+I_c^2)}^2},={\xi }_1^2{S}_h^2+{\xi }_2^2{I}_h^2+\frac{{\xi }_1^2{S}_h^2}{2{(S_h^2+I_h^2)}^2}+\frac{{\xi }_2^2{I}_h^2}{2{(S_h^2+I_h^2)}^2}+{\xi }_3^2{S}_c^2+{\xi }_4^2{I}_c^2+\frac{{\xi }_3^2{S}_c^2}{2{(S_c^2+I_c^2)}^2}+\frac{{\xi }_4^2{I}_c^2}{2{(S_c^2+I_c^2)}^2}.$$ (11)

$$\mathcal{V}={(S_h+I_h)}^2-log(S_h+I_h)+{(S_c+I_c)}^2-log(S_c+I_c)$$

$$\mathcal{A}=2(S_h+I_h)-\frac{1}{(S_h+I_h)},\mathcal{B}=2(S_c+I_c)-\frac{1}{(S_c+I_c)}$$

$${\mathcal{V}}_{x}gdB=\left(\begin{array}{cccc}\hfill \mathcal{A}\hfill & \hfill \mathcal{A}\hfill & \hfill \mathcal{B}\hfill & \hfill \mathcal{B}\hfill \end{array}\right)\left(\begin{array}{c}\hfill g_{1}d{B}_1\hfill \\ \hfill g_{2}d{B}_2\hfill \\ \hfill g_{3}d{B}_3\hfill \\ \hfill g_{4}d{B}_4\hfill \end{array}\right)$$

$${\mathcal{V}}_{x}gdB=(S_h-1){\xi }_{1}d{B}_1+(I_h-1){\xi }_{2}d{B}_2+(S_c-1){\xi }_{3}d{B}_3+(I_c-1){\xi }_{4}d{B}_4.$$ (12)

Using , (11), (12) in (4) we can get,

$$dV=LV+(S_h-1){\xi }_{1}d{B}_1+(I_h-1){\xi }_{2}d{B}_2+(S_c-1){\xi }_{3}d{B}_3+(I_c-1){\xi }_{4}d{B}_4.$$ (13)

Using (13), we get

$$LV=2\Delta S_h-2{\mu }_h{S}_h^2-2({\mu }_h+{\mu }_1+\varphi )I_h{S}_h+2\Delta I_h-2{\mu }_h{I}_h{S}_h-2({\mu }_h+{\mu }_1+\varphi )I_h^2-\frac{\Delta }{(S_h+I_h)}++\frac{{\mu }_h(S_h+I_h)}{(S_h+I_h)}+\frac{({\mu }_1+\varphi )I_h}{(S_h+I_h)}+2\Lambda S_c-2{\mu }_c{S}_c^2-2({\mu }_c+{\mu }_2)I_c{S}_c+2\Lambda I_c-2{\mu }_c{I}_c{S}_c-2({\mu }_c+{\mu }_2)I_c^2,-\frac{\Lambda }{(S_c+I_c)}+\frac{{\mu }_c(S_c+I_c)}{(S_c+I_c)}+\frac{{\mu }_2{I}_c}{(S_c+I_c)}+{\xi }_1^2{S}_h^2+{\xi }_2^2{I}_h^2+\frac{{\xi }_1^2{S}_h^2}{2{(S_h^2+I_h^2)}^2}+\frac{{\xi }_2^2{I}_h^2}{2{(S_h^2+I_h^2)}^2}+{\xi }_3^2{S}_c^2+{\xi }_4^2{I}_c^2++\frac{{\xi }_3^2{S}_c^2}{2{(S_c^2+I_c^2)}^2}+\frac{{\xi }_4^2{I}_c^2}{2{(S_c^2+I_c^2)}^2},=2\Delta S_h-2{\mu }_h{S}_h^2+{\xi }_1^2{S}_h^2-2({\mu }_h+{\mu }_1+\varphi )I_h{S}_h+2\Delta I_h-2{\mu }_h{I}_h{S}_h-2({\mu }_1+\varphi )I_h^2-2{\mu }_h{I}_h^2+{\xi }_2^2{I}_h^2+-\frac{\Delta }{(S_h+I_h)}+{\mu }_h+\frac{({\mu }_1+\varphi )I_h}{(S_h+I_h)}+\frac{{\xi }_1^2{S}_h^2+{\xi }_2^2{I}_h^2}{2{(S_h^2+I_h^2)}^2}+2\Lambda S_c-2{\mu }_c{S}_c^2+{\xi }_3^2{S}_c^2-2({\mu }_c+{\mu }_2)I_c{S}_c++2\Lambda I_c-2{\mu }_c{I}_c{S}_c-2{\mu }_2{I}_c^2-2{\mu }_c{I}_c^2+{\xi }_4^2{I}_c^2-\frac{\Lambda }{(S_c+I_c)}+{\mu }_c+\frac{{\mu }_2{I}_c}{(S_c+I_c)}+\frac{{\xi }_3^2{S}_c^2+{\xi }_4^2{I}_c^2}{2{(S_c^2+I_c^2)}^2}-(2{\mu }_h-{\xi }_1^2)S_h^2+2\Delta S_h-2({\mu }_h+{\mu }_1+\varphi )I_h{S}_h-2{\mu }_h{I}_h{S}_h-2({\mu }_1+\varphi )I_h^2-(2{\mu }_h-{\xi }_2^2)I_h^2+2\Delta I_h+-\frac{\Delta }{(S_h+I_h)}+{\mu }_h+\frac{({\mu }_1+\varphi )I_h}{(S_h+I_h)}+\frac{{\xi }_1^2{S}_h^2+{\xi }_2^2{I}_h^2}{2{(S_h^2+I_h^2)}^2}-(2{\mu }_c-{\xi }_3^2)S_c^2+2\Lambda S_c-{\mu }_2{I}_c{S}_c-2{\mu }_c{I}_c{S}_c-2{\mu }_2{I}_c^2+-(2{\mu }_c-{\xi }_4^2)I_c^2+2\Lambda I_c-\frac{\Lambda }{(S_c+I_c)}+{\mu }_c+\frac{{\mu }_2{I}_c}{(S_c+I_c)}+\frac{{\xi }_3^2{S}_c^2+{\xi }_4^2{I}_c^2}{2{(S_c^2+I_c^2)}^2}.$$ (14)

Using (14), we get

$$LV=-(2{\mu }_h-{\xi }_1^2){\left(S_h-\frac{\Delta }{(2{\mu }_h-{\xi }_1^2)}\right)}^2+\frac{{\Delta }^2}{2{\mu }_h-{\xi }_1^2}-2({\mu }_h+{\mu }_1+\varphi )I_h{S}_h-2{\mu }_h{I}_h{S}_h-2({\mu }_1+\varphi )I_h^2-(2{\mu }_h-{\xi }_2^2){\left(I_h-\frac{\Delta }{(2{\mu }_h-{\xi }_2^2)}\right)}^2+\frac{{\Delta }^2}{2{\mu }_h-{\xi }_2^2}-\frac{\Delta }{(S_h+I_h)}+{\mu }_h+\frac{({\mu }_1+\varphi )I_h}{(S_h+I_h)}+\frac{{\xi }_1^2{S}_h^2+{\xi }_2^2{I}_h^2}{2{(S_h^2+I_h^2)}^2}-(2{\mu }_c-{\xi }_3^2){\left(S_c-\frac{\Lambda }{(2{\mu }_c-{\xi }_3^2)}\right)}^2+\frac{{\Lambda }^2}{2{\mu }_c-{\xi }_3^2}-2{\mu }_2{I}_c{S}_c-2{\mu }_c{I}_c{S}_c-2{\mu }_2{I}_c^2-(2{\mu }_c-{\xi }_4^2){\left(I_c-\frac{\Lambda }{(2{\mu }_c-{\xi }_4^2)}\right)}^2+\frac{\Lambda }{2{\mu }_c-{\xi }_4^2}-\frac{\Lambda }{(S_c+I_c)}+{\mu }_c+\frac{{\mu }_2{I}_c}{(S_c+I_c)}+\frac{{\xi }_3^2{S}_c^2+{\xi }_4^2{I}_c^2}{2{(S_c^2+I_c^2)}^2},=-(2{\mu }_h-{\xi }_1^2){\left(S_h-\frac{\Delta }{(2{\mu }_h-{\xi }_1^2)}\right)}^2-(2{\mu }_h-{\xi }_2^2){\left(I_h-\frac{\Delta }{(2{\mu }_h-{\xi }_2^2)}\right)}^2+\frac{{\Delta }^2}{2{\mu }_h-{\xi }_1^2}+\frac{{\Delta }^2}{2{\mu }_h-{\xi }_2^2}-2({\mu }_h+{\mu }_1+\varphi )I_h{S}_h-2{\mu }_h{I}_h{S}_h-2({\mu }_1+\varphi )I_h^2-\frac{\Delta }{(S_h+I_h)}+{\mu }_h++\frac{({\mu }_1+\varphi )I_h}{(S_h+I_h)}+\frac{{\xi }_1^2{S}_h^2+{\xi }_2^2{I}_h^2}{2{(S_h^2+I_h^2)}^2}-(2{\mu }_c-{\xi }_3^2){\left(S_c-\frac{\Lambda }{(2{\mu }_c-{\xi }_3^2)}\right)}^2-(2{\mu }_c-{\xi }_4^2){\left(I_c-\frac{\Lambda }{(2{\mu }_c-{\xi }_4^2)}\right)}^2++\frac{{\Lambda }^2}{2{\mu }_c-{\xi }_3^2}+\frac{\Lambda }{2{\mu }_c-{\xi }_4^2}-\frac{\Lambda }{(S_c+I_c)}-2{\mu }_2{I}_c{S}_c-2{\mu }_c{I}_c{S}_c-2{\mu }_2{I}_c^2+{\mu }_c+\frac{{\mu }_2{I}_c}{(S_c+I_c)}+\frac{{\xi }_3^2{S}_c^2+{\xi }_4^2{I}_c^2}{2{(S_c^2+I_c^2)}^2}.=-(2{\mu }_h-{\xi }_1^2){\left(S_h-\frac{\Delta }{(2{\mu }_h-{\xi }_1^2)}\right)}^2-(2{\mu }_h-{\xi }_2^2){\left(I_h-\frac{\Delta }{(2{\mu }_h-{\xi }_2^2)}\right)}^2-(2{\mu }_c-{\xi }_3^2){\left(S_c-\frac{\Lambda }{(2{\mu }_c-{\xi }_3^2)}\right)}^2+-(2{\mu }_c-{\xi }_4^2){\left(I_c-\frac{\Lambda }{(2{\mu }_c-{\xi }_4^2)}\right)}^2+{\Delta }^2\left[\frac{1}{2{\mu }_h-{\xi }_1^2}+\frac{1}{2{\mu }_h-{\xi }_2^2}\right]+{\Lambda }^2\left[\frac{1}{2{\mu }_c-{\xi }_3^2}+\frac{1}{2{\mu }_c-{\xi }_4^2}\right]+\frac{{\xi }_1^2{S}_h^2+{\xi }_2^2{I}_h^2}{2{(S_h^2+I_h^2)}^2}+\frac{{\xi }_3^2{S}_c^2+{\xi }_4^2{I}_c^2}{2{(S_c^2+I_c^2)}^2}+{\mu }_h+\frac{({\mu }_1+\varphi )I_h}{(S_h+I_h)}+{\mu }_c+\frac{{\mu }_2{I}_c}{(S_c+I_c)}-2({\mu }_h+{\mu }_1+\varphi )I_h{S}_h-2{\mu }_h{I}_h{S}_h+-2({\mu }_1+\varphi )I_h^2-\frac{\Delta }{(S_h+I_h)}-\frac{\Lambda }{(S_c+I_c)}-2{\mu }_2{I}_c{S}_c-2{\mu }_c{I}_c{S}_c-2{\mu }_2{I}_c^2.$$ (15)

Simplifying (15) we obtain,

$$LV\le -(2{\mu }_h-{\xi }_1^2){\left(S_h-\frac{\Delta }{(2{\mu }_h-{\xi }_1^2)}\right)}^2-(2{\mu }_h-{\xi }_2^2){\left(I_h-\frac{\Delta }{(2{\mu }_h-{\xi }_2^2)}\right)}^2-(2{\mu }_c-{\xi }_3^2){\left(S_c-\frac{\Lambda }{(2{\mu }_c-{\xi }_3^2)}\right)}^2+-(2{\mu }_c-{\xi }_4^2){\left(I_c-\frac{\Lambda }{(2{\mu }_c-{\xi }_4^2)}\right)}^2+{\Delta }^2\left[\frac{1}{2{\mu }_h-{\xi }_1^2}+\frac{1}{2{\mu }_h-{\xi }_2^2}\right]+{\Lambda }^2\left[\frac{1}{2{\mu }_c-{\xi }_3^2}+\frac{1}{2{\mu }_c-{\xi }_4^2}\right]+\hat{\sigma }+\hat{\mu }-2({\mu }_h+{\mu }_1+\varphi )I_h{S}_h-2{\mu }_h{I}_h{S}_h-2({\mu }_1+\varphi )I_h^2-\frac{\Delta }{(S_h+I_h)}-\frac{\Lambda }{(S_c+I_c)}-2{\mu }_2{I}_c{S}_c-2{\mu }_c{I}_c{S}_c-2{\mu }_2{I}_c^2,\le -(2{\mu }_h-{\xi }_1^2){\left(S_h-\frac{\Delta }{(2{\mu }_h-{\xi }_1^2)}\right)}^2-(2{\mu }_h-{\xi }_2^2){\left(I_h-\frac{\Delta }{(2{\mu }_h-{\xi }_2^2)}\right)}^2-(2{\mu }_c-{\xi }_3^2){\left(S_c-\frac{\Lambda }{(2{\mu }_c-{\xi }_3^2)}\right)}^2+-(2{\mu }_c-{\xi }_4^2){\left(I_c-\frac{\Lambda }{(2{\mu }_c-{\xi }_4^2)}\right)}^2+{\Delta }^2\left[\frac{1}{2{\mu }_h-{\xi }_1^2}+\frac{1}{2{\mu }_h-{\xi }_2^2}\right]+{\Lambda }^2\left[\frac{1}{2{\mu }_c-{\xi }_3^2}+\frac{1}{2{\mu }_c-{\xi }_4^2}\right]+\hat{\xi }+\hat{\mu },$$

where $\hat{\xi }=max\{{\xi }_1^2,{\xi }_2^2,{\xi }_3^2,{\xi }_4^2\}$ and $\hat{\mu }=\{{\mu }_h,{\mu }_1,\varphi ,{\mu }_c,{\mu }_2\}$. Under the conditions of $2{\mu }_h>{\xi }_1^2$, $2{\mu }_h>{\xi }_2^2$, $2{\mu }_c>{\xi }_3^2$, and $2{\mu }_c>{\xi }_4^2$, it is not difficult to see that, for a sufficiently large number $M$,

$$LV(S_h,I_h,S_c,I_c)\le -1,(S_h,I_h,S_c,I_c)\in \left(0,\frac{1}{M}\right]\times \left(0,\frac{1}{M}\right]\times \left(0,\frac{1}{M}\right]\times \left(0,\frac{1}{M}\right]$$

and

$$LV(S_h,I_h,S_c,I_c)\le -1,(S_h,I_h,S_c,I_c)\in [M,\infty )\times [M,\infty )\times [M,\infty )\times [M,\infty ).$$

Therefore,

$$LV(S_h,I_h,S_c,I_c)\le -1,\forall (S_h,I_h,S_c,I_c)\in {\mathbb{R}}_{+}^4-H.$$ (16)

Let the initial value $(S_h\left(0\right),I_h\left(0\right),S_c\left(0\right),I_c\left(0\right))\in {\mathbb{R}}_{+}^4-H$ be arbitrary and let ${\tau }_H$ be the stopping time as defined in Lemma 1 By (13), (16), it follows that

$$0\ge V(S_h\left(0\right),I_h\left(0\right),S_c\left(0\right),I_c\left(0\right))-E(t\wedge {\tau }_H),\forall t\ge 0.$$

Letting $t\to \infty$ we obtain

$$E\left({\tau }_H\right)\le V(S_h\left(0\right),I_h\left(0\right),S_c\left(0\right),I_c\left(0\right)),\forall x\in {\mathbb{R}}_{+}^4-H.$$

This immediately implies condition (ii) in Lemma 1 The assertion hence follows from Lemma 1. The proof is complete.

Example 1

In this example, we chose some of the value of the parameters used in the proposed model form [35] and the other is assumed values: $\Lambda =10,{\beta }_1=0.000984,{\beta }_2=0.0142,{\mu }_h=0.001,{\mu }_1=0.02,\varphi =0.0759,\Delta =5,{\beta }_3=0.001,{\mu }_c=0.0992,{\mu }_2=0.003$. with initial values, $(S_h\left(0\right),I_h\left(0\right),S_c\left(0\right),I_c\left(0\right))=(100,50,80,30)$ and ${\sigma }_i=0.1,i=1,2,3,4;$. The graph of each approaches to endemic equilibrium point which is the justification of theorem 4 (see Fig. 1).

Fig. 1.

Behavior of different classes of epidemic model (2) at Endemic equilibrium point.

6. Extinction of the disease

It is know that the basic reproductive number $R_0^D$ outline the dynamical aspects of model (2). It also assure the persistence as well as extinction of an epidemic. For the underlying model (2) the associated threshold quantity is given as;

$${\mathcal{R}}_0^{\mathcal{S}}=max\left\{\frac{{\beta }_1{\Lambda }_h}{{\mu }_h({\mu }_h+{\mu }_1+\varphi )},\frac{{\beta }_3\Lambda }{{\mu }_c\left({\mu }_c+{\mu }_2+\frac{{\xi }_4}{2}\right)}\right\}=max\{{\mathcal{R}}_{0,1},{\mathcal{R}}_{0,2}\}.$$

Theorem 5

For the model (2) the class of infected individuals approach to extinction almost surely whenever,

$${\mathcal{C}}_1:{\mathcal{R}}_0^S<1,$$

$${\mathcal{C}}_2:{\xi }_2\ge \sqrt{\frac{{\beta }_2^2}{2({\mu }_h+{\mu }_1+\varphi )(1-{\mathcal{R}}_{0,1})}}.$$

Proof

Consider a solution $(S_h\left(t\right),I_h\left(t\right),S_c\left(t\right),I_c\left(t\right))$ of the model (2) along with initial condition

$(S_h\left(0\right),I_h\left(0\right),S_c\left(0\right),I_c\left(0\right))\in {\mathbb{R}}_{+}^4$. Use of $It\hat{o}$’s formula for the second equation of model (2) gives us,

$$dln{I}_h=\left({\beta }_1{S}_h+\frac{{\beta }_2{S}_h{I}_c}{I_h}-({\mu }_h+{\mu }_1+\varphi )-\frac{{\xi }_2^2}{2}\right)dt+{\xi }_{2}d{B}_2\left(t\right),=\left({\beta }_1{S}_h-\left(\frac{{\xi }_2^2}{2}-\frac{{\beta }_2{S}_h{I}_c}{I_h}\right)-({\mu }_h+{\mu }_1+\varphi )\right)dt+{\xi }_{2}d{B}_2\left(t\right),=\left({\beta }_1{S}_h-\frac{{\xi }_2^2}{2}\left(1-2\frac{{\beta }_2{S}_h{I}_c}{{\xi }_2^2}+{\left(\frac{{\beta }_2{S}_h{I}_c}{{\xi }_2^2{I}_h}\right)}^2-{\left(\frac{{\beta }_2{S}_h{I}_c}{{\xi }_2^2{I}_h}\right)}^2\right)-({\mu }_h+{\mu }_1+\varphi )\right)dt+{\xi }_{2}d{B}_2\left(t\right),=\left({\beta }_1{S}_h-\frac{{\xi }_2^2}{2}{\left(1-\frac{{\beta }_2{S}_h{I}_c}{{\xi }_2^2{I}_h}\right)}^2+\frac{{\xi }_2^2}{2}\left(\frac{{\beta }_2^2{S}_h^2{I}_c^2}{{\xi }_2^4{I}_h^2}\right)-({\mu }_h+{\mu }_1+\varphi )\right)dt+{\xi }_{2}d{B}_2\left(t\right),=\left({\beta }_1{S}_h-\frac{{\xi }_2^2}{2}{\left(1-\frac{{\beta }_2{S}_h{I}_c}{{\xi }_2^2{I}_h}\right)}^2+\frac{{\beta }_2^2{S}_h^2{I}_c^2}{2{\xi }_2^2{I}_h^2}-({\mu }_h+{\mu }_1+\varphi )\right)dt+{\xi }_{2}d{B}_2\left(t\right),\le \left(\frac{{\beta }_1\Lambda }{{\mu }_h}-\frac{{\xi }_2^2}{2}{\left(1-\frac{{\beta }_2{S}_h{I}_c}{{\xi }_2^2{I}_h}\right)}^2+\frac{{\beta }_2^2}{2{\xi }_2^2}-({\mu }_h+{\mu }_1+\varphi )\right)dt+{\xi }_{2}d{B}_2\left(t\right),=\left(\frac{{\beta }_1\Lambda }{{\mu }_h}+\frac{{\beta }_2^2}{2{\xi }_2^2}-\frac{{\xi }_2^2}{2}{\left(1-\frac{{\beta }_2{S}_h{I}_c}{{\xi }_2^2{I}_h}\right)}^2-({\mu }_h+{\mu }_1+\varphi )\right)dt+{\xi }_{2}d{B}_2\left(t\right).$$ (17)

Integrating both sides of (17) from $t$ to 0 gives,

$$ln{I}_h={\int }_0^t\left(\frac{{\beta }_1\Lambda }{{\mu }_h}+\frac{{\beta }_2^2}{2{\xi }_2^2}-\frac{{\xi }_2^2}{2}{\left(1-\frac{{\beta }_2{S}_h{I}_c}{{\xi }_2^2{I}_h}\right)}^2-({\mu }_h+{\mu }_1+\varphi )\right)dt+{\xi }_2{\int }_0^{t}d{B}_2\left(t\right),=\frac{{\beta }_1\Lambda }{{\mu }_h}t+\frac{{\beta }_2^2}{2{\xi }_2^2}t-\frac{{\xi }_2^2}{2}{\int }_0^t{\left(1-\frac{{\beta }_2{S}_h{I}_c}{{\xi }_2^2{I}_h}\right)}^{2}dt-({\mu }_h+{\mu }_1+\varphi )t+{\xi }_2{\int }_0^{t}d{B}_2\left(t\right)=\frac{{\beta }_1\Lambda }{{\mu }_h}t+\frac{{\beta }_2^2}{2{\xi }_2^2}t-\frac{{\xi }_2^2}{2}{\int }_0^t{\left(1-\frac{{\beta }_2{S}_h{I}_c}{{\xi }_2^2{I}_h}\right)}^{2}dt-({\mu }_h+{\mu }_1+\varphi )t+{\xi }_2{\int }_0^{t}d{B}_2\left(t\right).$$ (18)

Dividing both sides of (18) by $t$, we have

$$\frac{ln{I}_h}{t}\le \frac{{\beta }_1\Lambda }{{\mu }_h}+\frac{{\beta }_2^2}{2{\xi }_2^2}-({\mu }_h+{\mu }_1+\varphi )+{\xi }_2{\int }_0^{t}d{B}_2\left(t\right),=\frac{{\beta }_2^2}{2{\xi }_2^2}-({\mu }_h+{\mu }_1+\varphi )\left(1-\frac{{\beta }_1\Lambda }{{\mu }_h({\mu }_h+{\mu }_1+\varphi )}\right)+{\xi }_2{\int }_0^{t}d{B}_2\left(t\right),=\frac{{\beta }_2^2}{2{\xi }_2^2}-({\mu }_h+{\mu }_1+\varphi )(1-{\mathcal{R}}_{0,1})+{\xi }_2{\int }_0^{t}d{B}_2\left(t\right).$$

Now as $t\to \infty$, then

$$\underset{t\to \infty }{lim}\frac{ln{I}_h}{t}=\frac{{\beta }_2^2}{2{\xi }_2^2}-({\mu }_h+{\mu }_1+\varphi )(1-{\mathcal{R}}_{0,1}),$$

${lim}_{t\to \infty }\frac{ln{I}_h}{t}<0$ if ${\mathcal{C}}_2$ is satisfied, and ${lim}_{t\to \infty }\frac{1}{t}{\int }_0^{t}d{B}_2\left(t\right)=0$. Similarly, from the stochastic model (2) we get,

$$\frac{S_c\left(t\right)-S_c\left(0\right)}{t}=\Lambda -{\mu }_c\frac{{\int }_0^t{S}_c}{t}-{\beta }_3\frac{{\int }_0^t{S}_c{I}_c}{t}+{\xi }_3\frac{{\int }_0^t{S}_{c}d{B}_3\left(t\right)}{t},$$

$$\frac{I_c\left(t\right)-I_c\left(0\right)}{t}={\beta }_3\frac{{\int }_0^t{S}_c{I}_c}{t}-({\mu }_c+{\mu }_2)\frac{{\int }_0^t{I}_c}{t}+{\xi }_4\frac{{\int }_0^t{I}_{c}d{B}_4\left(t\right)}{t},$$

then

$${\mu }_c\frac{{\int }_0^t{S}_c\left(s\right)ds}{t}+({\mu }_c+{\mu }_2)\frac{{\int }_0^t{I}_c\left(s\right)ds}{t}=\Lambda -\frac{S_c\left(t\right)-S_c\left(0\right)}{t}-\frac{I_c\left(t\right)-I_c\left(0\right)}{t}+{\xi }_3\frac{{\int }_0^t{S}_{c}d{B}_3\left(t\right)}{t}+{\xi }_4\frac{{\int }_0^t{I}_{c}d{B}_4\left(t\right)}{t}.⟹{\mu }_c\frac{{\int }_0^t{S}_c\left(s\right)ds}{t}+({\mu }_c+{\mu }_2)\frac{{\int }_0^t{I}_c\left(s\right)ds}{t}≔\Lambda +\phi \left(t\right),$$

where $\phi \left(t\right)$ has the property that

$$\underset{t\to \infty }{lim}\phi \left(t\right)=0,$$ (19)

then

$$\frac{{\mu }_c{\int }_0^t{S}_c+({\mu }_c+{\mu }_2){\int }_0^t{I}_c}{t}=\Lambda .$$

In addition,

$$log{I}_c\left(t\right)=log{I}_c\left(0\right)+{\beta }_3{\int }_0^t{S}_c\left(s\right)ds-\left({\mu }_c+{\mu }_2+\frac{{\xi }_4}{2}\right)t+{\xi }_4{B}_4\left(t\right)=\frac{{\beta }_3\Lambda }{{\mu }_c}t-\frac{{\beta }_3({\mu }_c+{\mu }_2)}{{\mu }_c}{\int }_0^t{I}_c\left(s\right)ds-\left({\mu }_c+{\mu }_2+\frac{{\xi }_4}{2}\right)t+log{I}_c\left(0\right)+\frac{{\beta }_3}{{\mu }_c}\phi \left(t\right)t+{\xi }_4{B}_4\left(t\right).$$ (20)

and

$$\underset{t\to \infty }{lim}\left[log{I}_c\left(0\right)+\frac{{\beta }_3}{{\mu }_c}\phi \left(t\right)t+{\xi }_4{B}_4\left(t\right)\right]=0,a.s$$ (21)

by (19),(21) and the property of the Brown motion. If ${\mathcal{R}}_{0,2}<1$, from (20), we have

$$\underset{t\to \infty }{lim}sup\frac{log{I}_c\left(t\right)}{t}\le \frac{{\beta }_3\Lambda }{{\mu }_c}-\left({\mu }_c+{\mu }_2+\frac{{\xi }_4}{2}\right)=\left({\mu }_c+{\mu }_2+\frac{{\xi }_4}{2}\right)({\mathcal{R}}_{0,2}-1),$$

as required.

Example 2

In this example, we choose the values of the parameters used in the proposed model as: ${\beta }_1=9.84\times 10^{-5},{\beta }_2=0.0142,{\beta }_3=0.8.43\times 10^{-6},$ and $,\varphi =0.0759$ from [35]. While $\Lambda =10,{\mu }_h=0.04,{\mu }_1=0.5,\Delta =5,{\mu }_c=0.1,{\mu }_2=0.002$ are assumed with initial condition $(S_h\left(0\right),I_h\left(0\right),S_c\left(0\right),I_c\left(0\right))=(100,50,80,30)$. The graph show the stability of the system at disease free equilibrium point (see Fig. 2).

Fig. 2.

Behavior of different classes of epidemic model (2) at Disease free equilibrium point.

7. Conclusion

In this paper, we have considered a stochastic epidemic model of MERS-Cov (Middle East Respiratory Syndrome), where some stochasticity is presented in the death ratio. Stochastic differential equations give another option to model viral dynamics and stochastic effects. This approached introduce a more realistic way of modeling of infectious diseases. Here, we examined the existence and uniqueness of the solutions of the stochastic model and demonstrated positivity and boundedness, which is of foremost significance for the study of the dynamics of population models. Also, the positivity and boundedness of solutions are important to other nonlinear models that arise in sciences and engineering. Thus, a similar approach can be applied to other models from different areas. A Milstein numerical scheme is included to support our theoretical results. By computer simulation, we show how disease will dies out and when it will be prevailed, when the condition of Theorem Theorem 4, Theorem 5 is satisfied. Thus, health institutions can take measures to make the corresponding parameter changes to obtain a feasible way for MERS-COV prevention and control.