Transmission dynamics of mumps epidemic model through stochastic analysis with delay effect

Abstract

Stochastic delayed modeling (stochastic differential equations (SDEs) with delay parameters) has a significant non-pharmaceutical intervention to control transmission dynamics of infectious diseases and its results are close to the reality of nature. Mumps is a viral disease specified with swollen jaws and inflated cheeks. Direct contact with saliva or respiratory drop less from the mouth is the major causes of its outbreak. According to the World Health Organization (WHO), still, 20% of young adult males develop mumps worldwide. No doubt, the vaccination of Mumps exists. The main cause is to study the transmission dynamics of Mumps through stochastic with delay approaches. How is the stochastic delay the best strategy to study the dynamics of disease in a population? For this, we consider the existing deterministic model in literature, with the whole population, divided as susceptible human population S(t), exposed human population E(t), symptomatic infectious I(t), asymptomatic infectious A(t), isolated and treated symptomatic Q(t), recovered humans R(t). After that, we extend the deterministic model into a stochastic delay model (Stochastic delay differential equations (SDDEs) by using the transition probabilities and non-parametric perturbation ways. The positivity, boundedness, extinction, and persistence of disease study with essential properties of reproduction number rigorously. The mump-free equilibrium (MFE) and mumps existing equilibrium (MEE) are two states, local, and global stability of second order and sensitivity analysis of parameters analyzed to verify the model validations. Due to the highly nonlinear stochastic delay differential equations of the model, we used both standard and nonstandard methods such as Euler Maryama, stochastic Euler, stochastic Runge-Kutta, and stochastic nonstandard finite difference with a delayed sense to visualization of results. In the end, the comparison of the methods is presented to support the efficiency of non-standard methods in the sense of stochastic with delay parameters.

Literature review

Qu et al. demonstrated a dynamical model of the outspread of the mumps virus. This model is being used to control the mumps disease¹. Li et al.² described a mathematical model that is used to control the mumps virus in China. The disease was controlled by using this model to some extent. Mohammadi et al.³ introduced a model by using Caputo–Fabrizio fractional modelling which minimized the production of mumps virus in the population. In⁴ the writers proposed a model that can effectively cure the mumps virus permanently. This model proved to be a stone in the field of medicine. In⁵ the authors studied the mumps virus it proposed that medication is effective in controlling the disease. In⁶ scholars organized the program in Japan where mumps vaccination was provided to the people. In⁷ scholars studied the different cases of the mumps virus in England and briefed the population about precautionary measures to control the viral outspread. In⁸ the authors introduced the mathematical model by using the application of the homotopy perturbation method. This model is used to reduce the outbreaks of mumps virus. In⁹ provided a dynamical model which helps to control the disease in the human population. This model also elaborates the future changes and their control. In¹⁰ the authors developed a dynamical model to investigate the mumps-induced human immune response that also focuses on viral virus, which is to clarify the intricate relationship between the immune system and the virus in the hopes of advancing our knowledge of illness and developing new therapeutic approaches. In¹¹ the authors conducted a systematical analysis of mathematical modelling studies to investigate the effectiveness of integrated tactics for mumps response. In¹² the scholars conducted a critical analysis of stability in the mumps virus model with the saturation indices and treatment default. In¹³ authors, provide the mumps outbreak using a behavior-focused vaccination model. This strategy considers behavioural elements while evaluating immunization strategies. In¹⁴ an online tool using logistic regression to predict cerebrospinal meningitis incidence is currently being developed. The goal of this initiative is to enhance illness management and early diagnosis. Duru et al.¹⁵, provided a dynamical model used to control the epidemic of mumps in the population. In¹⁶ the authors developed a mathematical model in which mumps disease was controlled by vaccination. In¹⁷ scholars minimized the effect of mumps disease by using fractional order derivatives. In¹⁸, the authors introduced such a model which can control and cure the mumps virus at the same time. This model also elaborates the future changes and their control. In¹⁹ scholars developed a dynamic model which controls the mumps virus in children because this virus attacks children due to their weak immune system. In²⁰ the writers proposed a model that can effectively cure the mumps virus permanently. This model proved to be my stone in the field of medicine. In¹⁵ the authors provided the crucial factors influencing the spread of the virus, offering valuable insights for enhancing immunization campaigns and lessening the impact of outbreaks.

Analysis of stochastic modeling is more realistic than deterministic because of its close association with nature. That is the sole reason to study stochastic mumps disease in human beings. Moreover, computational techniques fail to restore the positivity, boundedness, and dynamical properties of the model because of its dependency on time. To examine the non-linear stochastic mumps epidemic model, we will recommend the nonstandard finite difference method in this case. In which our recommended methods are coherent with the manner of the continuous model.

The further paper is organized as follows: Sect. 2 for the dynamical properties of the delayed model, and Sects. 3 and 4 are for the local and global stabilities of the model respectively. Section 5, goes to the sensitivity of parameters of the model. Sections 6 and 7, go to the formulation of the stochastic model and its fundamental properties. Section 8, for the proposed method like the non-standard finite difference in the sense of stochastic, is implemented. Sections 9 and 10, go to discussion and conclusion.

Model formulation

In this section, we studied the dynamics of the mumps disease within a population. Consider the total population of humans, which is presented by $\:N\left(t\right),\:t\ge\:0$ as defined $\:N:[0,\infty\:)\to\:\mathcal{R}$. The differential function of each component of the population, including S, E, I, A, Q, and R, is non-negative and defined for values greater than or equal to zero, mapping to the set of real numbers. It divides the population into six compartments, distinguishing between susceptible S(t), Exposed E(t), Symptomatic Infectious I(t), Asymptomatic Infectious A(t), Isolated and Treated symptomatic Q(t), and recovered R(t) individuals (See Fig. 1). Also, the ingoing and outgoing ratios are the constants and present the transmission rates among the population as presented in Table 1.

Fig. 1

Flow chart of mumps epidemic model.

Table 1 Description of parameters.

The deterministic model, which describes the rate of change of each class, can now be written down.

$$\:{S}^{{\prime\:}}\left(t\right)={\Lambda\:}-\frac{{\alpha\:}_{1}{\beta\:}_{1}S\left(t-\tau\:\right)I\left(t-\tau\:\right){e}^{-\mu\:\tau\:}}{N}-\frac{{\alpha\:}_{2}{\beta\:}_{2}S\left(t-\tau\:\right)A\left(t-\tau\:\right){e}^{-\mu\:\tau\:}}{N}-\mu\:S\left(t\right),\:\forall\:\:t\ge\:0,\:\tau\:<\:t$$

(1)

$$\:{E}^{{\prime\:}}\left(t\right)=\frac{{\alpha\:}_{1}{\beta\:}_{1}S\left(t-\tau\:\right)I\left(t-\tau\:\right){e}^{-\mu\:\tau\:}}{N}+\frac{{\alpha\:}_{2}{\beta\:}_{2}S\left(t-\tau\:\right)A\left(t-\tau\:\right){e}^{-\mu\:\tau\:}}{N}-\left(\kappa\:+{\tau\:}_{1}\right)\delta\:E\left(t\right)-\mu\:E\left(t\right)\:,\:\forall\:\:t\ge\:0,\:\tau\:<t$$

(2)

$$\:I^{\prime\:}\left(t\right)=\kappa\:\delta\:E\left(t\right)-(\nu\:+\mu\:)I\left(t\right)\:,\:\forall\:\:t\ge\:0,$$

(3)

$$\:A^{\prime\:}\left(t\right)={\tau\:}_{1}\delta\:E\left(t\right)-(\omega\:+\phi\:+\mu\:)A\left(t\right)\:,\:\forall\:\:t\ge\:0,\:$$

(4)

$$\:Q^{\prime\:}\left(t\right)=\omega\:A\left(t\right)-(\theta\:+\mu\:)Q\left(t\right)\:,\:\forall\:\:t\ge\:0,\:$$

(5)

$$\:{R}^{{\prime\:}}\left(t\right)=\theta\:Q\left(t\right)+\phi\:A\left(t\right)+\nu\:I\left(t\right)-\mu\:R\left(t\right)\:,\:\forall\:\:t\ge\:0,\:$$

(6)

.

Here, $\:S\left(t\right)\ge\:0,\:E\left(t\right)\ge\:0,I\left(t\right)\ge\:0,A\left(t\right)\ge\:0,Q\left(t\right)\ge\:0\:R\left(t\right)\ge\:0$.

Dynamical properties

The state variables, as defined in Eqs. (1–6), provide a non-negative solution for all time $\:t\ge\:0$, and for $\:\tau\:<t$, given non-negative beginning circumstances.

Theorem 1

The system remains restricted and confined within the feasible region $\:\mathcal{M}$ for any given time “t”. if $\:\underset{t\to\:\infty\:}{\text{lim}}Sup\:N\left(t\right)\le\:\frac{{\Lambda\:}}{\mu\:}$.

Proof

The total population of the system (1–6) is given by:

$$\:\frac{dN}{dt}={\Lambda\:}-\mu\:N$$

.

$$\:N\left(t\right)=N\left(0\right){e}^{-\mu\:t}+\frac{{\Lambda\:}}{\mu\:}$$

.

Therefore, the solution of the system (1–6) remains within a finite range during any given time.

Theorem 2

The given region $\:\mathcal{M}=\left\{\left(S,E,I,A,Q,R\right)\in\:{{\mathcal{R}}_{+}}^{6}:\right.$ $S\left(t\right)+E\left(t\right)+I\left(t\right)+A\left(t\right)+Q\left(t\right)+R\left(t\right)=$ $\left. N\left(t\right)\le\:\frac{{\Lambda\:}}{\mu\:},\:S\left(t\right)\ge\:0,E\left(t\right)\ge\:0,I\left(t\right)\ge\:0,A\left(t\right)\ge\:0,Q\left(t\right)\ge\:0,R\left(t\right)\ge\:0\right\}$ is positive.

Proof

As $\:\text{t}\to\:{\infty\:}$, the resultant of the system (1–6) can be expressed as follows:

$$\:\underset{t\to\:\infty\:}{\text{lim}}Sup\:N\left(t\right)\le\:\frac{{\Lambda\:}}{\mu\:}$$

.

Therefore, the set $\:\mathcal{M}$ is positive.

Theorem 3

For any initial value $\:\left(S\left(0\right),E\left(0\right),I\left(0\right),A\left(0\right),Q\left(0\right),R\left(0\right)\right)\in\:{{\mathcal{R}}_{+}}^{6}$ then the solution of $\:\left(S\left(t\right),E\left(t\right),I\left(t\right),A\left(t\right),Q\left(t\right),R\left(t\right)\right)$ for the system is positive for all time t > 0, $\:\tau\:\le\:t$.

Proof

Let us suppose that, $\:S\left(t\right)\ge\:0,E\left(t\right)\ge\:0,I\left(t\right)\ge\:0,A\left(t\right)\ge\:0,Q\left(t\right)\ge\:0,R\left(t\right)\ge\:0$.

Then from Eq. (1). we have.

$$\:{S}^{{\prime\:}}={\Lambda\:}-\frac{{\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}}{N}-\frac{{\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}}{N}-\mu\:S$$

,

$$\:S^{\prime\:}\ge\:-\frac{{\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}}{N}-\frac{{\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}}{N}-\mu\:S$$

,

$$\:\text{ln}\left|S\right|\ge\:-\left(\frac{{\alpha\:}_{1}{\beta\:}_{1}{I}_{\infty\:}{e}^{-\mu\:\tau\:}}{N}+\frac{{\alpha\:}_{2}{\beta\:}_{2}{A}_{\infty\:}{e}^{-\mu\:\tau\:}}{N}+\mu\:\right)t+S\left(0\right)$$

.

$$\:S\left(t\right)\ge\:S\left(0\right)\times\:{e}^{-\left(\frac{{\alpha\:}_{1}{\beta\:}_{1}{I}_{\infty\:}{e}^{-\mu\:\tau\:}}{N}+\frac{{\alpha\:}_{2}{\beta\:}_{2}{A}_{\infty\:}{e}^{-\mu\:\tau\:}}{N}+\mu\:\right)t}\ge\:0,\:\:\forall\:t\ge\:0,\:\tau\:<t.$$

Since $\:S\left(0\right)\ge\:0$, it follows that ‘S’ is the result of multiplying positive terms, and hence it is always positive at any time $\:t>0$.

Furthermore, we can establish the following associations through Eqs. (2), (3), (4), (5) and (6).

$$\:E\left(t\right)\ge\:E\left(0\right){e}^{-\left(\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right)t}\ge\:0,\:\:\forall\:t\ge\:0$$

.

$$\:I\left(t\right)\ge\:I\left(0\right)\times\:{e}^{-(\nu\:+\mu\:)t}\ge\:0,\:\forall\:t\ge\:0$$

.

$$\:A\left(t\right)\ge\:A\left(0\right)\times\:{e}^{-(\omega\:+\phi\:+\mu\:)t}\ge\:0,\:\forall\:t\ge\:0$$

.

$$\:Q\left(t\right)\ge\:Q\left(0\right)\times\:{e}^{-(\theta\:+\mu\:)t}\ge\:0,\:\forall\:t\ge\:0$$

.

$$\:R\left(t\right)\ge\:R\left(0\right)\times\:{e}^{-\mu\:t}\ge\:0,\:\forall\:t\ge\:0.$$

Therefore, the solution of the model $\:S,E,\:I,A,Q,\:R$, according to the initial conditions $\:S\left(0\right)\ge\:0$, $\:E\left(0\right)\ge\:0$,$\:I\left(0\right)\ge\:0$, $\:A\left(0\right)\ge\:0,\text{Q}\left(0\right)\ge\:0,$ $\:R\left(0\right)\ge\:0$ remains positive for any time $\:t>0,\:\tau\:<t$.

Model equilibria

The system (1–6) demonstrates the following equilibrium states:

Mumps Free Equilibrium $\:\left(MFE\right)={M}_{0}=\left({S}_{0},{E}_{0},{I}_{0},{A}_{0},{Q}_{0},{R}_{0}\right)=\left(\frac{{\Lambda\:}}{\mu\:},\text{0,0},\text{0,0},0\right)$

Mumps Existing Equilibrium $\:\left(MEE\right)={M}^{*}=\left({S}^{*},{E}^{*},{I}^{*},{A}^{*},{Q}^{*},{R}^{*}\right)$,

As, $\:{S}^{*}=\frac{{\Lambda\:}}{\left(\frac{{\alpha\:}_{1}{\beta\:}_{1}{I}^{*}{e}^{-\mu\:\tau\:}}{{N}^{*}}+\frac{{\alpha\:}_{2}{\beta\:}_{2}{A}^{*}{e}^{-\mu\:\tau\:}}{{N}^{*}}+\mu\:\right)}$, $\:{E}^{*}=\frac{\left({\alpha\:}_{1}{\beta\:}_{1}{I}^{*}{e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}{A}^{*}{e}^{-\mu\:\tau\:}\right){S}^{*}}{{N}^{*}\left(\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right)}$, $\:{I}^{*}=\frac{\kappa\:\delta\:{E}^{*}}{\left(\nu\:+\mu\:\right)}$, $\:{A}^{*}=\frac{{\tau\:}_{1}\delta\:{E}^{*}}{\left(\omega\:+\phi\:+\mu\:\right)}$ ,

$$\:{Q}^{*}=\frac{\omega\:{A}^{*}}{\left(\theta\:+\mu\:\right)}\:,\:{R}^{*}=\frac{\theta\:{Q}^{*}+\phi\:{A}^{*}+\nu\:{I}^{*}}{\mu\:}$$

The reproduction number $\:{\mathcal{R}}_{0}$ is calculated by using the next-generation method. Then the reproduction number is given below:

$$\:{\mathcal{R}}_{0}=\frac{{\Lambda\:}\kappa\:\delta\:{\alpha\:}_{1}{\beta\:}_{1}{e}^{-\mu\:\tau\:}}{\mu\:\left(\nu\:+\mu\:\right)\left(\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right)}$$

.

Local stability analysis

In this section, we will analyze the local stability of the model around the equilibria. The Jacobian matrix of the system (1–6) and its components are provided below.

$$\:{J}_{M}=\left[\begin{array}{cccccc}{J}_{11}&\:{J}_{12}&\:{J}_{13}&\:{J}_{14}&\:{J}_{15}&\:{J}_{16}\\\:{J}_{21}&\:{J}_{22}&\:{J}_{23}&\:{J}_{24}&\:{J}_{25}&\:{J}_{26}\\\:{J}_{31}&\:{J}_{32}&\:{J}_{33}&\:{J}_{34}&\:{J}_{35}&\:{J}_{36}\\\:{J}_{41}&\:{J}_{42}&\:{J}_{43}&\:{J}_{44}&\:{J}_{45}&\:{J}_{46}\\\:{J}_{51}&\:{J}_{52}&\:{J}_{53}&\:{J}_{54}&\:{J}_{55}&\:{J}_{56}\\\:{J}_{61}&\:{J}_{62}&\:{J}_{63}&\:{J}_{64}&\:{J}_{65}&\:{J}_{66}\end{array}\right]$$

(7)

.

$\:{J}_{11}=-{\alpha\:}_{1}{\beta\:}_{1}I{e}^{-\mu\:\tau\:}-{\alpha\:}_{2}{\beta\:}_{2}A{e}^{-\mu\:\tau\:}-\mu\:$,$\:\:{J}_{12}=0$, $\:{J}_{13}=-{\alpha\:}_{1}{\beta\:}_{1}S{e}^{-\mu\:\tau\:},$ $\:{J}_{14}=-{\alpha\:}_{2}{\beta\:}_{2}S{e}^{-\mu\:\tau\:}$, $\:{J}_{15}=0,\:\:{J}_{16}=0$, $\:{J}_{21}={\alpha\:}_{1}{\beta\:}_{1}I{e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}A{e}^{-\mu\:\tau\:}$, $\:{J}_{22}=-\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]$, $\:{J}_{23}={\alpha\:}_{1}{\beta\:}_{1}S{e}^{-\mu\:\tau\:}$, $\:{J}_{24}={\alpha\:}_{2}{\beta\:}_{2}S{e}^{-\mu\:\tau\:}$, $\:{J}_{25}=0$, $\:{J}_{26}=0$ $\:{J}_{31}=0$, $\:{J}_{32}={\upkappa\:}{\updelta\:}$, $\:{J}_{33}=-(\nu\:+\mu\:)$, $\:{J}_{34}=0$, $\:{J}_{35}=0$, $\:{J}_{36}=0$, $\:{J}_{41}=0$, $\:{J}_{42}={\tau\:}_{1}\delta\:$,$\:\:{J}_{43}=0$, $\:{J}_{44}=-\left({\upomega\:}+{\upphi\:}+{\upmu\:}\right)$, $\:{J}_{45}=0$, $\:{J}_{46}=0$, $\:{J}_{51}=0$, $\:{J}_{52}=0$, $\:{J}_{53}=0$, $\:{J}_{54}={\upomega\:}$, $\:{J}_{55}=-\left(\theta\:+\mu\:\right)$, $\:{J}_{56}=0$, $\:{J}_{61}=0$, $\:{J}_{62}=0$, $\:{J}_{62}=0$, $\:{J}_{63}={\upnu\:}$, $\:{J}_{64}={\upphi\:}$, $\:{J}_{65}={\uptheta\:}$, $\:{J}_{61}=-{\upmu\:}$

Theorem 4

(Local stability at MFE): If $\:{\mathcal{R}}_{0}<1$, then the Mumps-free equilibrium $\:\left(MFE\right)={M}_{0}=\left({S}_{0},{E}_{0},{I}_{0},{A}_{0},{Q}_{0},{R}_{0}\right)=\left(\frac{{\Lambda\:}}{\mu\:},\text{0,0},\text{0,0},0\right)$ is stable.

Proof

The $\:{J}_{M}$ at the $\:{M}_{0}=\left({S}_{0},{E}_{0},{I}_{0},{A}_{0},{Q}_{0},{R}_{0}\right)$ is as follows:

$$\:{\left.{J}_{M}\right|}_{{M}_{0}}=\left[\begin{array}{cccccc}-\mu\:&\:0&\:-{\alpha\:}_{1}{\beta\:}_{1}{S}_{0}{e}^{-\mu\:\tau\:}&\:-{\alpha\:}_{2}{\beta\:}_{2}{S}_{0}{e}^{-\mu\:\tau\:}&\:0&\:0\\\:0&\:-\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]&\:{\alpha\:}_{1}{\beta\:}_{1}{S}_{0}{e}^{-\mu\:\tau\:}&\:{\alpha\:}_{2}{\beta\:}_{2}{S}_{0}{e}^{-\mu\:\tau\:}&\:0&\:0\\\:0&\:{\upkappa\:}{\updelta\:}&\:-(\nu\:+\mu\:)&\:0&\:0&\:0\\\:0&\:{\tau\:}_{1}\delta\:&\:0&\:-\left({\upomega\:}+{\upphi\:}+{\upmu\:}\right)&\:0&\:0\\\:0&\:0&\:0&\:\omega\:&\:-\left(\theta\:+\mu\:\right)&\:0\\\:0&\:0&\:\nu\:&\:\phi\:&\:\theta\:&\:-\mu\:\end{array}\right]$$

(8)

.

Consider, $\:\left|{\left.{J}_{M}\right|}_{{M}_{0}}-\lambda\:I\right|=0$

$$\:\left|{\left.{J}_{M}\right|}_{{M}_{0}}-\lambda\:I\right|=\left[\begin{array}{cccccc}-\mu\:&\:0&\:-{\alpha\:}_{1}{\beta\:}_{1}\left(\frac{{\Lambda\:}}{\mu\:}\right){e}^{-\mu\:\tau\:}&\:-{\alpha\:}_{2}{\beta\:}_{2}\left(\frac{{\Lambda\:}}{\mu\:}\right){e}^{-\mu\:\tau\:}&\:0&\:0\\\:0&\:-\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]&\:{\alpha\:}_{1}{\beta\:}_{1}\left(\frac{{\Lambda\:}}{\mu\:}\right){e}^{-\mu\:\tau\:}&\:{\alpha\:}_{2}{\beta\:}_{2}\left(\frac{{\Lambda\:}}{\mu\:}\right){e}^{-\mu\:\tau\:}&\:0&\:0\\\:0&\:{\upkappa\:}{\updelta\:}&\:-(\nu\:+\mu\:)&\:0&\:0&\:0\\\:0&\:{\tau\:}_{1}\delta\:&\:0&\:-\left({\upomega\:}+{\upphi\:}+{\upmu\:}\right)&\:0&\:0\\\:0&\:0&\:0&\:\omega\:&\:-\left(\theta\:+\mu\:\right)&\:0\\\:0&\:0&\:\nu\:&\:\phi\:&\:\theta\:&\:-\mu\:\end{array}\right]=0$$

.

By solving the above determinant, we got the negative Eigenvalues and polynomial of 3rd degree

$$\:{\lambda\:}_{1}=-{\upmu\:}<0\:,{\lambda\:}_{2}=-{\upmu\:}<0,{\lambda\:}_{3}=-\left(\theta\:+\mu\:\right)<0$$

.

$$\:{\lambda\:}^{3}+{a}_{2}{\lambda\:}^{2}+{a}_{1}\lambda\:+{a}_{0}=0$$

.

where $\:{a}_{2}=\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]+\left(\nu\:+\mu\:\right)+\left({\upomega\:}+{\upphi\:}+{\upmu\:}\right)$,

$$\:{a}_{1}=\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]\left(\nu\:+\mu\:\right)\left[1-{\mathcal{R}}_{0}\right]+\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]\left({\upomega\:}+{\upphi\:}+{\upmu\:}\right)+\left({\upomega\:}+{\upphi\:}+{\upmu\:}\right)\left(\nu\:+\mu\:\right)+{\tau\:}_{1}\delta\:{\alpha\:}_{2}{\beta\:}_{2}\left(\frac{{\Lambda\:}}{\mu\:}\right){e}^{-\mu\:\tau\:}\left(\nu\:+\mu\:\right)$$

$$\:{a}_{1}=\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]\left(\nu\:+\mu\:\right)\left({\upomega\:}+{\upphi\:}+{\upmu\:}\right)\left[1-{\mathcal{R}}_{0}\right]+{\tau\:}_{1}\delta\:{\alpha\:}_{2}{\beta\:}_{2}\left(\frac{{\Lambda\:}}{\mu\:}\right){e}^{-\mu\:\tau\:}\left(\nu\:+\mu\:\right)$$

.

Since, $\:{\mathcal{R}}_{0}$ < 1 and $\:{a}_{2}{a}_{1}>{a}_{1}$. Therefore, the Routh-Hurwitz criterion satisfied. Hence $\:{M}_{0}$ is stable.

Theorem 5

(Local stability at MEE): If $\:{\mathcal{R}}_{0}>\:1$, Then the Mumps Existing Equilibrium $\:\left(MEE\right)={M}^{*}=\left({S}^{*},{E}^{*},{I}^{*},{A}^{*},{Q}^{*},{R}^{*}\right)$ is stable.

Proof

The $\:{J}_{M}$ at the $\:{M}^{*}$ is as follows:

$$\:\:{\left.{J}_{M}\right|}_{{M}^{*}}=\left[\begin{array}{cccccc}-{\alpha\:}_{1}{\beta\:}_{1}{I}^{*}{e}^{-\mu\:\tau\:}-{\alpha\:}_{2}{\beta\:}_{2}{A}^{*}{e}^{-\mu\:\tau\:}-\mu\:&\:0&\:-{\alpha\:}_{1}{\beta\:}_{1}{S}^{*}{e}^{-\mu\:\tau\:}&\:-{\alpha\:}_{2}{\beta\:}_{2}{S}^{*}{e}^{-\mu\:\tau\:}&\:0&\:0\\\:{\alpha\:}_{1}{\beta\:}_{1}{I}^{*}{e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}{A}^{*}{e}^{-\mu\:\tau\:}&\:-\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]&\:{\alpha\:}_{1}{\beta\:}_{1}{S}^{*}{e}^{-\mu\:\tau\:}&\:{\alpha\:}_{2}{\beta\:}_{2}{S}^{*}{e}^{-\mu\:\tau\:}&\:0&\:0\\\:0&\:{\upkappa\:}{\updelta\:}&\:-(\nu\:+\mu\:)&\:0&\:0&\:0\\\:0&\:{\tau\:}_{1}\delta\:&\:0&\:-\left({\upomega\:}+{\upphi\:}+{\upmu\:}\right)&\:0&\:0\\\:0&\:0&\:0&\:\omega\:&\:-\left(\theta\:+\mu\:\right)&\:0\\\:0&\:0&\:\nu\:&\:\phi\:&\:\theta\:&\:-\mu\:\end{array}\right]$$

(9)

.

Consider, $\:\left|{\left.{J}_{M}\right|}_{{\text{M}}^{*}}-\lambda\:I\right|=0$

$$\:\left|{\left.{J}_{M}\right|}_{{\text{M}}^{*}}-\lambda\:I\right|=\left|\begin{array}{cccc}-{\alpha\:}_{1}{\beta\:}_{1}{I}^{*}{e}^{-\mu\:\tau\:}-{\alpha\:}_{2}{\beta\:}_{2}{A}^{*}{e}^{-\mu\:\tau\:}-\mu\:-\lambda\:&\:0&\:-{\alpha\:}_{1}{\beta\:}_{1}{S}^{*}{e}^{-\mu\:\tau\:}&\:-{\alpha\:}_{2}{\beta\:}_{2}{S}^{*}{e}^{-\mu\:\tau\:}\\\:{\alpha\:}_{1}{\beta\:}_{1}{I}^{*}{e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}{A}^{*}{e}^{-\mu\:\tau\:}&\:-\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]-\lambda\:&\:{\alpha\:}_{1}{\beta\:}_{1}{S}^{*}{e}^{-\mu\:\tau\:}&\:{\alpha\:}_{2}{\beta\:}_{2}{S}^{*}{e}^{-\mu\:\tau\:}\\\:0&\:{\upkappa\:}{\updelta\:}&\:-(\nu\:+\mu\:)-\lambda\:&\:0\\\:0&\:{\tau\:}_{1}\delta\:&\:0&\:-\left({\upomega\:}+{\upphi\:}+{\upmu\:}\right)-\lambda\:\end{array}\right|=0$$

$$\:{\lambda\:}^{4}+{a}_{3}{\lambda\:}^{3}+{a}_{2}{\lambda\:}^{2}+{a}_{1}\lambda\:+{a}_{0}=0.$$

where ${a}_{3}=A+C+G+K,$${a}_{2}=\text{A}\text{C}+\text{G}\text{K}-\text{F}\text{D}-\text{H}\text{E},$ ${a}_{1}=GK\left(A+C\right)+AC\left(G+K\right)-FD\left(A+K\right)-HE\left(A+G\right)-BFD-HE,$${a}_{0}=ACGK-FDAK-HEAG-BFDK-HEG.$

And

$\:A={\alpha\:}_{1}{\beta\:}_{1}{I}^{*}{e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}{A}^{*}{e}^{-\mu\:\tau\:}+\mu,$ $B={\alpha\:}_{1}{\beta\:}_{1}{I}^{*}{e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}{A}^{*}{e}^{-\mu\:\tau\:},$ $C=\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right],$ $D={\upkappa\:}{\updelta\:},\:E={\tau\:}_{1}\delta\:,\:F={\alpha\:}_{1}{\beta\:}_{1}{S}^{*}{e}^{-\mu\:\tau\:},$ $G=\left(\nu\:+\mu\:\right),\:H={\alpha\:}_{2}{\beta\:}_{2}{S}^{*}{e}^{-\mu\:\tau\:},K=\left({\upomega\:}+{\upphi\:}+{\upmu\:}\right).$

Since, $\:{\mathcal{R}}_{0}>\:1$ and, the Routh-Hurwitz criterion of the 4th order satisfied. Hence the $\:{M}^{*}$ is stable.

Global stability analysis

Definition

Lassalle’s Invariance Principal²¹.

Consider a dynamical system described by the ordinary differential equation:

$$\:\dot{x}=f\left(x\right),\:x\in\:{\mathbb{R}}^{n}$$

Where $\:f\left(x\right)$ is a continuously differentiable vector field. Lassalle’s invariance principle states that if there exists a Lyapunov function $\:V\left(x\right)$ such that:

• i. $\:V\left(x\right)$ is non-increasing along the trajectories of the system, i.e., $\:\dot{V}\left(x\right)\le\:0$,

• ii. The set $\:S=\left\{x\in\:{\mathbb{R}}^{n}|\dot{V}\left(x\right)=0\right\}$ contains no other invariant sets except equilibria, then the trajectory of the system will asymptotically approach the largest invariant set in $\:S$.

Theorem 6

(Global Stability at MFE): If $\:{\mathcal{R}}_{0}<1$, then the Mumps-free equilibrium $\:\left(MFE\right)={M}_{0}=\left({S}_{0},{E}_{0},{I}_{0},{A}_{0},{Q}_{0},{R}_{0}\right)=\left(\frac{{\Lambda\:}}{\mu\:},\text{0,0},\text{0,0},0\right)$ is stable.

Proof

The Lyapunov function $\:U:\mathcal{M}\to\:\mathbb{R}$ defined as

$$\:U=\left(S-{S}_{0}-{S}_{0}log\frac{S}{{S}_{0}}\right)+E+I+A+Q+R,\forall\:\:\left(S,E,I,\text{A},\text{Q},R\right)\: \in \:\mathcal{M}$$

.

$$\:\frac{dU}{dt}=\left(\frac{S-{S}_{0}}{S}\right)\frac{dS}{dt}+\frac{dE}{dt}+\frac{dI}{dt}+\frac{dA}{dt}+\frac{dQ}{dt}+\frac{dR}{dt}$$

(10)

.

$$\begin{aligned}\frac{dU}{dt} &=\left(\frac{\text{S}-{S}_{0}}{S}\right)\left[{\Lambda\:}-\frac{{\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}}{N}-\frac{{\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}}{N}-\mu\:S\right] \\ & \quad +\left[\frac{{\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}}{N}+\frac{{\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}}{N}-\left(\kappa\:+{\tau\:}_{1}\right)\delta\:E-\mu\:E\right]+\left[\kappa\:\delta\:E-(\nu\:+\mu\:)I\right] \\ & \quad +\left[{\tau\:}_{1}\delta\:E-(\omega\:+\phi\:+\mu\:)A\right]+\left[\omega\:A-(\theta\:+\mu\:)Q\right]+\left[\theta\:Q+\phi\:A+\nu\:I-\mu\:R\right]\end{aligned}$$

.

$$\:\frac{d\text{U}}{dt}\le\:-\frac{{\Lambda\:}{\left(S-{S}_{0}\right)}^{2}}{{SS}_{0}}-{\upmu\:}\text{I}\left(1-\frac{{\alpha\:}_{1}{\beta\:}_{1}S{e}^{-\mu\:\tau\:}}{\mu\:}\right)-\mu\:A\left(1-\frac{{\gamma\:}_{2}S{e}^{-\mu\:{\tau\:}_{2}}}{\mu\:}\right)-\delta\:{\tau\:}_{1}\mu\:E-\alpha\:A-\mu\:R$$

.

$\:\frac{dU}{dt}\le\:0$, if$\:\:{\mathcal{R}}_{0}<1$ and $\:\frac{dU}{dt}=0$ if$\:\:S={S}_{0},E=0,I=0\:,\text{A}=0,\text{Q}=0\:and\:R=0$. So, $\:{M}_{0}$ is stable and only independent path by Lassalle’s invariance principle.

Theorem 7

(Global Stability at MEE): If $\:{\mathcal{\:}\mathcal{R}}_{0}>\:1$, the Mumps Existing Equilibrium $\:\left(MEE\right)={M}^{*}=\left({S}^{*},{E}^{*},{I}^{*},{A}^{*},{Q}^{*},{R}^{*}\right)$ is stable.

Proof

The Lyapunov function Z$\::\mathcal{M}\to\:\mathbb{R}$ defined as

$$\begin{aligned}Z & =\left(S-{S}^{*}-{S}^{*}log\frac{S}{{S}^{*}}\right)+\left(E-{E}^{*}-{E}^{*}log\frac{E}{{E}^{*}}\right) \\ & \quad +\left(I-{I}^{*}-{I}^{*}log\frac{I}{{I}^{*}}\right)+\left(A-{A}^{*}-{A}^{*}log\frac{A}{{A}^{*}}\right) \\ & \quad +\left(Q-{Q}^{*}-{Q}^{*}log\frac{q}{{Q}^{*}}\right)+\left(R-{R}^{*}-{R}^{*}log\frac{R}{{R}^{*}}\right),\forall\:\:\left(S,E,I,\text{A},\text{Q},R\right)\: \in \:\mathcal{M} \end{aligned}$$

.

$$\begin{aligned}\frac{dZ}{dt} & =\left(1-\frac{{S}^{*}}{S}\right)\frac{dS}{dt}+\left(1-\frac{{E}^{*}}{E}\right)\frac{dE}{dt} \\ & \quad +\left(1-\frac{{I}^{*}}{I}\right)\frac{dI}{dt}+\left(1-\frac{{A}^{*}}{A}\right)\frac{dA}{dt} \\ & \quad +\left(1-\frac{{Q}^{*}}{Q}\right)\frac{dQ}{dt}+\left(1-\frac{{R}^{*}}{R}\right)\frac{dR}{dt} \end{aligned}$$

(11)

$$\begin{aligned}\frac{dZ}{dt} & =\left(\frac{S-{S}^{*}}{S}\right)\left[{\Lambda\:}-{\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}-{\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}-\mu\:S\right] \\ & \quad +\left(\frac{E-{E}^{*}}{E}\right)\left[{\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}-\left(\kappa\:+{\tau\:}_{1}\right)\delta\:E-\mu\:E\right] \\ & \quad +\left(\frac{I-{I}^{*}}{I}\right)\left[\kappa\:\delta\:E-(\nu\:+\mu\:)I\right]+\left(\frac{A-{A}^{*}}{A}\right)\left[{\tau\:}_{1}\delta\:E-(\omega\:+\phi\:+\mu\:)A\right] \\ & \quad +\left(\frac{Q-{Q}^{*}}{Q}\right)\left[\omega\:A-(\theta\:+\mu\:)Q\right]+\left(\frac{R-{R}^{*}}{R}\right)\left[\theta\:Q+\phi\:A+\nu\:I-\mu\:R\right] \end{aligned}$$

.

$$\begin{aligned}\frac{dZ}{dt} & =-\frac{{{\Lambda\:}\left(S-{S}^{*}\right)}^{2}}{S{S}^{*}}-\frac{{{\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}\left(E-{E}^{*}\right)}^{2}}{E{E}^{*}} \\ & \quad -\frac{{\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}\left(E-{E}^{*}\right)}{E{E}^{*}}-\frac{{\kappa\:\delta\:E\left(I-{I}^{*}\right)}^{2}}{I{I}^{*}}-\frac{{{\tau\:}_{1}\delta\:E\left(A-{A}^{*}\right)}^{2}}{A{A}^{*}} \\ & \quad -\frac{{\omega\:A\left(Q-{Q}^{*}\right)}^{2}}{Q{Q}^{*}}-\frac{{\theta\:Q\left(R-{R}^{*}\right)}^{2}}{R{R}^{*}}-\frac{{\phi\:A\left(R-{R}^{*}\right)}^{2}}{R{R}^{*}}-\frac{{\phi\:A\left(R-{R}^{*}\right)}^{2}}{R{R}^{*}}\le\:0 \end{aligned}$$

.

So, $\:{M}^{*}$ is a stable and only independent path by Lassalle’s invariance principle.

Theorem 8

(Second order stability at MFE): The Mumps-free equilibrium $\:\left(MFE\right)={M}_{0}=\left({S}_{0},{E}_{0},{I}_{0},{A}_{0},{Q}_{0},{R}_{0}\right)=\left(\frac{{\Lambda\:}}{\mu\:},\text{0,0},\text{0,0},0\right)$ is stable asymptotically in the sense of global if $\:{\mathcal{R}}_{0}<1$.

Proof

The Volterra Lyapunov function $\:U:\mathcal{M}\to\:\mathbb{R}$ is defined as follows:

$$\:{U}^{{\prime\:}}\left(I\right)=\frac{1}{I}\frac{dI}{dt}$$

$$\:{U}^{{\prime\:}{\prime\:}}\left(I\right)=\frac{1}{I}\frac{{d}^{2}I}{d{t}^{2}}-\frac{1}{{I}^{2}}{\left(\frac{dI}{dt}\right)}^{2}$$

(12)

$$\:{U}^{{\prime\:}{\prime\:}}\left(t\right)\le\:{\left(\nu\:+\mu\:\right)}^{2}{\left(1-{\mathcal{R}}_{0}\right)}^{2}-{\left[\left(\frac{\kappa\:\delta\:{\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}}{\left(\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right)}-(\nu\:+\mu\:)\right)\right]}^{2}$$

.

$\:{U}^{{\prime\:}{\prime\:}}\left(t\right)\le\:0$, if$\:\:{\mathcal{R}}_{0}<1$. Since by the Lassalle invariance principle $\:{M}_{0}$ is the unique trajectory of the system (1–6). Therefore, $\:{M}_{0}$ is stable asymptotically in the sense of global.

Theorem 9

(Second order stability at MEE): The Mumps Existing Equilibrium $\:\left(MEE\right)={M}^{*}=\left({S}^{*},{E}^{*},{I}^{*},{A}^{*},{Q}^{*},{R}^{*}\right)$ is stable asymptotically in the sense of global if $\:{\mathcal{\:}\mathcal{R}}_{0}\ge\:1$.

Proof

The Volterra Lyapunov function Z$\::\mathcal{M}\to\:\mathbb{R}$ defined as

$$\:\frac{dZ}{dt}=\left(1-\frac{{S}^{*}}{S}\right)\frac{dS}{dt}+\left(1-\frac{{E}^{*}}{E}\right)\frac{dE}{dt}+\left(1-\frac{{I}^{*}}{I}\right)\frac{dI}{dt}+\left(1-\frac{{A}^{*}}{A}\right)\frac{dA}{dt}+\left(1-\frac{{Q}^{*}}{Q}\right)\frac{dQ}{dt}+\left(1-\frac{{R}^{*}}{R}\right)\frac{dR}{dt}$$

.

$$\begin{aligned}\frac{{d}^{2}Z}{d{t}^{2}} & =\frac{{S}^{*}}{{S}^{2}}{\left(\frac{dS}{dt}\right)}^{2}+\left(1-\frac{{S}^{*}}{S}\right)\frac{{d}^{2}S}{d{t}^{2}}+\frac{{E}^{*}}{{E}^{2}}{\left(\frac{dE}{dt}\right)}^{2} \\ & \quad +\left(1-\frac{{E}^{*}}{E}\right)\frac{{d}^{2}E}{d{t}^{2}}+\frac{{I}^{*}}{{I}^{2}}{\left(\frac{dI}{dt}\right)}^{2}+\left(1-\frac{{I}^{*}}{I}\right)\frac{{d}^{2}I}{d{t}^{2}}+\frac{{A}^{*}}{{A}^{2}}{\left(\frac{dA}{dt}\right)}^{2} \\ & \quad +\left(1-\frac{{A}^{*}}{A}\right)\frac{{d}^{2}A}{d{t}^{2}}+\frac{{Q}^{*}}{{Q}^{2}}{\left(\frac{dQ}{dt}\right)}^{2}+\left(1-\frac{{Q}^{*}}{Q}\right)\frac{{d}^{2}Q}{d{t}^{2}}+\frac{{R}^{*}}{{R}^{2}}{\left(\frac{dR}{dt}\right)}^{2}+\left(1-\frac{{R}^{*}}{R}\right)\frac{{d}^{2}R}{d{t}^{2}}\end{aligned}$$

(13)

$$\begin{aligned}\frac{{d}^{2}Z}{d{t}^{2}} & =\left[{{\Lambda\:}}^{2}+{\left({\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}\right)}^{2}+{\left({\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}\right)}^{2}+{\left(\mu\:S\right)}^{2} \right. \\ & \left. +2\left({\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}\right)\left({\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}\right)+2\left({\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}\right)\left(\mu\:S\right)+2\left({\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}\right)\left(\mu\:S\right)\right]\frac{{S}^{*}}{{S}^{2}} \\ & \quad -\left[2{\Lambda\:}\left({\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}\right)+2{\Lambda\:}\left({\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}\right)+2{\Lambda\:}\left(\mu\:S\right)\right]\frac{{S}^{*}}{{S}^{2}}+\left[{\Lambda\:}\left({\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}\right)+{\Lambda\:}\left({\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}\right)+{\Lambda\:}\mu\:\right]\frac{{S}^{*}}{S} \\ & \quad -\left[\left({\alpha\:}_{1}{\beta\:}_{1}I{e}^{-\mu\:\tau\:}\right)\left({\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}\right)+\left({\alpha\:}_{1}{\beta\:}_{1}I{e}^{-\mu\:\tau\:}\right)\left({\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}\right)+\left({\alpha\:}_{1}{\beta\:}_{1}I{e}^{-\mu\:\tau\:}\right)\left(\mu\:S\right) \right. \\ & \quad \left. +\left({\alpha\:}_{2}{\beta\:}_{2}A{e}^{-\mu\:\tau\:}\right)\left({\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}\right)+\left({\alpha\:}_{2}{\beta\:}_{2}A{e}^{-\mu\:\tau\:}\right)\left({\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}\right)+\left({\alpha\:}_{2}{\beta\:}_{2}A{e}^{-\mu\:\tau\:}\right)\left(\mu\:S\right) \right. \\ & \quad \left. +\left({\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}\right)\mu\:+\left({\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}\right)\mu\:+\left({\mu\:}^{2}S\right)\right]\frac{{S}^{*}}{S} +\left[\left({\alpha\:}_{1}{\beta\:}_{1}I{e}^{-\mu\:\tau\:}\right)\left({\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}\right) \right. \\ & \quad \left. +\left({\alpha\:}_{1}{\beta\:}_{1}I{e}^{-\mu\:\tau\:}\right)\left({\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}\right)+\left({\alpha\:}_{1}{\beta\:}_{1}I{e}^{-\mu\:\tau\:}\right)\left(\mu\:S\right)+\left({\alpha\:}_{2}{\beta\:}_{2}A{e}^{-\mu\:\tau\:}\right)\left({\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}\right) \right. \\ & \quad \left. +\left({\alpha\:}_{2}{\beta\:}_{2}A{e}^{-\mu\:\tau\:}\right)\left({\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}\right)+\left({\alpha\:}_{2}{\beta\:}_{2}A{e}^{-\mu\:\tau\:}\right)\left(\mu\:S\right)+\left({\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}\right)\mu\:+\left({\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}\right)\mu\:+\left({\mu\:}^{2}S\right)\right] \\ & \quad -\left[{\Lambda\:}\left({\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}\right)+{\Lambda\:}\left({\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}\right)+{\Lambda\:}\mu\:\right]+\left[{\left({\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}\right)}^{2} \right. \\ & \quad \left. +{\left({\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}\right)}^{2}+{\left(\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]E\right)}^{2}+2\left({\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}\right)\left({\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}\right)\right]\frac{{E}^{*}}{{E}^{2}} \\ & \quad -\left[2\left({\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}\right)\left(\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]E\right)+2\left({\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}\right)\left(\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]E\right)\right]\frac{{E}^{*}}{{E}^{2}} \\ & \quad +\left[\left({\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}\right)\left(\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]E\right)+\left({\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}\right)\left(\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]E\right)\right]\frac{{E}^{*}}{E} \\ & \quad -\left[{\left(\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]E\right)}^{2}\right]\frac{{E}^{*}}{E}+\left[{\left(\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]E\right)}^{2}\right] \\ & \quad -\left[\left({\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}\right)\left(\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]E\right)+\left({\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}\right)\left(\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]E\right)\right] \\ & \quad +\left[{\left(\kappa\:\delta\:E\right)}^{2}+{\left(\left(\nu\:+\mu\:\right)I\right)}^{2}\right]\frac{{I}^{*}}{{I}^{2}}-\left[2\left(\kappa\:\delta\:E\right)\left(\left(\nu\:+\mu\:\right)I\right)\right]\frac{{I}^{*}}{{I}^{2}} \\ & \quad +\left[\left(\kappa\:\delta\:E\right)\left(\left(\nu\:+\mu\:\right)I\right)\right]\frac{{I}^{*}}{I}-\left[{\left(\left(\nu\:+\mu\:\right)\right)}^{2}I\right]\frac{{I}^{*}}{I}+\left[{\left(\left(\nu\:+\mu\:\right)\right)}^{2}I\right] \\ & \quad -\left[\left(\kappa\:\delta\:E\right)\left(\left(\nu\:+\mu\:\right)I\right)\right]+\left[{\left({\tau\:}_{1}\delta\:E\right)}^{2} +{\left((\omega\:+\phi\:+\mu\:)A\right)}^{2}\right]\frac{{A}^{*}}{{A}^{2}} \\ & \quad -\left[2\left({\tau\:}_{1}\delta\:E\right)(\omega\:+\phi\:+\mu\:)A\right]\frac{{A}^{*}}{{A}^{2}}+\left[\left({\tau\:}_{1}\delta\:E\right)(\omega\:+\phi\:+\mu\:)A\right]\frac{{A}^{*}}{A} \\ & \quad -\left[{(\omega\:+\phi\:+\mu\:)}^{2}A\right]\frac{{A}^{*}}{A}+\left[{(\omega\:+\phi\:+\mu\:)}^{2}A\right]-\left[\left({\tau\:}_{1}\delta\:E\right)(\omega\:+\phi\:+\mu\:)A\right] \\ & \quad +\left[{\left(\omega\:A\right)}^{2}+{\left(\left(\theta\:+\mu\:\right)Q\right)}^{2}\right]\frac{{Q}^{*}}{{Q}^{2}}-\left[2\left(\omega\:A\right)\left(\theta\:+\mu\:\right)Q\right]\frac{{Q}^{*}}{{Q}^{2}}+\left[\left(\omega\:A\right)\left(\theta\:+\mu\:\right)Q\right]\frac{{Q}^{*}}{Q} \\ & \quad -\left[{\left(\left(\theta\:+\mu\:\right)\right)}^{2}Q\right]\frac{{Q}^{*}}{Q}+\left[{\left(\left(\theta\:+\mu\:\right)\right)}^{2}Q\right]-\left[\left(\omega\:A\right)\left(\theta\:+\mu\:\right)Q\right] \\ & \quad +\left[{\left(\theta\:Q\right)}^{2}+{\left(\phi\:A\right)}^{2}+{\left(\nu\:I\right)}^{2}+{\left(\mu\:R\right)}^{2}+2\left(\theta\:Q\right)\left(\phi\:A\right)+2\left(\theta\:Q\right)\left(\nu\:I\right)+2\left(\phi\:A\right)\left(\nu\:I\right)+2\left(\mu\:R\right)\left(\nu\:I\right)\right]\frac{{R}^{*}}{{R}^{2}} \\ & \quad -\left[2\left(\theta\:Q\right)\left(\mu\:R\right)++2\left(\phi\:A\right)\left(\mu\:R\right)\right]\frac{{R}^{*}}{{R}^{2}}+\left[\mu\:\theta\:Q+\phi\:A+\mu\:I\right]\frac{{R}^{*}}{R}-\left[{\mu\:}^{2}R\right]\frac{{R}^{*}}{R}+\left[{\mu\:}^{2}R\right]-\left[\mu\:\theta\:Q+\phi\:A+\mu\:I\right]\end{aligned}$$

.

For simplicity, we choose

$$\:\frac{{d}^{2}Z}{d{t}^{2}}={{\Psi\:}}_{1}-{{\Psi\:}}_{2}$$

.

It can be seen that

$$\:{{\Psi\:}}_{1}>{{\Psi\:}}_{2},\:\:\frac{{d}^{2}Z}{d{t}^{2}}>0$$

$$\:{{\Psi\:}}_{1}<{{\Psi\:}}_{2},\:\:\frac{{d}^{2}Z}{d{t}^{2}}<0$$

.

$\:{{\Psi\:}}_{1}={{\Psi\:}}_{2},\:\:\frac{{d}^{2}Z}{d{t}^{2}}=0$. Since by the Lassalle invariance principle $\:{M}^{*}$is the unique trajectory of the system (1–6). Therefore, $\:{M}^{*}$ is stable asymptotically in the sense of global.

Sensitivity analysis

We use derivative-based local techniques to do sensitivity analysis by computing the partial derivatives of outcomes concerning inputs. The study emphasizes the importance of transmission rates in shifting the dynamics from a state of being free of mumps to one in which mumps exist.

$\:{\mathcal{M}}_{{\Lambda\:}}=\frac{{\Lambda\:}}{{\mathcal{R}}_{0}}\times\:\frac{\partial\:{\mathcal{R}}_{0}}{\partial\:{\Lambda\:}}=1>0,\:$ $\:{\mathcal{M}}_{{\alpha\:}_{1}}=\frac{{\alpha\:}_{1}}{{\mathcal{R}}_{0}}\times\:\frac{\partial\:{\mathcal{R}}_{0}}{\partial\:{\alpha\:}_{1}}=1>0$, $\:{\mathcal{M}}_{{\beta\:}_{1}}=\frac{{\beta\:}_{1}}{{\mathcal{R}}_{0}}\times\:\frac{\partial\:{\mathcal{R}}_{0}}{\partial\:{\beta\:}_{1}}=1>0$ $\:{\mathcal{M}}_{\nu\:}=\frac{\nu\:}{{\mathcal{R}}_{0}}\times\:\frac{\partial\:{\mathcal{R}}_{0}}{\partial\:\nu\:}=\frac{-1}{\mu\:+\nu\:}<0$, $\:{\mathcal{M}}_{{\tau\:}_{1}}=\frac{{\tau\:}_{1}}{{\mathcal{R}}_{0}}\times\:\frac{\partial\:{\mathcal{R}}_{0}}{\partial\:{\tau\:}_{1}}=\frac{-\delta\:}{\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:}<0$, $\:{\mathcal{M}}_{\kappa\:}=\frac{\kappa\:}{{\mathcal{R}}_{0}}\times\:\frac{\partial\:{\mathcal{R}}_{0}}{\partial\:\kappa\:}=-\text{v}\text{e}<0$, $\:{\mathcal{M}}_{\delta\:}=\frac{\delta\:}{{\mathcal{R}}_{0}}\times\:\frac{\partial\:{\mathcal{R}}_{0}}{\partial\:\delta\:}=-\text{v}\text{e}<0$

The results indicate that $\:\:{\Lambda\:},{\alpha\:}_{1},\:{and\:\beta\:}_{1}$are more sensitive than $\:{\updelta\:},{\upkappa\:}$, $\:\nu\:,{\:and\:\tau\:}_{1}$ which is less sensitive, as desired.

Stochastic formulation phase 1

Let’s examine the vector $\:\text{U}\left(\text{t}\right)={[S\left(t\right),\:\text{E}\left(\text{t}\right),\text{I}(t),\text{A}\left(\text{t}\right),\text{Q}\left(\text{t}\right),\text{R}(t\left)\right]}^{\text{T}},\:$ along with the corresponding event probabilities as outlined in Table 2:

Table 2 Transition probabilities of the model.

To calculate the expected value and variance for the drift and diffusion coefficients of the system (1–6), we will proceed as follows:

$$\:{\text{E}}^{\text{*}}\left[\varDelta\:\text{U}\right]=\sum\:_{\text{i}=1}^{15}{\text{P}}_{\text{i}}{\text{T}}_{\text{i}}=\left[\begin{array}{c}\begin{array}{c}{\Lambda\:}-{\alpha\:}_{1}{\beta\:}_{1}S\left(t\right)I\left(t\right){e}^{-\mu\:\tau\:}-{\alpha\:}_{2}{\beta\:}_{2}S\left(t\right)A\left(t\right){e}^{-\mu\:\tau\:}-\mu\:S\left(t\right)\\\:{\alpha\:}_{1}{\beta\:}_{1}S\left(t\right)I\left(t\right){e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}S\left(t\right)A\left(t\right){e}^{-\mu\:\tau\:}-\left(\kappa\:+{\tau\:}_{1}\right)\delta\:E\left(t\right)-\mu\:E\left(t\right)\\\:\begin{array}{c}\kappa\:\delta\:E\left(t\right)-(\nu\:+\mu\:)I\left(t\right)\\\:{\tau\:}_{1}\delta\:E\left(t\right)-(\omega\:+\phi\:+\mu\:)A\left(t\right)\\\:\omega\:A\left(t\right)-(\theta\:+\mu\:)Q\left(t\right)\\\:\theta\:Q\left(t\right)+\phi\:A\left(t\right)+\nu\:I\left(t\right)-\mu\:R\left(t\right)\end{array}\end{array}\end{array}\right]\varDelta\:t$$

.

$$\text{Var}=\:\:{\text{E}}^{\text{*}}\left[\varDelta\:\text{U}{\varDelta\:\text{U}}^{\text{T}}\right]=\sum\:_{\text{i}=1}^{15}{\text{P}}_{\text{i}}{[\text{T}}_{\text{i}}{{\left]\right[\text{T}}_{\text{i}}]}^{\text{T}}$$

$$\:=\left[\begin{array}{cccccc}{P}_{1}+{P}_{2}+{P}_{3}+{P}_{4}&\:-{P}_{2}-{P}_{3}&\:0&\:0&\:0&\:0\\\:-{P}_{2}-{P}_{3}&\:{P}_{2}+{P}_{3}+{P}_{5}+{P}_{6}+{P}_{7}&\:0&\:{-P}_{6}&\:0&\:0\\\:0&\:{-P}_{5}&\:{P}_{8}+{P}_{9}&\:0&\:0&\:{-P}_{8}\\\:0&\:-{P}_{6}&\:0&\:{P}_{6}+{P}_{10}+{P}_{11}+{P}_{12}&\:{-P}_{10}&\:{-P}_{11}\\\:0&\:0&\:0&\:-{P}_{10}+{P}_{13}+{P}_{14}&\:{P}_{10}+{P}_{13}&\:0\\\:0&\:0&\:{-P}_{8}&\:{-P}_{11}&\:{-P}_{13}&\:{P}_{8}+{P}_{11}+{P}_{13}\end{array}\right]\varDelta\:t$$

$$\:\text{d}\text{r}\text{i}\text{f}\text{t}\:=\text{H}\left(\text{U}\left(\text{t}\right),\text{t}\right)=\frac{{\text{E}}^{\text{*}}\left[\varDelta\:\text{U}\right]}{\varDelta\:\text{t}}\:,\:\text{d}\text{i}\text{f}\text{f}\text{u}\text{s}\text{i}\text{o}\text{n}\:=\text{K}\left(\text{U}\left(\text{t}\right),\text{t}\right)=\sqrt{\frac{{\text{E}}^{\text{*}}[\varDelta\:\text{U}{\varDelta\:\text{U}}^{\text{T}}]}{\varDelta\:\text{t}}}\:,\:\text{s}\text{o}$$

$$\:\text{d}\text{U}\left(\text{t}\right)=\text{H}\left(\text{U}\left(\text{t}\right),\text{t}\right)\text{d}\text{t}+\text{K}\left(\text{U}\left(\text{t}\right),\text{t}\right)\text{d}\text{B}\left(\text{t}\right)$$

(14)

.

Equation (14) is referred to as the stochastic differential equation, where $\:\text{B}\left(\text{t}\right)$ represents the Brownian motion.

The Euler Maruyama technique is utilized to simulate the outcomes of Eq. (14) by consulting the academic literature on the topic. The information is displayed in Table 2 and is as outlined:

$$\:{A}_{n+1}={A}_{n}+\text{G}\left({\text{A}}_{n},\text{t}\right){\Delta\:}\text{t}+\text{H}({\text{A}}_{n},\text{t})\text{d}\text{B}\left(\text{t}\right)$$

.

$$\begin{aligned} & \left[\begin{array}{c}{S}^{n+1}\\\:{E}^{n+1}\\\:{I}^{n+1}\\\:{A}^{n+1}\\\:\begin{array}{c}{Q}^{n+1}\\\:{A}^{n+1}\end{array}\end{array}\right] =\left[\begin{array}{c}{S}^{n}\\\:{E}^{n}\\\:{I}^{n}\\\:{A}^{n}\\\:\begin{array}{c}{Q}^{n}\\\:{A}^{n}\end{array}\end{array}\right]+\left[\begin{array}{c}\begin{array}{c}{\Lambda\:}-{\alpha\:}_{1}{\beta\:}_{1}S\left(t\right)I\left(t\right){e}^{-\mu\:\tau\:}-{\alpha\:}_{2}{\beta\:}_{2}S\left(t\right)A\left(t\right){e}^{-\mu\:\tau\:}-\mu\:S\left(t\right)\\\:{\alpha\:}_{1}{\beta\:}_{1}S\left(t\right)I\left(t\right){e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}S\left(t\right)A\left(t\right){e}^{-\mu\:\tau\:}-\left(\kappa\:+{\tau\:}_{1}\right)\delta\:E\left(t\right)-\mu\:E\left(t\right)\\\:\begin{array}{c}\kappa\:\delta\:E\left(t\right)-(\nu\:+\mu\:)I\left(t\right)\\\:{\tau\:}_{1}\delta\:E\left(t\right)-(\omega\:+\phi\:+\mu\:)A\left(t\right)\\\:\omega\:A\left(t\right)-(\theta\:+\mu\:)Q\left(t\right)\\\:\theta\:Q\left(t\right)+\phi\:A\left(t\right)+\nu\:I\left(t\right)-\mu\:R\left(t\right)\end{array}\end{array}\end{array}\right]{\Delta\:}t \\ & + \sqrt{\left[\left[\begin{array}{cccccc}{P}_{1}+{P}_{2}+{P}_{3}+{P}_{4}&\:-{P}_{2}-{P}_{3}&\:0&\:0&\:0&\:0\\\:-{P}_{2}-{P}_{3}&\:{P}_{2}+{P}_{3}+{P}_{5}+{P}_{6}+{P}_{7}&\:0&\:{-P}_{6}&\:0&\:0\\\:0&\:{-P}_{5}&\:{P}_{8}+{P}_{9}&\:0&\:0&\:{-P}_{8}\\\:0&\:-{P}_{6}&\:0&\:{P}_{6}+{P}_{10}+{P}_{11}+{P}_{12}&\:{-P}_{10}&\:{-P}_{11}\\\:0&\:0&\:0&\:-{P}_{10}+{P}_{13}+{P}_{14}&\:{P}_{10}+{P}_{13}&\:0\\\:0&\:0&\:{-P}_{8}&\:{-P}_{11}&\:{-P}_{13}&\:{P}_{8}+{P}_{11}+{P}_{13}\end{array}\right]{\Delta\:}t\right]}{\Delta\:}{B}_{n}\end{aligned}$$

where the discretization parameter is denoted by $\:\varDelta\:t$.

Stochastic formulation phase 2

The system of stochastic delay differential equations (SDDEs) is a mathematical model that describes the evolution of a set of variables over time, where the equations involve both deterministic time delays and stochastic (random) components. Where the stochastic term $\:\sigma\:d\left(B\left(t\right)\right)$ introduces randomness into the system of differential Eq. 

$$\:dS\left(t\right)=\left[{\Lambda\:}-\frac{{\alpha\:}_{1}{\beta\:}_{1}S\left(t\right)I\left(t\right){e}^{-\mu\:\tau\:}}{N}-\frac{{\alpha\:}_{2}{\beta\:}_{2}S\left(t\right)A\left(t\right){e}^{-\mu\:\tau\:}}{N}-\mu\:S\left(t\right)\right]dt+{\sigma\:}_{1}S\left(t\right)d\left(B\left(t\right)\right)$$

(15)

$$\:dE\left(t\right)=\left[\frac{{\alpha\:}_{1}{\beta\:}_{1}S\left(t\right)I\left(t\right){e}^{-\mu\:\tau\:}}{N}+\frac{{\alpha\:}_{2}{\beta\:}_{2}S\left(t\right)A\left(t\right){e}^{-\mu\:\tau\:}}{N}-\left(\kappa\:+{\tau\:}_{1}\right)\delta\:E\left(t\right)-\mu\:E\left(t\right)\right]dt+{\sigma\:}_{2}E\left(t\right)d\left(B\left(t\right)\right)$$

(16)

$$\:dI\left(t\right)=\left[\kappa\:\delta\:E\left(t\right)-(\nu\:+\mu\:)I\left(t\right)\right]dt+{\sigma\:}_{3}I\left(t\right)d\left(B\left(t\right)\right)$$

(17)

$$\:dA\left(t\right)=\left[{\tau\:}_{1}\delta\:E\left(t\right)-(\omega\:+\phi\:+\mu\:)A\left(t\right)\right]dt+{\sigma\:}_{4}A\left(t\right)d\left(B\left(t\right)\right)$$

(18)

$$\:dQ\left(t\right)=\left[\omega\:A\left(t\right)-(\theta\:+\mu\:)Q\left(t\right)\right]dt+{\sigma\:}_{5}Q\left(t\right)d\left(B\left(t\right)\right)$$

(19)

$$\:dR\left(t\right)=\left[\theta\:Q\left(t\right)+\phi\:A\left(t\right)+\nu\:I\left(t\right)-\mu\:R\left(t\right)\right]dt+{\sigma\:}_{6}R\left(t\right)d\left(B\left(t\right)\right)$$

(20)

.

Subject to non-negative initial conditions $\:S\left(t\right)={S}_{0}\ge\:0$,$\:\:E\left(t\right)={E}_{0}\ge\:0,I\left(t\right)={I}_{0}\ge\:0,A\left(t\right)={A}_{0}\ge\:0,Q\left(t\right)={Q}_{0}\ge\:0,\:R\left(t\right)={R}_{0}\ge\:0.$

where $\:{\sigma\:}_{i}=\text{1,2},\text{3,4},5,6$ denote the randomness of each compartment and B(t) indicates the Brownian motion.

Feasible properties

Let us examine a probability space represented as $\:(\mathcal{M},F,P)$ and filtered by $\:{\left\{{F}_{t}\right\}}_{t \in R}$. All P-null sets that fulfill the criteria of being both right-continuous and growing are encompassed inside $\:{F}_{o}$. Represent

$$\:U\left(t\right)=(S\left(t\right),E\left(t\right),I\left(t\right),A\left(t\right),\:Q\left(t\right),R(t\left)\right)$$

(21)

.

And the norm $\:\left|U\left(t\right)\right|=\sqrt{{S}^{2}\left(t\right)+{E}^{2}\left(t\right)+{I}^{2}\left(t\right)+{A}^{2}\left(t\right)+{Q}^{2}\left(t\right)+{R}^{2}\left(t\right)}$. And denote $\:{O}^{5,1}({R}^{6}\times\:\left(0,\infty\:\right);{R}_{+})$

We analyze the collection of all non-negative functions V (U, t) defined on $\:{R}^{6}\times\:\left(0,\infty\:\right)$, which possess continuous second-order differentiability in U and first-order differentiability in t. Within this specific framework, we define the differential operator Q precisely, which is associated with a three-dimensional stochastic delayed differential Eq. 

$$\:dU\left(t\right)=H\left(U,t\right)dt+K\left(U,t\right)dW\left(t\right)$$

(22)

.

As, $\:Q=\frac{\partial\:}{\partial\:t}+\sum\:_{i=1}^{6}{H}_{i}(U,t)\frac{\partial\:}{\partial\:{u}_{i}}+\frac{1}{2}\sum\:_{i,j=1}^{6}{\left({K}^{T}\left(U,t\right)K\left(U,t\right)\right)}_{i,j}\times\:\frac{{\partial\:}^{2}}{\partial\:{U}_{i}\partial\:{U}_{j}}$

If Q acts on a function $\:V\in\:{O}^{\text{5,1}}({R}^{6}\times\:\left(0,\infty\:\right);{R}_{+})$ then we denote

$$\:QV\left(U,t\right)={V}_{t}\left(U,t\right)+{V}_{U}\left(U,t\right)H\left(U,t\right)+\frac{1}{2}Trace\left({K}^{T}\left(U,t\right){V}_{UU}\left(U,t\right)K\left(U,t\right)\right),$$

.

Where $\:T$ means transportation.

Theorem 9

For a given model (15–20) and initial values $\:(S\left(0\right),E\left(0\right),I\left(0\right),A\left(0\right),Q\left(0\right),R(0\left)\right)\in\:{R}_{+}^{6}$, there exists a single solution $\:(S\left(t\right),E\left(t\right),I\left(t\right),A\left(t\right),Q\left(t\right),R(t\left)\right)$ on $\:t\ge\:0,\tau\:<t$, and this solution will always remain in $\:{R}_{+}^{6}$ with a probability of one.

Proof

According to Ito’s formula, the model (15–20) has a unique local positive solution in the interval $\:\left[0,{\tau\:}_{e}\right]$, and the time at which it explodes is represented by $\:{\tau\:}_{e}$. Since all the coefficients in the aforementioned model satisfy the local Lipschitz criterion.

Next, we will demonstrate that the provided model (15–20) possesses a solution in the global sense, specifically when $\:{\tau\:}_{e}$ approaches infinity with practically certain probability.

Assume that $\:{m}_{o}=0$ is a suitably high value such that $\:S\left(0\right),E\left(0\right),I\left(0\right),A\left(0\right),Q\left(0\right)$ and $\:R\left(0\right)$ fall within the range $\:[\frac{1}{{m}_{o}},{m}_{o}]$. Let’s define a sequence of interrupting times for each integer $\:m\ge\:{m}_{o}$.

$$\begin{aligned}{\tau\:}_{m} & =\text{inf}\left\{t \in \left[0,{\tau\:}_{e}\right]:S\left(t\right)\notin\:\left(\frac{1}{m},m\right)ore \; \right. \\ & \quad \left. E\left(t\right)\notin\:\left(\frac{1}{m},m\right)\:or\:I\left(t\right)\notin\:\left(\frac{1}{m},m\right) \right. \\ & \quad \left. or\:A\left(t\right)\notin\:\left(\frac{1}{m},m\right)\:or\:Q\left(t\right)\notin\:\left(\frac{1}{m},m\right)\:or\:R\left(t\right)\notin\:\left(\frac{1}{m},m\right)\right\}\end{aligned}$$

(23)

.

Where we set $\:inf\varphi\:=\infty\:\left(\varphi\:\:\text{r}\text{e}\text{p}\text{r}\text{e}\text{s}\text{e}\text{n}\text{t}\text{s}\:\text{t}\text{h}\text{e}\:\text{e}\text{m}\text{p}\text{t}\text{y}\:\text{s}\text{e}\text{t}\right)$. Since $\:{\tau\:}_{m}$ is non-decreasing as $\:m\to\:\infty\:,$

$$\:{\tau\:}_{\infty\:}\:=\underset{m\to\:\infty\:}{\text{lim}}{\tau\:}_{m}$$

(24)

.

Then the limit $\:{\tau\:}_{\infty\:}$ is less than or equal to $\:{\tau\:}_{e}$. Now, we must demonstrate that $\:{\tau\:}_{\infty\:}$ approaches infinity with almost certain probability.

If this condition is not met, then there exist positive values T and ε, where T is more than zero and ε is between 0 and 1, such that

$$\:P\left\{{\tau\:}_{\infty\:}\le\:T\right\}>\epsilon\:$$

(25)

.

this, there is an integer $\:{m}_{1}>{m}_{0}$ such that

$$\:P\left\{{\tau\:}_{m}\le\:T\right\}\ge\:\epsilon\:\:\:\:\:\:\:\:\forall\:\:m\ge\:{m}_{1}$$

(26)

$$\:\text{D}\text{e}\text{f}\text{i}\text{n}\text{e}\:\text{a}\:{O}^{6}-\text{f}\text{u}\text{n}\text{c}\text{t}\text{i}\text{o}\text{n}\:f:{R}_{+}^{6}\to\:{R}_{+}\:\text{b}\text{y}$$

$$\:f\left(S,E,I,A,Q,R\right)=\left(S-1-\text{ln}S\right)+\left(E-1-\text{ln}E\right)+\left(I-1-\text{ln}I\right)+\left(A-1-\text{ln}A\right)+\left(Q-1-\text{ln}Q\right)+\left(R-1-\text{ln}R\right)$$

(27)

.

By using Ito’s formula, we calculate

$$\begin{aligned}df\left(S,E,I,A,Q,R\right) & =\left(1-\frac{1}{S}\right)dS+\left(1-\frac{1}{E}\right)dE \\ & \quad +\left(1-\frac{1}{I}\right)dI+\left(1-\frac{1}{A}\right)dA+\left(1-\frac{1}{Q}\right)dQ \\& \quad +\left(1-\frac{1}{R}\right)dR+\frac{{\sigma\:}_{1}^{2}+{\sigma\:}_{2}^{2}+{\sigma\:}_{3}^{2}+{\sigma\:}_{4}^{2}+{\sigma\:}_{5}^{2}+{\sigma\:}_{6}^{2}}{2}dt\end{aligned}$$

$$\begin{aligned}df\left(S,E,I,A,Q,R\right) & =\left(1-\frac{1}{S}\right)\left(\left({\Lambda\:}-\frac{{\alpha\:}_{1}{\beta\:}_{1}S\left(t\right)I\left(t\right){e}^{-\mu\:\tau\:}}{N}-\frac{{\alpha\:}_{2}{\beta\:}_{2}S\left(t\right)A\left(t\right){e}^{-\mu\:\tau\:}}{N}-\mu\:S\left(t\right)\right)dt+{\sigma\:}_{1}S\left(t\right)d\left(B\left(t\right)\right)\right) \\ & \quad +\left(1-\frac{1}{E}\right)\left(\left(\frac{{\alpha\:}_{1}{\beta\:}_{1}S\left(t\right)I\left(t\right){e}^{-\mu\:\tau\:}}{N}+\frac{{\alpha\:}_{2}{\beta\:}_{2}S\left(t\right)A\left(t\right){e}^{-\mu\:\tau\:}}{N}-\left(\kappa\:+{\tau\:}_{1}\right)\delta\:E\left(t\right)-\mu\:E\left(t\right)\right)dt+{\sigma\:}_{2}E\left(t\right)d\left(B\left(t\right)\right)\right) \\ & \quad +\left(1-\frac{1}{I}\right)\left(\left(\kappa\:\delta\:E\left(t\right)-(\nu\:+\mu\:)I\left(t\right)\right)dt+{\sigma\:}_{3}I\left(t\right)d\left(B\left(t\right)\right)\right) \\ & \quad +\left(1-\frac{1}{A}\right)\left(\left({\tau\:}_{1}\delta\:E\left(t\right)-(\omega\:+\phi\:+\mu\:)A\left(t\right)\right)dt+{\sigma\:}_{4}A\left(t\right)d\left(B\left(t\right)\right)\right) \\ & \quad +\left(1-\frac{1}{Q}\right)\left(\left(\omega\:A\left(t\right)-(\theta\:+\mu\:)Q\left(t\right)\right)dt+{\sigma\:}_{5}Q\left(t\right)d\left(B\left(t\right)\right)\right) \\ & \quad +\left(1-\frac{1}{R}\right)\left(\theta\:Q\left(t\right)+\phi\:A\left(t\right)+\nu\:I\left(t\right)-\mu\:R\left(t\right)\right)dt+{\sigma\:}_{6}R\left(t\right)d\left(B\left(t\right)\right)\end{aligned}$$

.

$$\begin{aligned}df\left(S,E,I,A,Q,R\right) & \le\:\left[{\Lambda\:}+6\mu\:+\kappa\:\delta\:+{\tau\:}_{1}\delta\:+\nu\:+\theta\:+\phi\:+\omega\:+\frac{{\sigma\:}_{1}^{2}+{\sigma\:}_{2}^{2}+{\sigma\:}_{3}^{2}+{\sigma\:}_{4}^{2}+{\sigma\:}_{5}^{2}+{\sigma\:}_{6}^{2}}{2}\right]dt \\ & \quad +\:[{\sigma\:}_{1}S\left(t\right)+{\sigma\:}_{2}E\left(t\right)+{\sigma\:}_{3}I\left(t\right)+{\sigma\:}_{4}A\left(t\right)+{\sigma\:}_{5}Q\left(t\right)+{\sigma\:}_{6}R\left(t\right)]d\left(B\left(t\right)\right)\end{aligned}$$

(28)

.

For simplification, we assume.

$\:{N}_{1}={\Lambda\:}+6\mu\:+\kappa\:\delta\:+{\tau\:}_{1}\delta\:+\nu\:+\theta\:+\phi\:+\omega\:+\frac{{\sigma\:}_{1}^{2}+{\sigma\:}_{2}^{2}+{\sigma\:}_{3}^{2}+{\sigma\:}_{4}^{2}+{\sigma\:}_{5}^{2}+{\sigma\:}_{6}^{2}}{2}$, Then Eq. (28) could be written as:

$$\:df\left(S,E,I,A,Q,R\right)\le\:{N}_{1}dt+\:[{\sigma\:}_{1}S\left(t\right)+{\sigma\:}_{2}E\left(t\right)+{\sigma\:}_{3}I\left(t\right)+{\sigma\:}_{4}A\left(t\right)+{\sigma\:}_{5}Q\left(t\right)+{\sigma\:}_{6}R\left(t\right)]d\left(B\left(t\right)\right)$$

(29)

.

Assume $\:{N}_{1}$ to be a positive constant. By integrating from 0 to $\:{\tau\:}_{m}\wedge\:\tau\:$, we obtain

$$\begin{aligned}\underset{0}{\overset{{\tau\:}_{m}\wedge\:\tau\:}{\int\:}}df(S\left(t\right),E\left(t\right),I\left(t\right),A\left(t\right),Q\left(t\right),R(t\left)\right) & \le\:\underset{0}{\overset{{\tau\:}_{m}\wedge\:\tau\:}{\int\:}}{N}_{1}ds+\underset{0}{\overset{{\tau\:}_{m}\wedge\:\tau\:}{\int\:}}[{\sigma\:}_{1}S\left(t\right)+{\sigma\:}_{2}E\left(t\right) \\ & \quad +{\sigma\:}_{3}I\left(t\right)+{\sigma\:}_{4}A\left(t\right)+{\sigma\:}_{5}Q\left(t\right)+{\sigma\:}_{6}R\left(t\right)]d\left(B\left(t\right)\right)\end{aligned}$$

(30)

.

where $\:{\tau\:}_{m}\wedge\:\tau\:=\text{min}\left({\tau\:}_{m},\:T\right),$ taking the expectations lead to

$$\:Ef\left(S\left({\tau\:}_{m}\wedge\:\tau\:\right),E\left({\tau\:}_{m}\wedge\:\tau\:\right),I\left({\tau\:}_{m}\wedge\:\tau\:\right),A\left({\tau\:}_{m}\wedge\:\tau\:\right),Q\left({\tau\:}_{m}\wedge\:\tau\:\right),R\left({\tau\:}_{m}\wedge\:\tau\:\right)\right)\le\:f\left(S\left(0\right),E\left(0\right),I\left(0\right),A\left(0\right),Q\left(0\right)R\left(0\right)\right)+{N}_{1}T$$

(31)

.

Set $\:{{\Omega\:}}_{m}=\{{\tau\:}_{m}\le\:T\}$ for $\:m>{m}_{1}$ and from (26), we have $\:P\left({{\Omega\:}}_{m}\ge\:\epsilon\:\right).\:$For every $\:l\in\:{{\Omega\:}}_{m}\:$there are some $\:i$ such that $\:{u}_{i}({\tau\:}_{m},l)$ equals either $\:m$ or $\:\frac{1}{m}$ for $\:i=\text{1,2},\text{3,4},\text{5,6};$

Hence, $\:f\left(S\left({\tau\:}_{m},l\right),E\left({\tau\:}_{m},l\right),I\left({\tau\:}_{m},l\right),A\left({\tau\:}_{m},l\right),Q\left({\tau\:}_{m},l\right),R\left({\tau\:}_{m},l\right)\right)$ is less than $\:\text{min}\left\{m-1-\text{ln}m,\frac{1}{m}-1-\text{ln}\frac{1}{m}\right\}.$

Then we obtain,

$$\begin{aligned} & f\left(S\left(0\right),E\left(0\right),I\left(0\right),A\left(0\right),Q\left(0\right),R\left(0\right)\right)+{N}_{1}T \\ & \ge\:E\left({I}_{{{\Omega\:}}_{m}\left(l\right)}V\left(S\left({\tau\:}_{m}\right),E\left({\tau\:}_{m}\right),I\left({\tau\:}_{m}\right),A\left({\tau\:}_{m}\right),Q\left({\tau\:}_{m}\right),R\left({\tau\:}_{m}\right)\right)\right) \\ & \ge\:\left\{\text{min}\left(m-1-\text{ln}m,\frac{1}{m}-1-\text{ln}\frac{1}{m}\right)\:\right\} \end{aligned}$$

(32)

.

The indicator function is represented by $\:{I}_{{{\Omega\:}}_{m}\left(l\right)}$ of $\:{{\Omega\:}}_{m}$. Letting $\:m\to\:\infty\:$ leads to the contradiction $\:\infty\:=f\left(S\left(0\right),E\left(0\right),I\left(0\right),A\left(0\right),Q\left(0\right),R\left(0\right)\right)+{N}_{1}T<\infty\:$, as desired.

Stochastic nonstandard finite difference scheme with delay parameter

In our non-parametric perturbation model’s first Eq. (15) can be stated using an unconventional computing technique; specifically, Eqs. (15–20) could be solved using a stochastic non-standard finite difference technique with delay parameter.

$$\:dS\left(t\right)={\Lambda\:}-{\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}-{\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}-\mu\:S+{\sigma\:}_{1}SdB$$

(33)

.

For the Stochastic NSFD approach, the structure of the equation is as follows:

$$\:\frac{{S}^{n+1}-{S}^{n}}{h}={\Lambda\:}-{\alpha\:}_{1}{\beta\:}_{1}{S}^{n+1}{I}^{n}{e}^{-\mu\:\tau\:}-{\alpha\:}_{2}{\beta\:}_{2}{S}^{n+1}{A}^{n}{e}^{-\mu\:\tau\:}-\mu\:{S}^{n+1}+{\sigma\:}_{1}{S}^{n}\varDelta\:{B}_{n}$$

(34)

.

The system (15–20) can be broken down using the stochastic NSFD process and the complete system can then be written as follows:

$$\:{S}^{n+1}=\frac{{S}^{n}+h\left[{\Lambda\:}+{\sigma\:}_{1}{S}^{n}\varDelta\:{B}_{n}\right]}{1+h\left[{\alpha\:}_{1}{\beta\:}_{1}{I}^{n}{e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}{A}^{n}{e}^{-\mu\:\tau\:}+\mu\:\right]}$$

(35)

$$\:{E}^{n+1}=\frac{{E}^{n}+h\left[{\alpha\:}_{1}{\beta\:}_{1}{I}^{n}{e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}{A}^{n}{e}^{-\mu\:\tau\:}+{\sigma\:}_{2}{E}^{n}\varDelta\:{B}_{n}\right]}{1+h\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]}$$

(36)

$$\:{I}^{n+1}=\frac{{I}^{n}+h\left[\kappa\:\delta\:{E}^{n}+{\sigma\:}_{3}{I}^{n}\varDelta\:{B}_{n}\right]}{1+h(\nu\:+\mu\:)}$$

(37)

$$\:{A}^{n+1}=\frac{{A}^{n}+h\left[{\tau\:}_{1}\delta\:{E}^{n}+{\sigma\:}_{4}{A}^{n}\varDelta\:{B}_{n}\right]}{1+h(\omega\:+\phi\:+\mu\:)}$$

(38)

$$\:{Q}^{n+1}=\frac{{Q}^{n}+h\left[\omega\:{A}^{n}+{\sigma\:}_{5}{Q}^{n}\varDelta\:{B}_{n}\right]}{1+h(\theta\:+\mu\:)}$$

(39)

$$\:{R}^{n+1}=\frac{{R}^{n}+h\left[\theta\:{Q}^{n}+\phi\:{A}^{n}+\nu\:{I}^{n}+{\sigma\:}_{6}{R}^{n}\varDelta\:{B}_{n}\right]}{1+h\mu\:}$$

(40)

.

where,$\:\:n=\text{0,1},2,\dots\:$ and discretization gap is denoted by “$\:h$”.

where h represents a discretization parameter and $\:n=\text{0,1},2,\dots\:$ and $\:\varDelta\:{\text{B}}_{n}=\varDelta\:{\text{B}}_{{t}_{n+1}}-\varDelta\:{\text{B}}_{{t}_{n}}$ is a general normal distribution. i.e. $\:\varDelta\:{\text{B}}_{\text{n}}\sim\text{N}(0,\:1$).

Stability analysis

Assuming $\:\varDelta\:{B}_{n}=0$, the system (35–40) consists of functions $\:U,\:V,\:W,\:X,\:Y,\:and\:Z.$

$\:U=\frac{S+h\left[{\Lambda\:}\right]}{1+h\left[{\alpha\:}_{1}{\beta\:}_{1}I{e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}A{e}^{-\mu\:\tau\:}+\mu\:\right]}$, $\:V=\frac{E+h\left[{\alpha\:}_{1}{\beta\:}_{1}SI{e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}SA{e}^{-\mu\:\tau\:}\right]}{1+h\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]}$, $\:W=\frac{I+h\left[\kappa\:\delta\:E\right]}{1+h(\nu\:+\mu\:)}$, $\:X=\frac{A+h\left[{\tau\:}_{1}\delta\:E\right]}{1+h(\omega\:+\phi\:+\mu\:)}$, $\:Y=\frac{Q+h\left[\omega\:A\right]}{1+h(\theta\:+\mu\:)}$, $\:Z=\frac{R+h\left[\theta\:Q+\phi\:A+\nu\:I\right]}{1+h\mu\:}$

It is commonly understood that a system of the forms (35–40) converges to the model’s optimal state if and only if the Jacobian’s spectral radius, (J), lie in the unit circle.

$$\:{J}_{M}=\left[\begin{array}{cccccc}\frac{\partial\:U}{\partial\:S}&\:\frac{\partial\:U}{\partial\:E}&\:\frac{\partial\:U}{\partial\:I}&\:\frac{\partial\:U}{\partial\:A}&\:\frac{\partial\:U}{\partial\:Q}&\:\frac{\partial\:U}{\partial\:R}\\\:\frac{\partial\:V}{\partial\:S}&\:\frac{\partial\:V}{\partial\:E}&\:\frac{\partial\:V}{\partial\:I}&\:\frac{\partial\:V}{\partial\:A}&\:\frac{\partial\:V}{\partial\:Q}&\:\frac{\partial\:V}{\partial\:R}\\\:\frac{\partial\:W}{\partial\:S}&\:\frac{\partial\:W}{\partial\:E}&\:\frac{\partial\:W}{\partial\:I}&\:\frac{\partial\:W}{\partial\:A}&\:\frac{\partial\:W}{\partial\:Q}&\:\frac{\partial\:W}{\partial\:R}\\\:\frac{\partial\:X}{\partial\:S}&\:\frac{\partial\:X}{\partial\:E}&\:\frac{\partial\:X}{\partial\:I}&\:\frac{\partial\:X}{\partial\:A}&\:\frac{\partial\:X}{\partial\:Q}&\:\frac{\partial\:X}{\partial\:R}\\\:\frac{\partial\:Y}{\partial\:S}&\:\frac{\partial\:Y}{\partial\:E}&\:\frac{\partial\:Y}{\partial\:I}&\:\frac{\partial\:Y}{\partial\:A}&\:\frac{\partial\:Y}{\partial\:Q}&\:\frac{\partial\:Y}{\partial\:R}\\\:\frac{\partial\:Z}{\partial\:S}&\:\frac{\partial\:Z}{\partial\:E}&\:\frac{\partial\:Z}{\partial\:I}&\:\frac{\partial\:Z}{\partial\:A}&\:\frac{\partial\:Z}{\partial\:Q}&\:\frac{\partial\:Z}{\partial\:R}\end{array}\right]$$

(41)

.

The elements of the Jacobian associated with the method are given by

$$\begin{aligned}\frac{\partial\:U}{\partial\:S} & =\frac{1}{1+h\left[{\alpha\:}_{1}{\beta\:}_{1}I{e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}A{e}^{-\mu\:\tau\:}+\mu\:\right]},\frac{\partial\:U}{\partial\:E}=0, \\ \:\frac{\partial\:U}{\partial\:I} & =\frac{-\left(S+h\left[{\Lambda\:}\right]\right)\left(h{\alpha\:}_{1}{\beta\:}_{1}{e}^{-\mu\:\tau\:}\right)}{{\left(1+h\left[{\alpha\:}_{1}{\beta\:}_{1}I{e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}A{e}^{-\mu\:\tau\:}+\mu\:\right]\right)}^{2}}, \\ \frac{\partial\:U}{\partial\:A} & =\frac{-\left(S+h\left[{\Lambda\:}\right]\right)\left(h{\alpha\:}_{2}{\beta\:}_{2}{e}^{-\mu\:\tau\:}\right)}{{\left(1+h\left[{\alpha\:}_{1}{\beta\:}_{1}I{e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}A{e}^{-\mu\:\tau\:}+\mu\:\right]\right)}^{2}},\frac{\partial\:U}{\partial\:Q}=0,\:\frac{\partial\:U}{\partial\:R}=0,\\ \frac{\partial\:V}{\partial\:S} &=\frac{h\left[{\alpha\:}_{1}{\beta\:}_{1}I{e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}A{e}^{-\mu\:\tau\:}\right]}{1+h\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]}, \\ \frac{\partial\:V}{\partial\:E} & =\frac{1}{1+h\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]},\:\frac{\partial\:V}{\partial\:I}=\frac{h\left[{\alpha\:}_{1}{\beta\:}_{1}{e}^{-\mu\:\tau\:}\right]}{1+h\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]}, \\ \frac{\partial\:V}{\partial\:A} & =\frac{h\left[{\alpha\:}_{2}{\beta\:}_{2}{e}^{-\mu\:\tau\:}\right]}{1+h\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]},\frac{\partial\:V}{\partial\:Q}=0,\:\frac{\partial\:V}{\partial\:R}=0,\frac{\partial\:W}{\partial\:S}=0, \\ \frac{\partial\:W}{\partial\:E} & =\frac{h\left[\kappa\:\delta\:\right]}{1+h(\nu\:+\mu\:)},\:\frac{\partial\:W}{\partial\:I}=\frac{1}{1+h(\nu\:+\mu\:)},\frac{\partial\:W}{\partial\:A}=0,\frac{\partial\:W}{\partial\:Q}=0, \\ \:\frac{\partial\:W}{\partial\:R} & =0,\frac{\partial\:X}{\partial\:S}=0,\frac{\partial\:X}{\partial\:E}=\frac{h\left[{\tau\:}_{1}\delta\:\right]}{1+h(\omega\:+\phi\:+\mu\:)},\:\frac{\partial\:X}{\partial\:I}=0, \\ \frac{\partial\:X}{\partial\:A}& =\frac{1}{1+h(\omega\:+\phi\:+\mu\:)},\frac{\partial\:X}{\partial\:Q}=0,\:\frac{\partial\:X}{\partial\:R}=0,\frac{\partial\:Y}{\partial\:S}=0, \\ \frac{\partial\:Y}{\partial\:E}&=0,\:\frac{\partial\:Y}{\partial\:I}=0,\frac{\partial\:Y}{\partial\:A}=\frac{h\left[\omega\:\right]}{1+h(\theta\:+\mu\:)},\frac{\partial\:Y}{\partial\:Q}=\frac{1}{1+h(\theta\:+\mu\:)}, \\ \frac{\partial\:Y}{\partial\:R} & =0,\frac{\partial\:Z}{\partial\:S}=0,\frac{\partial\:Z}{\partial\:E}=0,\:\frac{\partial\:Z}{\partial\:I}=\frac{h\left[\nu\:\right]}{1+h\mu\:},\frac{\partial\:Z}{\partial\:A}=\frac{h\left[\phi\:\right]}{1+h\mu\:},\frac{\partial\:Z}{\partial\:Q}=\frac{h\left[\theta\:\right]}{1+h\mu\:},\:\frac{\partial\:Z}{\partial\:R}=\frac{1}{1+h\mu\:},\end{aligned}$$

.

Theorem 13

If $\:{\mathcal{R}}_{0}<1$, then the mumps-free equilibrium ($\:MFE-{\text{M}}_{0}$) of the NSFD is locally asymptotically stable. Furthermore, $\:{\mathcal{R}}_{0}\ge\:1$ then $\:{\text{M}}_{0}$ is unstable.

Proof

The Jacobian matrix (41) at $\:{\text{M}}_{0}$ is

$$\:{J}_{{\text{M}}_{0}}=\left[\begin{array}{cccccc}\frac{1}{1+h\left[\mu\:\right]}&\:0&\:\frac{-\left(\left(\frac{{\Lambda\:}}{\mu\:}\right)+h\left[{\Lambda\:}\right]\right)\left(h{\alpha\:}_{1}{\beta\:}_{1}{e}^{-\mu\:\tau\:}\right)}{{\left(1+h\left[\mu\:\right]\right)}^{2}}&\:\frac{-\left(\left(\frac{{\Lambda\:}}{\mu\:}\right)+h\left[{\Lambda\:}\right]\right)\left(h{\alpha\:}_{2}{\beta\:}_{2}{e}^{-\mu\:\tau\:}\right)}{{\left(1+h\left[\mu\:\right]\right)}^{2}}&\:0&\:0\\\:0&\:\frac{1}{1+h\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]}&\:\frac{h\left[{\alpha\:}_{1}{\beta\:}_{1}{e}^{-\mu\:\tau\:}\right]}{1+h\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]}&\:\frac{h\left[{\alpha\:}_{2}{\beta\:}_{2}{e}^{-\mu\:\tau\:}\right]}{1+h\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]}&\:0&\:0\\\:0&\:\frac{h\left[\kappa\:\delta\:\right]}{1+h(\nu\:+\mu\:)}&\:\frac{1}{1+h(\nu\:+\mu\:)}&\:0&\:0&\:0\\\:0&\:\frac{h\left[{\tau\:}_{1}\delta\:\right]}{1+h(\omega\:+\phi\:+\mu\:)}&\:0&\:\frac{1}{1+h(\omega\:+\phi\:+\mu\:)}&\:0&\:0\\\:0&\:0&\:0&\:\frac{h\left[\omega\:\right]}{1+h(\theta\:+\mu\:)}&\:\frac{1}{1+h(\theta\:+\mu\:)}&\:0\\\:0&\:0&\:\frac{h\left[\nu\:\right]}{1+h\mu\:}&\:\frac{h\left[\phi\:\right]}{1+h\mu\:}&\:\frac{h\left[\theta\:\right]}{1+h\mu\:}&\:\frac{1}{1+h\mu\:}\end{array}\right]$$

(42)

$$\:{\lambda\:}_{1}=\frac{1}{1+h\mu\:}<1,{\lambda\:}_{2}=\frac{1}{1+h(\theta\:+\mu\:)}<1,{\lambda\:}_{3}=\frac{1}{1+h(\theta\:+\mu\:)}<1,{\lambda\:}_{4}=\frac{1}{1+h(\nu\:+\mu\:)}<1,\:{\lambda\:}_{5}=\frac{1-{h}^{2}\left[{\alpha\:}_{1}{\beta\:}_{1}{e}^{-\mu\:\tau\:}\right]}{1+h\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]}<1$$

.

Using the definition of $\:{\mathcal{R}}_{0}$, we can show that if $\:{\mathcal{R}}_{0}<1,\:$then $\:{\lambda\:}_{5}<1$, and $\:{\text{M}}_{0}$is L.A.S. on the contrary, it is obviously to verify that $\:{\lambda\:}_{5}>1$, if $\:{\mathcal{R}}_{0}\ge\:1$, which shows that $\:{\text{M}}_{0}$ is unstable.

Theorem 14

If $\:{\mathcal{R}}_{0}\ge\:1$, then mumps existing equilibrium ($\:MEE-{M}^{*}$) $\:\left({S}^{*},{E}^{*},{I}^{*},{A}^{*},{Q}^{*},{R}^{*}\right)$ of the Stochastic NSFD is locally asymptotically stable.

Proof

The Jacobian matrix (41) at $\:{M}^{*}$ is

$$\begin{aligned} & {J}_{{M}^{*}} \\ & =\left[\begin{array}{cccccc}\frac{1}{1+h\left[{\alpha\:}_{1}{\beta\:}_{1}{I}^{*}{e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}{A}^{*}{e}^{-\mu\:\tau\:}+\mu\:\right]}&\:0&\:\frac{-\left({S}^{*}+h\left[{\Lambda\:}\right]\right)\left(h{\alpha\:}_{1}{\beta\:}_{1}{e}^{-\mu\:\tau\:}\right)}{{\left(1+h\left[{\alpha\:}_{1}{\beta\:}_{1}{I}^{*}{e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}{A}^{*}{e}^{-\mu\:\tau\:}+\mu\:\right]\right)}^{2}}&\:\frac{-\left({S}^{*}+h\left[{\Lambda\:}\right]\right)\left(h{\alpha\:}_{2}{\beta\:}_{2}{e}^{-\mu\:\tau\:}\right)}{{\left(1+h\left[{\alpha\:}_{1}{\beta\:}_{1}{I}^{*}{e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}{A}^{*}{e}^{-\mu\:\tau\:}+\mu\:\right]\right)}^{2}}&\:0&\:0\\\:\frac{h\left[{\alpha\:}_{1}{\beta\:}_{1}{I}^{*}{e}^{-\mu\:\tau\:}+{\alpha\:}_{2}{\beta\:}_{2}{A}^{*}{e}^{-\mu\:\tau\:}\right]}{1+h\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]}&\:\frac{1}{1+h\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]}&\:\frac{h\left[{\alpha\:}_{1}{\beta\:}_{1}{e}^{-\mu\:\tau\:}\right]}{1+h\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]}&\:\frac{h\left[{\alpha\:}_{2}{\beta\:}_{2}{e}^{-\mu\:\tau\:}\right]}{1+h\left[\left(\kappa\:+{\tau\:}_{1}\right)\delta\:+\mu\:\right]}&\:0&\:0\\\:0&\:\frac{h\left[\kappa\:\delta\:\right]}{1+h(\nu\:+\mu\:)}&\:\frac{1}{1+h(\nu\:+\mu\:)}&\:0&\:0&\:0\\\:0&\:\frac{h\left[{\tau\:}_{1}\delta\:\right]}{1+h(\omega\:+\phi\:+\mu\:)}&\:0&\:\frac{1}{1+h(\omega\:+\phi\:+\mu\:)}&\:0&\:0\\\:0&\:0&\:0&\:\frac{h\left[\omega\:\right]}{1+h(\theta\:+\mu\:)}&\:\frac{1}{1+h(\theta\:+\mu\:)}&\:0\\\:0&\:0&\:\frac{h\left[\nu\:\right]}{1+h\mu\:}&\:\frac{h\left[\phi\:\right]}{1+h\mu\:}&\:\frac{h\left[\theta\:\right]}{1+h\mu\:}&\:\frac{1}{1+h\mu\:}\end{array}\right]\end{aligned}$$

.

Using MATLAB and the data presented in Table 3, we schemed the largest eigenvalue for Mump’s existing equilibrium as shown in Fig. 2. Hence, all the remaining Eigenvalues should be less than one. Thus, the proposed method NSFD is stable around the MEE.

Fig. 2

The largest eigenvalues of mumps existing equilibrium (MEE).

Results

To obtain the numerical results, we consider the system (35–40) with reported mumps cases in¹⁵. Time is defined in days, and parameter values presented in Table 3, are fitted using the nonlinear least-square curve technique. The initial conditions used in graphical results are $\:S\left(0\right)=0.3,\:\:E\left(0\right)=0.3,\:I\left(0\right)=0.1,\:A\left(0\right)=0.1,\:Q\left(0\right)=0.1,\:R\left(0\right)=0.1$. This section examines the characteristics of the graphs representing the number of infected persons using existing methods in literature like Euler Maruyama, stochastic Euler, and stochastic Runge Kutta schemes, in comparison to the newly developed construction for the particular model as stochastic NSFD scheme, across various step sizes.

Table 3 Numerical values¹⁵ of the model parameters.

Fig. 3

Comparison graph of computational methods at the mumps-existing equilibrium of the model (a) The convergent behavior of the symptomatic infectious population using Euler Maruyama and stochastic NSFD at h = 0.01 .(b) The divergent behavior of the symptomatic infectious population using Euler Maruyama and stochastic NSFD at h = 0.7. (c) The convergent behavior of symptomatic infectious population using stochastic Euler and stochastic NSFD at h = 0.01. (d) The divergent behavior of symptomatic infectious population using stochastic Euler and Stochastic NSFD at h = 1. (e) The convergent behavior of symptomatic infectious population using stochastic Runge Kutta and stochastic NSFD at h = 0.01. (f) The divergent behavior of symptomatic infectious population using stochastic Runge Kutta and stochastic NSFD at h = 2.

Fig. 4

Time-Plot with the time delay on susceptible and symptomatic infected population. (a) Efficiency of different delay parameters on susceptible population of the model. (b) Efficiency of different delay parameters on symptomatic infectious population of the model.

Fig. 5

Time plot of the effect of time delay ($\:\tau\:$) with reproduction number $\:\left({\mathcal{R}}_{0}\right)$.

Discussion

Figures 3(a) and (b) provide a comparison of the infected class of the stochastic NSFD, Euler Maryama methods, and its deterministic solution. Figure 3(a) demonstrates the convergence of both techniques at ℎ = 0.01, and the deterministic solution is the mean of the random solution. When the step size was increased to ℎ = 0.7, the Euler Maryama method diverged and violated the dynamical properties of the model and its results are not consistent with the biological nature, whereas the stochastic NSFD method remained convergent, as illustrated in Fig. 3 (b). In the same way, Figs. 3(c) and 3 (d) compare the infected class of the stochastic NSFD to the stochastic Euler method. Figure 3(c) shows convergence for both approaches at ℎ = 0.01. When the step size was raised to ℎ = 1.0, the stochastic Euler method diverged, whereas the stochastic NSFD method remained convergent, as seen in Fig. 3(d). Additionally, Figs. 3 (e) and (f) show the infected class of the stochastic NSFD and stochastic Runge Kutta method. At ℎ = 0.01, both techniques converge and are consistent, as seen in Fig. 3(e). When the step size was raised to ℎ = 2.0, the stochastic Runge Kutta technique diverged, whereas the stochastic NSFD technique converged, as seen in Fig. 3(f). Figures 4(a) and 4(b) illustrate the effect of delay on the subpopulation of the model they are inversely to each other. Like susceptible and symptomatic humans. So, for the different values of the delay we observe the infectivity of the disease reduce and eventually converge to zero for large-scale use of delays and meanwhile, the susceptibility of humans increases at various τ values (0.1, 0.2, 0.3, 0.4, 0.5). Figure 5 shows the effect of the delay factor with the corresponding reproduction number.

Conclusion

In this paper, we discussed the transmission dynamics of mumps disease through stochastic systems (stochastic differential equations (SDEs)) by incorporating a delay factor in place of nonlinearity in the model. The six subpopulations are included during the modeling processes as susceptible, exposed, symptomatic infectious, asymptomatic infectious, isolated symptomatic, and recovered. The threshold parameter, equilibria, boundedness, and positivity are all taken into account in the model’s dynamic analysis. The study’s conclusions include an assessment of the model’s parameter susceptibility. The model is linearized using well-established approaches such as the Routh-Hurwitz and Jacobian criteria. The Lassalle invariance principle and Lyapunov theory were used to assure the model’s global stability.

The stochastic NSFD (Nonstandard Finite Difference) method proved the most effective, precise, and efficient. For such models to eliminate unexpected behavior and produce reliable findings, stability is required. Even with extended time intervals, the stochastic NSFD approach excels in terms of solidity, cynicism, and adherence to normal bounds. Other methods, such as Euler Maryama, stochastic Euler, and stochastic Runge Kutta, are still useful methods in our computational techniques, but they suffer from severe limits at large time steps, which makes them more unstable and consistent. The results of the comparisons showed that the stochastic NSFD approach was the most stable and reliable.

Furthermore, this study helps the policymakers and practitioners who work for pharmaceutical interventions to control disease. Without non-pharmacological techniques for diseases, no organization overcomes the strain of disease with pharmaceutical steps. In conclusion, stochastic with delay modeling is a realistic way to understand the transmission dynamics of diseases and this type of analysis could be more appropriate to other types of real-world problems.