Estimating the Spread of Generalized Compartmental Model of Monkeypox Virus Using a Fuzzy Fractional Laplace Transform Method

Abstract

The main objective of this work is to develop the fuzzy fractional mathematical model that will be used to examine the dynamics of monkeypox viral transmission. The proposed dynamical model consists of human and rodents individuals and this monkeypox infection model is mathematically formulated by fuzzy fractional differential equation defined in Caputo’s sense. We provide results that demonstrate the existence and uniqueness of the considered model’s solution. We observe that our results are accurate, and that our method is applicable to the fuzzy system of fractional ordinary differential equations (ODEs). Furthermore, this monkeypox virus model has been identified as a generalization of SEIQR and SEI models. The results show that keeping diseased rodents apart from the human population reduces the spread of disease. Finally, we present brief discussions and numerical simulations to illustrate our findings.

1. Introduction

The first case of monkeypox in humans was detected in the Congo region in 1970, where smallpox had been eradicated in that region in 1968 [1]. Monkeypox is a zoonotic viral disease that is connected to smallpox and other orthopoxviruses [2]. Although early research revealed that human-to-human transmission was unusual, more current research indicates significant attack rates, raising the prospect of an epidemic or pandemic [3,4]. However, compared to smallpox, monkeypox has a much lower mortality rate [5]. Additionally, it appears that those who have received the smallpox vaccination tolerate the sickness better than the general public who are not vaccinated (see [2,3,6]). However, in Europe, where smallpox was successfully eradicated, only the older age groups of the population received smallpox vaccinations [7]. This problem was resolved by the swift procurement of smallpox vaccines in the United Kingdom, which will mostly be spread to close relatives of people with monkeypox diagnoses and to healthcare professionals who will look after such patients [6]. Although there are various cutting-edge antivirals, such as Tecovirimat, vaccinia immune globulin, and Brincindofovir that can be used to stop the transmission of the disease, there are currently no established treatments for monkeypox infection.

In human history, smallpox was the first viral disease to be eradicated. Following a successful global vaccination program, the World Health Organization (WHO) proclaimed the elimination of the Variola virus in 1980. Since that time, the only known smallpox virus samples have been stored in particular repositories in Russia and the USA [8,9,10]. Smallpox is a highly contagious illness that only affects humans and has an extreme fatality rate of up to 40. Between 300 and 500 million people died of smallpox throughout the 20th century [11]. All nations stopped routinely delivering vaccinia-based smallpox vaccinations at least 40 years ago. Unvaccinated populations are now more vulnerable to infection with the monkeypox virus, whereas vaccination provided protection against the disease in the west and central Africa [1].

A non-endemic outbreak of human monkeypox was reported in 50 different nations as of the end of June 2022, with fewer than 4900 cases worldwide. Transmission has moved from human-to-human in each nation beyond instances connected to original exposure in Africa [12]. The monkeypox virus spreads slowly and only through close contact, there are licensed vaccines and treatments are available, and the circulating virus is a member of the Western African clade of monkeypox viruses, which is known to be less virulent. These criteria suggest that the outbreak is manageable [12,13,14,15,16]. Therapeutics and vaccinations for smallpox and monkeypox that have just received approval are now being used in the real world, and revealing gaps in our arsenal [17]. Numerous vaccinations and antivirals have been tested in this model thus far [18], including medicines licensed by the Drug Administration (FDA) and US Food [19]. The spread of monkeypox virus is provided in [20].

Monkeypox outbreaks were reported in many non-endemic nations in May 2022. Research is now being conducted to better understand the epidemiology, patterns of transmission, and sources of illness [1]. The yearwise monkeypox virus outbreaks are mentioned in

Table 1. The global spread of the monkeypox virus is depicted below.

Region	Year
Congo	1997 and 2020
African nations	1997
Central African	2016
Nigeria	2017 and 2018
Cameroon	2018
Singapore	2019
U.K and Northern Ireland	2021
U.S.A	2003 and 2021

.

The WHO has classified the ongoing monkeypox outbreak as a public health emergency of international concern [21]. This is in accordance with the surveillance report published by the European Center for Disease Prevention and Control (ECDC) up to the second of August 2022, which listed 15,926 MPXV cases detected in 38 different countries throughout Europe [22]. From 1 January 2022, to 15 August 2022, the conformed global cases and confirmed global deaths are 31,799 and 12, respectively, [23].

Recently, ref. [24] The European Medicines Agency (EMA) approved Tecovirimat as the first oral therapy for monkeypox in January 2022. The most recent information on the phylogenesis, pathophysiology, prevention, and treatment of this alarming disease is available in [25].

Modern genetic research on orthologous immunogenic vaccinia-virus proteins is examined in [26]. The article [27] describes a morphogenic composition-characterized preparation of EV-enriched monkeypox virus. The authors in [28] describe the illness brought on by the Zaire strain of pure monkeypox virus [29]. The thorough explanation of an early instance, supported by clinical findings as well as results from Whole-Genome Sequencing (WGS) data and studies. This report illustrates one of the current outbreak’s first well-documented instances. A panel of closely related pathogens were being quickly and accurately identified using the RANS technique [30].

Since the disease has not received much attention in the past, we do not know enough about how it spreads. To explore the dynamics of the monkeypox virus, mathematical modeling has only been used in a small number of research. Our aim is to applying the fuzzy fractional differential equations (FFDE) in the monkeypox virus model and then investigating the disease transmission for the time t and effects of such factors on a mathematical approach. One can refer the different types of solving technique for the fractional order epidemic models in [31,32,33,34,35].

In our work, the fractional order (Caputo’s sense) SEIQRSEI epidemic model is investigated for Monkeypox virus system mathematically. Since the fractional order differential operators are non-local operators, they can better represent some dynamic system processes and natural physics processes when compared to integer order differential equation. The Caputo fractional operator is more flexible for analysis and handles the initial and boundary value problems. It is also widely used to define the time-fractional derivatives in fractional partial differential equations. This motivates us to solve the fuzzy fractional differential equations in Caputo sense. The fractional differential equations with fuzzy solutions, as well as fuzzy boundary and initial value problems can be solved using the fuzzy Laplace transform technique. Another significant benefit is that it offers direct problem-solving without first generating non-homogeneous differential equations and then figuring out a general solution. In addition, the numerical results from the fuzzy Laplace transform based on the Adomian decomposition are helpful in understanding the physical behaviour of Monkeypox virus with dynamical structures. The monotonicity theorem and the numerical simulations for the inter-valued fractional order differential equations can refer in the recent articles such as [36,37,38,39,40].

The following is how the paper is organized: Section 2 describes model formulation and analysis. Next, we analyze the existence and uniqueness of the solutions of fuzzy fractional differential equation system for the model. Section 3.2 is followed by numerical simulations and results, followed by Section 4, and then finally conclusion.

2. Mathematical Modelling

We present a deterministic compartmental model of monkeypox transmission dynamics based on two individuals: humans and rodents [41]. In the given model (1), the human population is subdivided into five compartment such as suspected $S_h(t)$, exposed $E_h(t)$, infected $I_h(t)$, Quarantine $Q_h(t)$, recovered $R_h(t)$ and rodent individuals is subdivided into three compartment such as suspected $S_r(t)$, exposed $E_r(t)$, infected $I_r(t)$. The corresponding system is taken as

$$\left\{\begin{array}{c}{S}_h^{{}^{\prime }}(t)={\theta }_h-{\displaystyle \frac{({\beta }_1{I}_r+{\beta }_2{I}_h)S_h}{N_h}}-{\mu }_h{S}_h+\varphi Q_h\hfill \\ E_h^{{}^{\prime }}(t)={\displaystyle \frac{({\beta }_1{I}_r+{\beta }_2{I}_h)S_h}{N_h}}-({\alpha }_1+{\alpha }_2+{\mu }_h)E_h\hfill \\ I_h^{{}^{\prime }}(t)={\alpha }_1{E}_h-({\mu }_h+{\delta }_h+\nu )I_h\hfill \\ Q_h^{{}^{\prime }}(t)={\alpha }_2{E}_h-(\varphi +\tau +{\delta }_h+{\mu }_h)Q_h\hfill \\ R_h^{{}^{\prime }}(t)=\nu I_h+\tau Q_h-{\mu }_h{R}_h\hfill \\ S_r^{{}^{\prime }}(t)={\theta }_r-{\displaystyle \frac{{\beta }_3{S}_r{I}_r}{N_r}}-{\mu }_r{S}_r\hfill \\ E_r^{{}^{\prime }}(t)={\displaystyle \frac{{\beta }_3{S}_r{I}_r}{N_r}}-({\mu }_r+{\alpha }_3)E_r\hfill \\ I_r^{{}^{\prime }}(t)={\alpha }_3{E}_r-({\mu }_r+{\delta }_r)I_r\hfill \end{array}\right.$$

(1)

where $N_h=S_h+E_h+I_h+Q_h+R_h$.

The transmission parameters for human population and the rodent individuals is given in

Table 2. The parameter (transmission rate) values are mentioned as follows [41].

Notations	Description	Values
${\theta }_h$	Human recruitment rate	0.029
${\beta }_1$	Human-rodent contact rate	$2.5\times {10}^{-4}$
${\beta }_2$	Human-humans contact rate	$6\times {10}^{-4}$
${\alpha }_1$	Proportion of infected humans from exposed humans	0.2
${\alpha }_2$	Proportion of suspected cases detected	2.0
$\varphi$	Proportion not detected after diagnosis	2.0
$\tau$	Progression from isolated class to recovered class	0.52
$\nu$	Humans recovery rate	0.83
${\mu }_h$	Natural (human) death rate	1.5
${\delta }_h$	Disease (human) induced death rate	0.2
${\theta }_r$	Rodents recruitment rate	0.2
${\beta }_3$	Rodent-rodent contact rate	0.027
${\alpha }_3$	Proportion of infected rodents from exposed rodents	2.0
${\mu }_r$	Natural (rodents) death rate	$2\times {10}^{-2}$
${\delta }_r$	Disease (rodents) induced death rate	0.5

and their Schemmatic representation of monkeypox virus model is given in Figure 1.

Since we are dealing with the caputo derivative, we should solve the right side of Equation (1) in the caputo sense by applying the caputo fractional derivative to the left side of Equation (1). Using the Caputo fractional derivative for fractional order $0<y\le 1$, Equation (1) can be formulated as

$$\left\{\begin{array}{c}{\displaystyle \frac{d^y}{d{t}^y}}\mathcal{D}{S}_h={\theta }_h-{\displaystyle \frac{({\beta }_1{I}_r+{\beta }_2{I}_h)S_h}{N_h}}-{\mu }_h{S}_h+\varphi Q_h\hfill \\ {\displaystyle \frac{d^y}{d{t}^y}}{E}_h={\displaystyle \frac{({\beta }_1{I}_r+{\beta }_2{I}_h)S_h}{N_h}}-({\alpha }_1+{\alpha }_2+{\mu }_h)E_h\hfill \\ {\displaystyle \frac{d^y}{d{t}^y}}{I}_h={\alpha }_1{E}_h-({\mu }_h+{\delta }_h+\nu )I_h\hfill \\ {\displaystyle \frac{d^y}{d{t}^y}}{Q}_h={\alpha }_2{E}_h-(\varphi +\tau +{\delta }_h+{\mu }_h)Q_h\hfill \\ {\displaystyle \frac{d^y}{d{t}^y}}{R}_h=\nu I_h+\tau Q_h-{\mu }_h{R}_h\hfill \\ {\displaystyle \frac{d^y}{d{t}^y}}{S}_r={\theta }_r-{\displaystyle \frac{{\beta }_3{S}_r{I}_r}{N_r}}-{\mu }_r{S}_r\hfill \\ {\displaystyle \frac{d^y}{d{t}^y}}{E}_r={\displaystyle \frac{{\beta }_3{S}_r{I}_r}{N_r}}-({\mu }_r+{\alpha }_3)E_r\hfill \\ {\displaystyle \frac{d^y}{d{t}^y}}{I}_r={\alpha }_3{E}_r-({\mu }_r+{\delta }_r)I_r\hfill \end{array}\right.$$

(2)

Fuzzy calculus and FODEs have been developed in recent years by extending modern calculus and DEs [42] and then it was extended to fuzzy FODEs [43]. In order to establish the uniqueness and existence theory of solutions, several academics have explored FODEs and fuzzy integral equations [44,45,46]. Numerous efforts have been undertaken by mathematicians to solve fuzzy FODEs using a variety of strategies, including spectral techniques, integral transform methods, perturbation methods and stability analysis [47,48,49].

The fuzzy fractional operator in caputo sense of Equation (2) becomes

$$\left\{\begin{array}{c}{}_t{\mathcal{D}}^y{S}_h=\widetilde{{\theta }_h}-{\displaystyle \frac{(\widetilde{{\beta }_1}{I}_r+\widetilde{{\beta }_2}{I}_h)S_h}{N_h}}-\widetilde{{\mu }_h}{S}_h+\tilde{\varphi }{Q}_h\hfill \\ {}_t{\mathcal{D}}^y{E}_h={\displaystyle \frac{(\widetilde{{\beta }_1}{I}_r+\widetilde{{\beta }_2}{I}_h)S_h}{N_h}}-(\widetilde{{\alpha }_1}+\widetilde{{\alpha }_2}+\widetilde{{\mu }_h})E_h\hfill \\ {}_t{\mathcal{D}}^y{I}_h=\widetilde{{\alpha }_1}{E}_h-(\widetilde{{\mu }_h}+\widetilde{{\delta }_h}+\tilde{\nu })I_h\hfill \\ {}_t{\mathcal{D}}^y{Q}_h=\widetilde{{\alpha }_2}{E}_h-(\tilde{\varphi }+\tilde{\tau }+\widetilde{{\delta }_h}+\widetilde{{\mu }_h})Q_h\hfill \\ {}_t{\mathcal{D}}^y{R}_h=\tilde{\nu }{I}_h+\tilde{\tau }{Q}_h-\widetilde{{\mu }_h}{R}_h\hfill \\ {}_t{\mathcal{D}}^y{S}_r=\widetilde{{\theta }_r}-{\displaystyle \frac{\widetilde{{\beta }_3}{S}_r{I}_r}{N_r}}-\widetilde{{\mu }_r}{S}_r\hfill \\ {}_t{\mathcal{D}}^y{E}_r={\displaystyle \frac{\widetilde{{\beta }_3}{S}_r{I}_r}{N_r}}-(\widetilde{{\mu }_r}+\widetilde{{\alpha }_3})E_r\hfill \\ {}_t{\mathcal{D}}^y{R}_r=\widetilde{{\alpha }_3}{E}_r-(\widetilde{{\mu }_r}+\widetilde{{\delta }_r})I_r\hfill \end{array}\right.$$

(3)

with fuzzy initial conditions for $\zeta \in [0,1]$.

$$\begin{array}{cc}{\tilde{S}}_h(0,\zeta )=({\underline{S}}_h(0,\zeta ),{\overline{S}}_h(0,\zeta ))\hfill & {\tilde{S}}_r(0,\zeta )=({\underline{S}}_r(0,\zeta ),{\overline{S}}_r(0,\zeta ))\hfill \\ {\tilde{E}}_h(0,\zeta )=({\underline{E}}_h(0,\zeta ),{\overline{E}}_h(0,\zeta ))\hfill & {\tilde{E}}_r(0,\zeta )=({\underline{E}}_r(0,\zeta ),{\overline{E}}_r(0,\zeta ))\hfill \\ {\tilde{I}}_h(0,\zeta )=({\underline{I}}_h(0,\zeta ),{\overline{I}}_h(0,\zeta )\hfill & {\tilde{I}}_r(0,\zeta )=({\underline{I}}_r(0,\zeta ),{\overline{I}}_r(0,\zeta ))\hfill \\ {\tilde{Q}}_h(0,\zeta )=({\underline{Q}}_h(0,\zeta ),{\overline{Q}}_h(0,\zeta ))\hfill \\ {\tilde{R}}_h(0,\zeta )=({\underline{R}}_h(0,\zeta ),{\overline{R}}_h(0,\zeta )).\hfill \end{array}$$

3. Fuzzy Fractional Analysis in the Monkeypox Model

This section takes into account the solution’s existence and uniqueness for the fuzzy fractional model.

3.1. A Fuzzy Fractional Model’S Existence Furthermore, Uniqueness

It is significant to remember the following points, if the fuzzy fractional differential equation represents a physical problem mathematically, i.e.,

• The mathematical model’s beginning conditions should have a solution.
• We want every mathematical model to have a single solution that is determined by the initial conditions. Now, rewriting Equation (3) in the form of

  $$\left\{\begin{array}{c}\mathfrak{A}(t,S_h(t))=\widetilde{{\theta }_h}-{\displaystyle \frac{(\widetilde{{\beta }_1}{I}_r+\widetilde{{\beta }_2}{I}_h)S_h}{N_h}}-\widetilde{{\mu }_h}{S}_h+\tilde{\varphi }{Q}_h\hfill \\ \mathfrak{B}(t,E_h(t))={\displaystyle \frac{(\widetilde{{\beta }_1}{I}_r+\widetilde{{\beta }_2}{I}_h)S_h}{N_h}}-(\widetilde{{\alpha }_1}+\widetilde{{\alpha }_2}+\widetilde{{\mu }_h})E_h\hfill \\ \mathfrak{C}(t,I_h(t))=\widetilde{{\alpha }_1}{E}_h-(\widetilde{{\mu }_h}+\widetilde{{\delta }_h}+\tilde{\nu })I_h\hfill \\ \mathfrak{D}(t,Q_h(t))=\widetilde{{\alpha }_2}{E}_h-(\tilde{\varphi }+\tilde{\tau }+\widetilde{{\delta }_h}+\widetilde{{\mu }_h})Q_h\hfill \\ \mathfrak{E}(t,R_h(t))=\tilde{\nu }{I}_h+\tilde{\tau }{Q}_h-\widetilde{{\mu }_h}{R}_h\hfill \\ \mathfrak{F}(t,S_r(t))=\widetilde{{\theta }_r}-{\displaystyle \frac{\widetilde{{\beta }_3}{S}_r{I}_r}{N_r}}-\widetilde{{\mu }_r}{S}_r\hfill \\ \mathfrak{G}(t,E_r(t))={\displaystyle \frac{\widetilde{{\beta }_3}{S}_r{I}_r}{N_r}}-(\widetilde{{\mu }_r}+\widetilde{{\alpha }_3})E_r\hfill \\ \mathfrak{H}(t,I_r(t))=\widetilde{{\alpha }_3}{E}_r-(\widetilde{{\mu }_r}+\widetilde{{\delta }_r})I_r,\hfill \end{array}\right.$$

  (4)

  where the fuzzy functions are $\mathfrak{A},\mathfrak{B},\mathfrak{C},\mathfrak{D},\mathfrak{E},\mathfrak{F},\mathfrak{G},\mathfrak{H}$. Therefore,

  $$\left\{\begin{array}{c}{}_t{\mathcal{D}}^y{S}_h=\mathfrak{A}(t,S_h(t)){}_t{\mathcal{D}}^y{S}_r=\mathfrak{F}(t,S_r(t))\hfill \\ {}_t{\mathcal{D}}^y{E}_h=\mathfrak{B}(t,E_h(t)){}_t{\mathcal{D}}^y{E}_r=\mathfrak{G}(t,E_r(t))\hfill \\ {}_t{\mathcal{D}}^y{I}_h=\mathfrak{C}(t,I_h(t)){}_t{\mathcal{D}}^y{I}_r=\mathfrak{H}(t,I_r(t))\hfill \\ {}_t{\mathcal{D}}^y{Q}_h=\mathfrak{D}(t,Q_h(t))\hfill \\ {}_t{\mathcal{D}}^y{R}_h=\mathfrak{E}(t,R_r(t))\hfill \end{array}\right.$$

  (5)

  which subject to the initial conditions

  $$\begin{array}{cc}{\tilde{S}}_h(0,\zeta )=(\underline{S_h}(0,\zeta ),\overline{S_h}(0,\zeta ))\hfill & {\tilde{S}}_r(0,\zeta )=(\underline{S_r}(0,\zeta ),\overline{S_r}(0,\zeta ))\hfill \\ \widetilde{E_h}(0,\zeta )=(\underline{E_h}(0,\zeta ),\overline{E_h}(0,\zeta ))\hfill & {\tilde{E}}_r(0,\zeta )=(\underline{E_r}(0,\zeta ),\overline{E_r}(0,\zeta ))\hfill \\ {\tilde{I}}_h(0,\zeta )=(\underline{I_h}(0,\zeta ),\overline{I_h}(0,\zeta ))\hfill & {\tilde{I}}_r(0,\zeta )=(\underline{I_r}(0,\zeta ),\overline{I_r}(0,\zeta ))\hfill \\ \widetilde{Q_h}(0,\zeta )=(\underline{Q_h}(0,\zeta ),\overline{Q_h}(0,\zeta ))\hfill \\ \widetilde{R_h}(0,\zeta )=(\underline{R_h}(0,\zeta ),\overline{R_h}(0,\zeta )).\hfill \end{array}$$

Now, applying the fuzzy fractional Integration $I^y$ on Equation (5), we obtain

$$\left\{\begin{array}{c}{S}_h(t)=\widetilde{S_h}(0,\zeta )+(1/\mathsf{\Gamma }(y))\underset{0}{\overset{t}{\int }}{(t-s)}^{y-1}\mathfrak{A}(s,S_h(s))ds\hfill \\ E_h(t)=\widetilde{E_h}(0,\zeta )+(1/\mathsf{\Gamma }(y))\underset{0}{\overset{t}{\int }}{(t-s)}^{y-1}\mathfrak{B}(s,E_h(s))ds\hfill \\ I_h(t)=\widetilde{I_h}(0,\zeta )+(1/\mathsf{\Gamma }(y))\underset{0}{\overset{t}{\int }}{(t-s)}^{y-1}\mathfrak{C}(s,I_h(s))ds\hfill \\ Q_h(t)=\widetilde{Q_h}(0,\zeta )+(1/\mathsf{\Gamma }(y))\underset{0}{\overset{t}{\int }}{(t-s)}^{y-1}\mathfrak{D}(s,Q_h(s))ds\hfill \\ R_h(t)=\widetilde{R_h}(0,\zeta )+(1/\mathsf{\Gamma }(y))\underset{0}{\overset{t}{\int }}{(t-s)}^{y-1}\mathfrak{E}(s,R_h(s))ds\hfill \\ S_r(t)=\widetilde{S_r}(0,\zeta )+(1/\mathsf{\Gamma }(y))\underset{0}{\overset{t}{\int }}{(t-s)}^{y-1}\mathfrak{F}(s,S_r(s))ds\hfill \\ E_r(t)=\widetilde{E_r}(0,\zeta )+(1/\mathsf{\Gamma }(y))\underset{0}{\overset{t}{\int }}{(t-s)}^{y-1}\mathfrak{G}(s,E_r(s))ds\hfill \\ I_r(t)=\widetilde{I_r}(0,\zeta )+(1/\mathsf{\Gamma }(y))\underset{0}{\overset{t}{\int }}{(t-s)}^{y-1}\mathfrak{H}(s,I_r(s))ds\hfill \end{array}\right.$$

(6)

3.2. Scheme of the Solution

Applying the fuzzy Laplace transform on Equation (5), one can easily find

$$\left\{\begin{array}{c}\mathfrak{L}\left[{}_t{\mathcal{D}}^y{S}_h\right]=\mathfrak{L}\left[\mathfrak{A}(t,S_h(t))\right]\mathfrak{L}\left[{}_t{\mathcal{D}}^y{S}_r\right]=\mathfrak{L}\left[\mathfrak{F}(t,S_r(t))\right]\hfill \\ \mathfrak{L}\left[{}_t{\mathcal{D}}^y{E}_h\right]=\mathfrak{L}\left[\mathfrak{B}(t,E_h(t))\right]\mathfrak{L}\left[{}_t{\mathcal{D}}^y{E}_r\right]=\mathfrak{L}\left[\mathfrak{G}(t,E_r(t))\right]\hfill \\ \mathfrak{L}\left[{}_t{\mathcal{D}}^y{I}_h\right]=\mathfrak{L}\left[\mathfrak{C}(t,I_h(t))\right]\mathfrak{L}\left[{}_t{\mathcal{D}}^y{I}_r\right]=\mathfrak{L}\left[\mathfrak{H}(t,I_r(t))\right]\hfill \\ \mathfrak{L}\left[{}_t{\mathcal{D}}^y{Q}_h\right]=\mathfrak{L}\left[\mathfrak{D}(t,Q_h(t))\right]\hfill \\ \mathfrak{L}\left[{}_t{\mathcal{D}}^y{R}_h\right]=\mathfrak{L}\left[\mathfrak{E}(t,R_r(t))\right].\hfill \end{array}\right.$$

(7)

Substituting the initial conditions in Equation (7) and by Equation (6), we arrive at

$\left\{\begin{array}{c}{s}^y\mathfrak{L}\left[{}_t{\mathcal{D}}^y{S}_h\right]=s^{y-1}{\tilde{S}}_h(o,\zeta )+\mathfrak{L}\left[\mathfrak{A}(t,S_h(t))\right]s^y\mathfrak{L}\left[{}_t{\mathcal{D}}^y{S}_r\right]=s^{y-1}{\tilde{S}}_r(o,\zeta )+\mathfrak{L}\left[\mathfrak{F}(t,S_r(t))\right]\hfill \\ s^y\mathfrak{L}\left[{}_t{\mathcal{D}}^y{E}_h\right]=s^{y-1}{\tilde{E}}_h(o,\zeta )+\mathfrak{L}\left[\mathfrak{B}(t,E_h(t))\right]s^y\mathfrak{L}\left[{}_t{\mathcal{D}}^y{E}_r\right]=s^{y-1}{\tilde{E}}_r(o,\zeta )+\mathfrak{L}\left[\mathfrak{G}(t,E_r(t))\right]\hfill \\ s^y\mathfrak{L}\left[{}_t{\mathcal{D}}^y{I}_h\right]=s^{y-1}{\tilde{I}}_h(o,\zeta )+\mathfrak{L}\left[\mathfrak{C}(t,I_h(t))\right]s^y\mathfrak{L}\left[{}_t{\mathcal{D}}^y{I}_r\right]=s^{y-1}{\tilde{I}}_r(o,\zeta )+\mathfrak{L}\left[\mathfrak{H}(t,I_r(t))\right]\hfill \\ s^y\mathfrak{L}\left[{}_t{\mathcal{D}}^y{Q}_h\right]=s^{y-1}{\tilde{Q}}_h(o,\zeta )+\mathfrak{L}\left[\mathfrak{D}(t,Q_h(t))\right]\hfill \\ s^y\mathfrak{L}\left[{}_t{\mathcal{D}}^y{R}_h\right]=s^{y-1}{\tilde{R}}_h(o,\zeta )+\mathfrak{L}\left[\mathfrak{E}(t,R_h(t))\right].\hfill \end{array}\right.$

The above expressions can be written as the form of

$\left\{\begin{array}{c}\mathfrak{L}\left[{}_t{\mathcal{D}}^y{S}_h\right]=\left(1s\right){\tilde{S}}_h(o,\zeta )+\left({\displaystyle \frac{1}{s^y}}\right)\mathfrak{L}\left[\mathfrak{A}(t,S_h(t))\right]\mathfrak{L}\left[{}_t{\mathcal{D}}^y{S}_r\right]=\left({\displaystyle \frac{1}{s}}\right){\tilde{S}}_r(o,\zeta )+\left({\displaystyle \frac{1}{s^y}}\right)\mathfrak{L}\left[\mathfrak{F}(t,S_r(t))\right]\hfill \\ \mathfrak{L}\left[{}_t{\mathcal{D}}^y{E}_h\right]=\left({\displaystyle \frac{1}{s}}\right){\tilde{E}}_h(o,\zeta )+\left({\displaystyle \frac{1}{s^y}}\right)\mathfrak{L}\left[\mathfrak{B}(t,E_h(t))\right]\mathfrak{L}\left[{}_t{\mathcal{D}}^y{E}_r\right]=\left({\displaystyle \frac{1}{s}}\right){\tilde{E}}_r(o,\zeta )+\left({\displaystyle \frac{1}{s^y}}\right)\mathfrak{L}\left[\mathfrak{G}(t,E_r(t))\right]\hfill \\ \mathfrak{L}\left[{}_t{\mathcal{D}}^y{I}_h\right]=\left({\displaystyle \frac{1}{s}}\right){\tilde{I}}_h(o,\zeta )+\left({\displaystyle \frac{1}{s^y}}\right)\mathfrak{L}\left[\mathfrak{C}(t,I_h(t))\right]\mathfrak{L}\left[{}_t{\mathcal{D}}^y{I}_r\right]=\left({\displaystyle \frac{1}{s}}\right){\tilde{I}}_r(o,\zeta )+\left({\displaystyle \frac{1}{s^y}}\right)\mathfrak{L}\left[\mathfrak{H}(t,I_r(t))\right]\hfill \\ \mathfrak{L}\left[{}_t{\mathcal{D}}^y{Q}_h\right]=\left({\displaystyle \frac{1}{s}}\right){\tilde{Q}}_h(o,\zeta )+\left({\displaystyle \frac{1}{s^y}}\right)\mathfrak{L}\left[\mathfrak{D}(t,Q_h(t))\right]\hfill \\ \mathfrak{L}\left[{}_t{\mathcal{D}}^y{R}_h\right]=\left({\displaystyle \frac{1}{s}}\right){\tilde{R}}_h(o,\zeta )+\left({\displaystyle \frac{1}{s^y}}\right)\mathfrak{L}\left[\mathfrak{E}(t,R_h(t))\right]\hfill \end{array}\right.$

By substituting ${}_t{\mathcal{D}}^y{Z}_k=Z_k(t)$ and then performing the inverse Laplace transform on both sides, we obtain

$$\begin{array}{l}{S}_h(t)={\tilde{S}}_h(o,\zeta ){\mathfrak{L}}^{-\mathfrak{1}}(1/s)+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}\left[\mathfrak{A}(t,S_h(t))\right]\right\}\\ E_h(t)={\tilde{E}}_h(o,\zeta ){\mathfrak{L}}^{-\mathfrak{1}}(1/s)+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}\left[\mathfrak{B}(t,E_h(t))\right]\right\}\\ I_h(t)={\tilde{I}}_h(o,\zeta ){\mathfrak{L}}^{-\mathfrak{1}}(1/s)+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}\left[\mathfrak{C}(t,I_h(t))\right]\right\}\\ Q_h(t)={\tilde{Q}}_h(o,\zeta ){\mathfrak{L}}^{-\mathfrak{1}}(1/s)+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}\left[\mathfrak{D}(t,Q_h(t))\right]\right\}\\ R_h(t)={\tilde{R}}_h(o,\zeta ){\mathfrak{L}}^{-\mathfrak{1}}(1/s)+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}\left[\mathfrak{E}(t,R_h(t))\right]\right\}\\ S_r(t)={\tilde{S}}_r(o,\zeta ){\mathfrak{L}}^{-\mathfrak{1}}(1/s)+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}\left[\mathfrak{F}(t,S_r(t))\right]\right\}\\ E_r(t)={\tilde{E}}_r(o,\zeta ){\mathfrak{L}}^{-\mathfrak{1}}(1/s)+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}\left[\mathfrak{G}(t,E_r(t))\right]\right\}\\ I_r(t)={\tilde{I}}_r(o,\zeta ){\mathfrak{L}}^{-\mathfrak{1}}(1/s)+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}\left[\mathfrak{H}(t,I_r(t))\right]\right\}\end{array}$$

If $\mathfrak{L}(1)=1/s$, then ${\mathfrak{L}}^{-1}(1/s)=1$, then the aforementioned relation can be expressed as

$$\begin{array}{l}{S}_h(t)={\tilde{S}}_h(o,\zeta )+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}\left[\mathfrak{A}(t,S_h(t))\right]\right\}\\ E_h(t)={\tilde{E}}_h(o,\zeta )+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}\left[\mathfrak{B}(t,E_h(t))\right]\right\}\\ I_h(t)={\tilde{I}}_h(o,\zeta )+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}\left[\mathfrak{C}(t,I_h(t))\right]\right\}\\ Q_h(t)={\tilde{Q}}_h(o,\zeta )+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}\left[\mathfrak{D}(t,Q_h(t))\right]\right\}\\ R_h(t)={\tilde{R}}_h(o,\zeta )+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}\left[\mathfrak{E}(t,R_h(t))\right]\right\}\\ S_r(t)={\tilde{S}}_r(o,\zeta )+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}\left[\mathfrak{F}(t,S_r(t))\right]\right\}\\ E_r(t)={\tilde{E}}_r(o,\zeta )+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}\left[\mathfrak{G}(t,E_r(t))\right]\right\}\\ I_r(t)={\tilde{I}}_r(o,\zeta )+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}\left[\mathfrak{H}(t,I_r(t))\right]\right\}\end{array}$$

Considering the infinite series solution, we arrive at

$\begin{array}{c}{S}_h(t)={\displaystyle \sum _{n=0}^{\infty }}{S}_n^h(t)\hspace{1em}\hspace{1em}{E}_h(t)={\displaystyle \sum _{n=0}^{\infty }}{E}_n^h(t)\hspace{1em}\hspace{1em}{I}_h(t)={\displaystyle \sum _{n=0}^{\infty }}{I}_n^h(t)\hspace{1em}\hspace{1em}{Q}_h(t)={\displaystyle \sum _{n=0}^{\infty }}{Q}_n^h(t)\\ R_h(t)={\displaystyle \sum _{n=0}^{\infty }}{H}_n^h(t)\hspace{1em}\hspace{1em}{S}_r(t)={\displaystyle \sum _{n=0}^{\infty }}{S}_n^r(t)\hspace{1em}\hspace{1em}{E}_r(t)={\displaystyle \sum _{n=0}^{\infty }}{E}_n^r(t)\hspace{1em}\hspace{1em}{I}_h(t)={\displaystyle \sum _{n=0}^{\infty }}{I}_n^h(t).\end{array}$

Now, substituting these infinite series solution in the previous equations, we obtain

$$\left\{\begin{array}{c}{\displaystyle \sum _{n=0}^{\infty }}{S}_n^h(t)={\tilde{S}}_h(o,\zeta )+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}[\mathfrak{A}(t,{\displaystyle \sum _{n=0}^{\infty }}{S}_n^h(t))]\right\}\hfill \\ {\displaystyle \sum _{n=0}^{\infty }}{E}_n^h(t)={\tilde{E}}_h(o,\zeta )+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}[\mathfrak{B}(t,{\displaystyle \sum _{n=0}^{\infty }}{E}_n^h(t))]\right\}\hfill \\ {\displaystyle \sum _{n=0}^{\infty }}{I}_n^h(t)={\tilde{I}}_h(o,\zeta )+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}[\mathfrak{C}(t,{\displaystyle \sum _{n=0}^{\infty }}{I}_n^h(t))]\right\}\hfill \\ {\displaystyle \sum _{n=0}^{\infty }}{Q}_n^h(t)={\tilde{Q}}_h(o,\zeta )+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}[\mathfrak{D}(t,{\displaystyle \sum _{n=0}^{\infty }}{Q}_n^h(t))]\right\}\hfill \\ {\displaystyle \sum _{n=0}^{\infty }}{R}_n^h(t)={\tilde{R}}_h(o,\zeta )+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}[\mathfrak{E}(t,{\displaystyle \sum _{n=0}^{\infty }}{R}_n^h(t))]\right\}\hfill \\ {\displaystyle \sum _{n=0}^{\infty }}{S}_n^r(t)={\tilde{S}}_r(o,\zeta )+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}[\mathfrak{F}(t,{\displaystyle \sum _{n=0}^{\infty }}{S}_n^r(t))]\right\}\hfill \\ {\displaystyle \sum _{n=0}^{\infty }}{E}_n^r(t)={\tilde{E}}_r(o,\zeta )+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}[\mathfrak{G}(t,{\displaystyle \sum _{n=0}^{\infty }}{E}_n^r(t))]\right\}\hfill \\ {\displaystyle \sum _{n=0}^{\infty }}{I}_n^r(t)={\tilde{I}}_r(o,\zeta )+{\mathfrak{L}}^{-\mathfrak{1}}\left\{(1/s^y)\mathfrak{L}[\mathfrak{H}(t,{\displaystyle \sum _{n=0}^{\infty }}{I}_n^r(t))]\right\}.\hfill \end{array}\right.$$

(8)

When equating the terms in the parametric form, then Equation (8) becomes

$$\begin{array}{c}{\stackrel{̲}{S}}_0^h={\stackrel{̲}{S}}_h(0,\zeta );{\underline{S}}_0^h={\underline{S}}_h(0,\zeta )\hspace{1em}\hspace{1em} {\stackrel{̲}{S}}_0^r={\stackrel{̲}{S}}_r(0,\zeta );{\underline{S}}_0^r={\underline{S}}_r(0,\zeta )\hfill \\ {\stackrel{̲}{E}}_0^h={\stackrel{̲}{S}}_h(0,\zeta );{\underline{E}}_0^h={\underline{E}}_h(0,\zeta )\hspace{1em}\hspace{1em}{\stackrel{̲}{E}}_0^r={\stackrel{̲}{E}}_r(0,\zeta );{\underline{E}}_0^r={\underline{E}}_r(0,\zeta )\hfill \\ {\stackrel{̲}{I}}_0^h={\stackrel{̲}{I}}_h(0,\zeta );{\underline{I}}_0^h={\underline{I}}_h(0,\zeta )\hspace{1em}\hspace{1em}\hspace{1em}{\stackrel{̲}{I}}_0^r={\stackrel{̲}{I}}_r(0,\zeta );{\underline{I}}_0^r={\underline{I}}_r(0,\zeta )\hfill \\ {\stackrel{̲}{Q}}_0^h={\stackrel{̲}{Q}}_h(0,\zeta );{\underline{Q}}_0^h={\underline{Q}}_h(0,\zeta )\hfill \\ {\stackrel{̲}{R}}_0^h={\stackrel{̲}{R}}_h(0,\zeta );{\underline{R}}_0^h={\underline{R}}_h(0,\zeta )\hfill \end{array}$$

The second term of the human suspected case of infinite sum of Equation (8) will be

$$\left\{\begin{array}{l}{\overline{S}}_1^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[{\tilde{\theta }}_h-(({\tilde{\beta }}_1{\overline{I}}_0^r(t)+{\tilde{\beta }}_2{\overline{I}}_0^h(t)){\overline{S}}_0^h(t)/{\overline{N}}_0^h(t))-{\tilde{\mu }}_h{\overline{S}}_0^h(t)+\tilde{\varphi }{\overline{Q}}_0^h(t)\right]\right)\hfill \\ {\underline{S}}_1^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[{\tilde{\theta }}_h-(({\tilde{\beta }}_1{\underline{I}}_0^r(t)+{\tilde{\beta }}_2{\underline{I}}_0^h(t)){\underline{S}}_0^h(t)/{\underline{N}}_0^h(t))-{\tilde{\mu }}_h{\underline{S}}_0^h(t)+\tilde{\varphi }{\underline{Q}}_0^h(t)\right]\right)\hfill \end{array}\right.$$

(9)

The second term of the human exposed case of infinite sum of Equation (8) will be

$$\left\{\begin{array}{c}{\overline{E}}_1^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[((\widetilde{{\beta }_1}{\overline{I}}_0^r(t)+\widetilde{{\beta }_2}{\overline{I}}_0^h(t)){\overline{S}}_0^h(t)/{\overline{N}}_0^h(t))-(\widetilde{{\alpha }_1}+\widetilde{{\alpha }_2}+\widetilde{{\mu }_h}){\overline{E}}_0^h(t)\right]\right)\hfill \\ {\underline{E}}_1^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[((\widetilde{{\beta }_1}{\underline{I}}_0^r(t)+\widetilde{{\beta }_2}{\underline{I}}_0^h(t)){\underline{S}}_0^h(t)/{\underline{N}}_0^h(t))-(\widetilde{{\alpha }_1}+\widetilde{{\alpha }_2}+\widetilde{{\mu }_h}){\underline{E}}_0^h(t)\right]\right)\hfill \end{array}\right.$$

(10)

Similarly, the second term of the human-infected, quarantined, and recovered case of infinite sum of Equation (8) are given in Equations (11)–(13), respectively.

$$\left\{\begin{array}{c}{\overline{I}}_1^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[{\tilde{\alpha }}_1{\overline{E}}_0^h(t)-(\widetilde{{\mu }_h}+\widetilde{{\delta }_h}+\tilde{\nu }){\overline{I}}_0^h(t)\right]\right)\hfill \\ {\underline{I}}_1^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[{\tilde{\alpha }}_1{\underline{E}}_0^h(t)-(\widetilde{{\mu }_h}+\widetilde{{\delta }_h}+\tilde{\nu }){\underline{I}}_0^h(t)\right]\right)\hfill \end{array}\right.$$

(11)

$$\left\{\begin{array}{c}{\overline{Q}}_1^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[\widetilde{{\alpha }_2}{\overline{E}}_0^h(t)-(\tilde{\varphi }+\tilde{\tau }+\widetilde{{\delta }_h}+\widetilde{{\mu }_h}){\overline{Q}}_0^h(t)\right]\right)\hfill \\ {\underline{Q}}_1^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[\widetilde{{\alpha }_2}{\underline{E}}_0^h(t)-(\tilde{\varphi }+\tilde{\tau }+\widetilde{{\delta }_h}+\widetilde{{\mu }_h}){\underline{Q}}_0^h(t)\right]\right)\hfill \end{array}\right.$$

(12)

$$\left\{\begin{array}{c}{\overline{R}}_1^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[\tilde{\nu }{\overline{I}}_0^h(t)+\tilde{\tau }{\overline{Q}}_0^h(t)-\widetilde{{\mu }_h}{\overline{R}}_0^h(t)\right]\right)\hfill \\ {\underline{R}}_1^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[\tilde{\nu }{\underline{I}}_0^h(t)+\tilde{\tau }{\underline{Q}}_0^h(t)-\widetilde{{\mu }_h}{\underline{R}}_0^h(t)\right]\right)\hfill \end{array}\right.$$

(13)

In the similar manner, the second term of the rodent-infected, quarentine and recovered case of infinite sum of Equation (8) are, respectively, given in Equations (14)–(16).

$$\left\{\begin{array}{c}{\overline{S}}_1^r(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[\widetilde{{\theta }_r}-(\widetilde{{\beta }_3}{\overline{S}}_0^r(t){\overline{I}}_0^r(t)/{\overline{N}}_0^r(t))-\widetilde{{\mu }_r}{\overline{S}}_0^r(t)\right]\right)\hfill \\ {\underline{S}}_1^r(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[\widetilde{{\theta }_r}-(\widetilde{{\beta }_3}{\underline{S}}_0^r(t){\underline{I}}_0^r(t)/{\underline{N}}_0^r(t))-\widetilde{{\mu }_r}{\underline{S}}_0^r(t)\right]\right)\hfill \end{array}\right.$$

(14)

$$\left\{\begin{array}{c}{\overline{E}}_1^r(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[(\widetilde{{\beta }_3}{\overline{S}}_0^r(t){\overline{I}}_0^r(t)/{\overline{N}}_0^r(t))-(\widetilde{{\mu }_r}+\widetilde{{\alpha }_3}){\overline{E}}_0^r(t)\right]\right)\hfill \\ {\underline{E}}_1^r(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[(\widetilde{{\beta }_3}{\underline{S}}_0^r(t){\underline{I}}_0^r(t)/{\underline{N}}_0^r(t))-(\widetilde{{\mu }_r}+\widetilde{{\alpha }_3}){\underline{E}}_0^r(t)\right]\right)\hfill \end{array}\right.$$

(15)

$$\left\{\begin{array}{c}{\overline{I}}_1^r(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[\widetilde{{\alpha }_3}{\overline{E}}_0^r(t)-(\widetilde{{\mu }_r}+\widetilde{{\delta }_r}){\overline{I}}_0^r(t)\right]\right)\hfill \\ {\underline{I}}_1^r(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[\widetilde{{\alpha }_3}{\underline{E}}_0^r(t)-(\widetilde{{\mu }_r}+\widetilde{{\delta }_r}){\underline{I}}_0^r(t)\right]\right).\hfill \end{array}\right.$$

(16)

The third term of the human suspected, exposed, infected, quarantined, and recovered case are mentioned in Equations (17)–(21).

$$\left\{\begin{array}{c}{\overline{S}}_2^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[{\tilde{\theta }}_h-(({\tilde{\beta }}_1{\overline{I}}_1^r(t)+{\tilde{\beta }}_2{\overline{I}}_1^h(t)){\overline{S}}_1^h(t)/{\overline{N}}_1^h(t))-{\tilde{\mu }}_h{\overline{S}}_1^h(t)+\tilde{\varphi }{\overline{Q}}_1^h(t)\right]\right)\hfill \\ {\underline{S}}_2^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[{\tilde{\theta }}_h-(({\tilde{\beta }}_1{\underline{I}}_1^r(t)+{\tilde{\beta }}_2{\underline{I}}_1^h(t)){\underline{S}}_1^h(t)/{\underline{N}}_1^h(t))-{\tilde{\mu }}_h{\underline{S}}_1^h(t)+\tilde{\varphi }{\underline{Q}}_1^h(t)\right]\right)\hfill \end{array}\right.$$

(17)

$$\left\{\begin{array}{c}{\overline{E}}_2^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[((\widetilde{{\beta }_1}{\overline{I}}_1^r(t)+\widetilde{{\beta }_2}{\overline{I}}_1^h(t)){\overline{S}}_1^h(t)/{\overline{N}}_1^h(t))-(\widetilde{{\alpha }_1}+\widetilde{{\alpha }_2}+\widetilde{{\mu }_h}){\overline{E}}_1^h(t)\right]\right)\hfill \\ {\underline{E}}_2^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[((\widetilde{{\beta }_1}{\underline{I}}_1^r(t)+\widetilde{{\beta }_2}{\underline{I}}_1^h(t)){\underline{S}}_1^h(t)/{\underline{N}}_1^h(t))-(\widetilde{{\alpha }_1}+\widetilde{{\alpha }_2}+\widetilde{{\mu }_h}){\underline{E}}_1^h(t)\right]\right)\hfill \end{array}\right.$$

(18)

$$\left\{\begin{array}{c}{\overline{I}}_2^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[{\tilde{\alpha }}_1{\overline{E}}_1^h(t)-(\widetilde{{\mu }_h}+\widetilde{{\delta }_h}+\tilde{\nu }){\overline{I}}_1^h(t)\right]\right)\hfill \\ {\underline{I}}_2^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[{\tilde{\alpha }}_1{\underline{E}}_1^h(t)-(\widetilde{{\mu }_h}+\widetilde{{\delta }_h}+\tilde{\nu }){\underline{I}}_1^h(t)\right]\right)\hfill \end{array}\right.$$

(19)

$$\left\{\begin{array}{c}{\overline{Q}}_2^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[\widetilde{{\alpha }_2}{\overline{E}}_1^h(t)-(\tilde{\varphi }+\tilde{\tau }+\widetilde{{\delta }_h}+\widetilde{{\mu }_h}){\overline{Q}}_1^h(t)\right]\right)\hfill \\ {\underline{Q}}_2^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[\widetilde{{\alpha }_2}{\underline{E}}_1^h(t)-(\tilde{\varphi }+\tilde{\tau }+\widetilde{{\delta }_h}+\widetilde{{\mu }_h}){\underline{Q}}_1^h(t)\right]\right)\hfill \end{array}\right.$$

(20)

$$\left\{\begin{array}{c}{\overline{R}}_2^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[\tilde{\nu }{\overline{I}}_1^h(t)+\tilde{\tau }{\overline{Q}}_1^h(t)-\widetilde{{\mu }_h}{\overline{R}}_1^h(t)\right]\right)\hfill \\ {\underline{R}}_2^h(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[\tilde{\nu }{\underline{I}}_1^h(t)+\tilde{\tau }{\underline{Q}}_1^h(t)-\widetilde{{\mu }_h}{\underline{R}}_1^h(t)\right]\right)\hfill \end{array}\right.$$

(21)

The third term of the rodent suspected, exposed, or infected cases are mentioned in Equations (22)–(24).

$$\left\{\begin{array}{c}{\overline{S}}_2^r(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[\widetilde{{\theta }_r}-(\widetilde{{\beta }_3}{\overline{S}}_1^r(t){\overline{I}}_1^r(t)/{\overline{N}}_1^r(t))-\widetilde{{\mu }_r}{\overline{S}}_1^r(t)\right]\right)\hfill \\ {\underline{S}}_2^r(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[\widetilde{{\theta }_r}-(\widetilde{{\beta }_3}{\underline{S}}_1^r(t){\underline{I}}_1^r(t)/{\underline{N}}_1^r(t))-\widetilde{{\mu }_r}{\underline{S}}_1^r(t)\right]\right)\hfill \end{array}\right.$$

(22)

$$\left\{\begin{array}{c}{\overline{E}}_2^r(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[(\widetilde{{\beta }_3}{\overline{S}}_1^r(t){\overline{I}}_1^r(t)/{\overline{N}}_1^r(t))-(\widetilde{{\mu }_r}+\widetilde{{\alpha }_3}){\overline{E}}_1^r(t)\right]\right)\hfill \\ {\underline{E}}_2^r(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[(\widetilde{{\beta }_3}{\underline{S}}_1^r(t){\underline{I}}_1^r(t)/{\underline{N}}_1^r(t))-(\widetilde{{\mu }_r}+\widetilde{{\alpha }_3}){\underline{E}}_1^r(t)\right]\right)\hfill \end{array}\right.$$

(23)

$$\left\{\begin{array}{c}{\overline{I}}_2^r(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[\widetilde{{\alpha }_3}{\overline{E}}_1^r(t)-(\widetilde{{\mu }_r}+\widetilde{{\delta }_r}){\overline{I}}_1^r(t)\right]\right)\hfill \\ {\underline{I}}_2^r(t)={\mathfrak{L}}^{-\mathfrak{1}}\left((1/s^y)\mathfrak{L}\left[\widetilde{{\alpha }_3}{\underline{E}}_1^r(t)-(\widetilde{{\mu }_r}+\widetilde{{\delta }_r}){\underline{I}}_1^r(t)\right]\right).\hfill \end{array}\right.$$

(24)

Similarly, we can find the third, fourth, fifth, ⋯ terms. Finally, the general solution for the SEIQRSEI epidemic model for the monkeypox virus infection is given below.

(i)

Human suspected case:

$$\left\{\begin{array}{c}{\overline{S}}_h(t)={\overline{S}}_0^h(t)+{\overline{S}}_1^h(t)+{\overline{S}}_2^h(t)+{\overline{S}}_3^h(t)+{\overline{S}}_4^h(t)+{\overline{S}}_4^h(t)+{\overline{S}}_5^h(t)+\dots \hfill \\ {\underline{S}}_h(t)={\underline{S}}_0^h(t)+{\underline{S}}_1^h(t)+{\underline{S}}_2^h(t)+{\underline{S}}_3^h(t)+{\underline{S}}_4^h(t)+{\underline{S}}_4^h(t)+{\underline{S}}_5^h(t)+\dots \hfill \end{array}\right.$$

(25)

(ii)

Human-exposed case:

$$\left\{\begin{array}{c}{\overline{E}}_h(t)={\overline{E}}_0^h(t)+{\overline{E}}_1^h(t)+{\overline{E}}_2^h(t)+{\overline{E}}_3^h(t)+{\overline{E}}_4^h(t)+{\overline{E}}_4^h(t)+{\overline{E}}_5^h(t)+\dots \hfill \\ {\underline{E}}_h(t)={\underline{E}}_0^h(t)+{\underline{E}}_1^h(t)+{\underline{E}}_2^h(t)+{\underline{E}}_3^h(t)+{\underline{E}}_4^h(t)+{\underline{E}}_4^h(t)+{\underline{E}}_5^h(t)+\dots \hfill \end{array}\right.$$

(26)

(iii)

Human-infected case:

$$\left\{\begin{array}{c}{\overline{I}}_h(t)={\overline{I}}_0^h(t)+{\overline{I}}_1^h(t)+{\overline{I}}_2^h(t)+{\overline{I}}_3^h(t)+{\overline{I}}_4^h(t)+{\overline{I}}_4^h(t)+{\overline{I}}_5^h(t)+\dots \hfill \\ {\underline{I}}_h(t)={\underline{I}}_0^h(t)+{\underline{I}}_1^h(t)+{\underline{I}}_2^h(t)+{\underline{I}}_3^h(t)+{\underline{I}}_4^h(t)+{\underline{I}}_4^h(t)+{\underline{I}}_5^h(t)+\dots \hfill \end{array}\right.$$

(27)

(iv)

Human-quarantined case:

$$\left\{\begin{array}{c}{\overline{Q}}_h(t)={\overline{Q}}_0^h(t)+{\overline{Q}}_1^h(t)+{\overline{Q}}_2^h(t)+{\overline{Q}}_3^h(t)+{\overline{Q}}_4^h(t)+{\overline{Q}}_4^h(t)+{\overline{Q}}_5^h(t)+\dots \hfill \\ {\underline{Q}}_h(t)={\underline{Q}}_0^h(t)+{\underline{Q}}_1^h(t)+{\underline{Q}}_2^h(t)+{\underline{Q}}_3^h(t)+{\underline{Q}}_4^h(t)+{\underline{Q}}_4^h(t)+{\underline{Q}}_5^h(t)+\dots \hfill \end{array}\right.$$

(28)

(v)

Human recovered case:

$$\left\{\begin{array}{c}{\overline{R}}_h(t)={\overline{R}}_0^h(t)+{\overline{R}}_1^h(t)+{\overline{R}}_2^h(t)+{\overline{R}}_3^h(t)+{\overline{R}}_4^h(t)+{\overline{R}}_4^h(t)+{\overline{R}}_5^h(t)+\dots \hfill \\ {\underline{R}}_h(t)={\underline{R}}_0^h(t)+{\underline{R}}_1^h(t)+{\underline{R}}_2^h(t)+{\underline{R}}_3^h(t)+{\underline{R}}_4^h(t)+{\underline{R}}_4^h(t)+{\underline{R}}_5^h(t)+\dots \hfill \end{array}\right.$$

(29)

(vi)

Rodent suspected case:

$$\left\{\begin{array}{c}{\overline{S}}_r(t)={\overline{S}}_0^r(t)+{\overline{S}}_1^r(t)+{\overline{S}}_2^r(t)+{\overline{S}}_3^r(t)+{\overline{S}}_4^r(t)+{\overline{S}}_4^r(t)+{\overline{S}}_5^r(t)+\dots \hfill \\ {\underline{S}}_r(t)={\underline{S}}_0^r(t)+{\underline{S}}_1^r(t)+{\underline{S}}_2^r(t)+{\underline{S}}_3^r(t)+{\underline{S}}_4^r(t)+{\underline{S}}_4^r(t)+{\underline{S}}_5^r(t)+\dots \hfill \end{array}\right.$$

(30)

(vii)

Rodent exposed case:

$$\left\{\begin{array}{c}{\overline{E}}_r(t)={\overline{E}}_0^r(t)+{\overline{E}}_1^r(t)+{\overline{E}}_2^r(t)+{\overline{E}}_3^r(t)+{\overline{E}}_4^r(t)+{\overline{E}}_4^r(t)+{\overline{E}}_5^r(t)+\dots \hfill \\ {\underline{E}}_r(t)={\underline{E}}_0^r(t)+{\underline{E}}_1^r(t)+{\underline{E}}_2^r(t)+{\underline{E}}_3^r(t)+{\underline{E}}_4^r(t)+{\underline{E}}_4^r(t)+{\underline{E}}_5^r(t)+\dots \hfill \end{array}\right.$$

(31)

(viii)

Rodent-infected case:

$$\left\{\begin{array}{c}{\overline{I}}_r(t)={\overline{I}}_0^r(t)+{\overline{I}}_1^r(t)+{\overline{I}}_2^r(t)+{\overline{I}}_3^r(t)+{\overline{I}}_4^r(t)+{\overline{I}}_4^r(t)+{\overline{I}}_5^r(t)+\dots \hfill \\ {\underline{I}}_r(t)={\underline{I}}_0^r(t)+{\underline{I}}_1^r(t)+{\underline{I}}_2^r(t)+{\underline{I}}_3^r(t)+{\underline{I}}_4^r(t)+{\underline{I}}_4^r(t)+{\underline{I}}_5^r(t)+\dots \hfill \end{array}\right.$$

(32)

To determine the spread of the monkeypox virus at a specific period t, one can use Equations (25)–(32).

4. Results Furthermore, Discussion

In this section, the series solution for a particular triangular fuzzy number are computed. Take the suggested model’s initial conditions as

$$\begin{array}{l}\widetilde{S_h}(0,\zeta )=(5\zeta -1,1-5\zeta )\hspace{1em}\hspace{1em}\widetilde{S_r}(0,\zeta )=(5\zeta -1,1-5\zeta )\\ \widetilde{E_h}(0,\zeta )=(5\zeta -1,1-5\zeta )\hspace{1em} \widetilde{E_r}(0,\zeta )=(5\zeta -1,1-5\zeta )\\ \widetilde{I_h}(0,\zeta )=(5\zeta -1,1-5\zeta )\hspace{1em} \widetilde{I_r}(0,\zeta )=(5\zeta -1,1-5\zeta )\\ \widetilde{Q_h}(0,\zeta )=(5\zeta -1,1-5\zeta )\\ \widetilde{R_h}(0,\zeta )=(5\zeta -1,1-5\zeta )\end{array}$$

Using the approach described above and with the initial conditions, we obtain the first term as

$$\begin{array}{l}{\overline{S}}_0^h=1-5\zeta ;{\underline{S}}_0^h=5\zeta -1\hspace{1em}\hspace{1em}{\overline{S}}_0^r=1-5\zeta ;{\underline{S}}_0^r=5\zeta -1\\ {\overline{E}}_0^h=1-5\zeta ;{\underline{E}}_0^h=5\zeta -1\hspace{1em}\hspace{1em}{\overline{E}}_0^r=1-5\zeta ;{\underline{E}}_0^r=5\zeta -1\\ {\overline{I}}_0^h=1-5\zeta ;{\underline{I}}_0^h=5\zeta -1\hspace{1em}\hspace{1em} {\overline{I}}_0^r=1-5\zeta ;{\underline{I}}_0^r=5\zeta -1\\ {\overline{Q}}_0^h=1-5\zeta ;{\underline{Q}}_0^h=5\zeta -1\\ {\overline{R}}_0^h=1-5\zeta ;{\underline{R}}_0^h=5\zeta -1\end{array}$$

The second term of the infinite sum will be

$$\left\{\begin{array}{c}{\overline{S}}_1^h(t)=({\tilde{\theta }}_h-[({\tilde{\beta }}_1+{\tilde{\beta }}_2/5)+{\tilde{\mu }}_h-\tilde{\varphi }](1-5\zeta ))(t^y/\mathsf{\Gamma }(y+1))\hfill \\ {\underline{S}}_1^h(t)=({\tilde{\theta }}_h-[({\tilde{\beta }}_1+{\tilde{\beta }}_2/5)+{\tilde{\mu }}_h-\tilde{\varphi }](5\zeta -1))(t^y/\mathsf{\Gamma }(y+1)).\hfill \end{array}\right.$$

(33)

$$\left\{\begin{array}{c}{\overline{E}}_1^h(t)=([({\tilde{\beta }}_1+{\tilde{\beta }}_2/5)-({\tilde{\alpha }}_1+{\tilde{\alpha }}_2+{\tilde{\mu }}_h)](1-5\zeta ))(t^y/\mathsf{\Gamma }(y+1))\hfill \\ {\underline{E}}_1^h(t)=([({\tilde{\beta }}_1+{\tilde{\beta }}_2/5)-({\tilde{\alpha }}_1+{\tilde{\alpha }}_2+{\tilde{\mu }}_h)](5\zeta -1))(t^y/\mathsf{\Gamma }(y+1)).\hfill \end{array}\right.$$

(34)

$$\left\{\begin{array}{c}{\overline{I}}_1^h(t)=([{\tilde{\alpha }}_1-(\widetilde{{\mu }_h}+\widetilde{{\delta }_h}+\tilde{\nu })](1-5\zeta ))(t^y/\mathsf{\Gamma }(y+1))\hfill \\ {\underline{I}}_1^h(t)=([{\tilde{\alpha }}_1-(\widetilde{{\mu }_h}+\widetilde{{\delta }_h}+\tilde{\nu })](5\zeta -1))(t^y/\mathsf{\Gamma }(y+1)).\hfill \end{array}\right.$$

(35)

$$\left\{\begin{array}{c}{\overline{Q}}_1^h(t)=([\widetilde{{\alpha }_2}-(\tilde{\varphi }+\tilde{\tau }+\widetilde{{\delta }_h}+\widetilde{{\mu }_h})](1-5\zeta ))(t^y/\mathsf{\Gamma }(y+1))\hfill \\ {\underline{Q}}_1^h(t)=([\widetilde{{\alpha }_2}-(\tilde{\varphi }+\tilde{\tau }+\widetilde{{\delta }_h}+\widetilde{{\mu }_h})](5\zeta -1))(t^y/\mathsf{\Gamma }(y+1)).\hfill \end{array}\right.$$

(36)

$$\left\{\begin{array}{c}{\overline{R}}_1^h(t)=\left([\tilde{\nu }+\tilde{\tau }-\widetilde{{\mu }_h}](1-\zeta )\right)(t^y/\mathsf{\Gamma }(y+1))\hfill \\ {\underline{R}}_1^h(t)=\left([\tilde{\nu }+\tilde{\tau }-\widetilde{{\mu }_h}](\zeta -1)\right)(t^y/\mathsf{\Gamma }(y+1)).\hfill \end{array}\right.$$

(37)

$$\left\{\begin{array}{c}{\overline{S}}_1^r(t)=({\theta }_r-[({\tilde{\beta }}_3/5)+{\tilde{\mu }}_r](1-5\zeta ))(t^y/\mathsf{\Gamma }(y+1))\hfill \\ {\underline{S}}_1^r(t)=({\theta }_r-[({\tilde{\beta }}_3/5)+{\tilde{\mu }}_r](5\zeta -1))(t^y/\mathsf{\Gamma }(y+1))\hfill \end{array}\right.$$

(38)

$$\left\{\begin{array}{c}{\overline{E}}_1^r(t)=([({\tilde{\beta }}_3/5)-({\tilde{\mu }}_r+{\tilde{\alpha }}_3)](1-5\zeta ))(t^y/\mathsf{\Gamma }(y+1))\hfill \\ {\underline{E}}_1^r(t)=([({\tilde{\beta }}_3/5)-({\tilde{\mu }}_r+{\tilde{\alpha }}_3)](5\zeta -1))(t^y/\mathsf{\Gamma }(y+1)).\hfill \end{array}\right.$$

(39)

$$\left\{\begin{array}{c}{\overline{I}}_1^r(t)=([\widetilde{{\alpha }_3}-(\widetilde{{\mu }_r}+\widetilde{{\delta }_r})](1-5\zeta ))(t^y/\mathsf{\Gamma }(y+1))\hfill \\ {\underline{I}}_1^r(t)=([\widetilde{{\alpha }_3}-(\widetilde{{\mu }_r}+\widetilde{{\delta }_r})](5\zeta -1))(t^y/\mathsf{\Gamma }(y+1)).\hfill \end{array}\right.$$

(40)

The third term of the infinite sum will be

$$\left\{\begin{array}{c}{\overline{S}}_2^h(t)=\left({\tilde{\theta }}_h-{\displaystyle \frac{({\tilde{\beta }}_1{\overline{I}}_1^r(t)+{\tilde{\beta }}_2{\overline{I}}_1^h(t)){\overline{S}}_1^h(t)}{({\theta }_h-3.400058(1-5\zeta ))(t^y/\mathsf{\Gamma }(y+1))}}-{\tilde{\mu }}_h{\overline{S}}_1^h(t)+\tilde{\varphi }{\overline{Q}}_1^h(t)\right)(t^y/\mathsf{\Gamma }(y+1))\hfill \\ {\underline{S}}_2^h(t)=\left({\tilde{\theta }}_h-{\displaystyle \frac{({\tilde{\beta }}_1{\underline{I}}_1^r(t)+{\tilde{\beta }}_2{\underline{I}}_1^h(t)){\underline{S}}_1^h(t)}{({\theta }_h-3.400058(5\zeta -1))(t^y/\mathsf{\Gamma }(y+1))}}-{\tilde{\mu }}_h{\underline{S}}_1^h(t)+\tilde{\varphi }{\underline{Q}}_1^h(t)\right)(t^y/\mathsf{\Gamma }(y+1)).\hfill \end{array}\right.$$

(41)

$$\left\{\begin{array}{c}{\overline{E}}_2^h(t)=\left({\displaystyle \frac{(\widetilde{{\beta }_1}{\overline{I}}_1^r(t)+\widetilde{{\beta }_2}{\overline{I}}_1^h(t)){\overline{S}}_1^h(t)}{({\theta }_h-3.400058(1-5\zeta ))(t^y/\mathsf{\Gamma }(y+1))}}-(\widetilde{{\alpha }_1}+\widetilde{{\alpha }_2}+\widetilde{{\mu }_h}){\overline{E}}_1^h(t)\right)(t^y/\mathsf{\Gamma }(y+1))\hfill \\ {\underline{E}}_2^h(t)=\left({\displaystyle \frac{(\widetilde{{\beta }_1}{\underline{I}}_1^r(t)+\widetilde{{\beta }_2}{\underline{I}}_1^h(t)){\underline{S}}_1^h(t)}{({\theta }_h-3.400058(5\zeta -1))(t^y/\mathsf{\Gamma }(y+1))}}-(\widetilde{{\alpha }_1}+\widetilde{{\alpha }_2}+\widetilde{{\mu }_h}){\underline{E}}_1^h(t)\right)(t^y/\mathsf{\Gamma }(y+1)).\hfill \end{array}\right.$$

(42)

$$\left\{\begin{array}{c}{\overline{I}}_2^h(t)=\left({\tilde{\alpha }}_1{\overline{E}}_1^h(t)-(\widetilde{{\mu }_h}+\widetilde{{\delta }_h}+\tilde{\nu }){\overline{I}}_1^h(t)\right)(t^y/\mathsf{\Gamma }(y+1))\hfill \\ {\underline{I}}_2^h(t)=\left({\tilde{\alpha }}_1{\underline{E}}_1^h(t)-(\widetilde{{\mu }_h}+\widetilde{{\delta }_h}+\tilde{\nu }){\underline{I}}_1^h(t)\right)(t^y/\mathsf{\Gamma }(y+1)).\hfill \end{array}\right.$$

(43)

$$\left\{\begin{array}{c}{\overline{Q}}_2^h(t)=\left(\widetilde{{\alpha }_2}{\overline{E}}_1^h(t)-(\tilde{\varphi }+\tilde{\tau }+\widetilde{{\delta }_h}+\widetilde{{\mu }_h}){\overline{Q}}_1^h(t)\right)(t^y/\mathsf{\Gamma }(y+1))\hfill \\ {\underline{Q}}_2^h(t)=\left(\widetilde{{\alpha }_2}{\underline{E}}_1^h(t)-(\tilde{\varphi }+\tilde{\tau }+\widetilde{{\delta }_h}+\widetilde{{\mu }_h}){\underline{Q}}_1^h(t)\right)(t^y/\mathsf{\Gamma }(y+1)).\hfill \end{array}\right.$$

(44)

$$\left\{\begin{array}{c}{\overline{R}}_2^h(t)=\left(\tilde{\nu }{\overline{I}}_1^h(t)+\tilde{\tau }{\overline{Q}}_1^h(t)-\widetilde{{\mu }_h}{\overline{R}}_1^h(t)\right)(t^y/\mathsf{\Gamma }(y+1))\hfill \\ {\underline{R}}_2^h(t)=\left(\tilde{\nu }{\underline{I}}_1^h(t)+\tilde{\tau }{\underline{Q}}_1^h(t)-\widetilde{{\mu }_h}{\overline{R}}_1^h(t)\right)(t^y/\mathsf{\Gamma }(y+1)).\hfill \end{array}\right.$$

(45)

$$\left\{\begin{array}{c}{\overline{S}}_2^r(t)=\left(\widetilde{{\theta }_r}-{\displaystyle \frac{\widetilde{{\beta }_3}{\overline{S}}_1^r(t){\overline{I}}_1^r(t)}{({\theta }_r+0.4912(1-5\zeta ))(t^y/\mathsf{\Gamma }(y+1))}}-\widetilde{{\mu }_r}{\overline{S}}_1^r(t)\right)(t^y/\mathsf{\Gamma }(y+1))\hfill \\ {\underline{S}}_2^r(t)=\left(\widetilde{{\theta }_r}-{\displaystyle \frac{\widetilde{{\beta }_3}{\underline{S}}_1^r(t){\underline{I}}_1^r(t)}{({\theta }_r+0.4912(5\zeta -1))(t^y/\mathsf{\Gamma }(y+1))}}-\widetilde{{\mu }_r}{\underline{S}}_1^r(t)\right)(t^y/\mathsf{\Gamma }(y+1)).\hfill \end{array}\right.$$

(46)

$$\left\{\begin{array}{c}{\overline{E}}_2^r(t)=\left({\displaystyle \frac{\widetilde{{\beta }_3}{\overline{S}}_1^r(t){\overline{I}}_1^r(t)}{({\theta }_r+0.4912(1-5\zeta ))(t^y/\mathsf{\Gamma }(y+1))}}-(\widetilde{{\mu }_r}+\widetilde{{\alpha }_3}){\overline{E}}_1^r(t)\right)(t^y/\mathsf{\Gamma }(y+1))\hfill \\ {\underline{E}}_2^r(t)=\left({\displaystyle \frac{\widetilde{{\beta }_3}{\underline{S}}_1^r(t){\underline{I}}_1^r(t)}{({\theta }_r+0.4912(5\zeta -1))(t^y/\mathsf{\Gamma }(y+1))}}-(\widetilde{{\mu }_r}+\widetilde{{\alpha }_3}){\underline{E}}_1^r(t)\right)(t^y/\mathsf{\Gamma }(y+1)).\hfill \end{array}\right.$$

(47)

$$\left\{\begin{array}{c}{\overline{I}}_2^r(t)=\left(\widetilde{{\alpha }_3}{\overline{E}}_1^r(t)-(\widetilde{{\mu }_r}+\widetilde{{\delta }_r}){\overline{I}}_1^r(t)){\overline{E}}_1^r(t)\right)(t^y/\mathsf{\Gamma }(y+1))\hfill \\ {\underline{I}}_2^r(t)=\left(\widetilde{{\alpha }_3}{\underline{E}}_1^r(t)-(\widetilde{{\mu }_r}+\widetilde{{\delta }_r}){\underline{I}}_1^r(t)\right)(t^y/\mathsf{\Gamma }(y+1)).\hfill \end{array}\right.$$

(48)

Now we can solve Equations (25)–(32) by substituting the values from Table 2 in the first term, second term, third term, etc. relations, and we obtain

$\left\{\begin{array}{c}{\overline{S}}_h(t)=(1-5\zeta )+((0.029+0.499938(1-5\zeta ))(1-5\zeta )+0.029)(t^y/\mathsf{\Gamma }(y+1))-\hfill \\ ({\displaystyle \frac{6.786\times {10}^{-6}+0.000117(1-5\zeta )}{0.029-3.4000558(1-5\zeta )}}(1-5\zeta )-0.0435(1-5\zeta )\hfill \\ -5.189907(1-5\zeta )){(t^y/\mathsf{\Gamma }(y+1))}^2+\dots \hfill \\ {\underline{S}}_h(t)=(5\zeta -1)+((0.029+0.499938(5\zeta -1))(5\zeta -1)+0.029)(t^y/\mathsf{\Gamma }(y+1))-\hfill \\ ({\displaystyle \frac{6.786\times {10}^{-6}+0.000117(1-5\zeta )}{0.029-3.4000558(5\zeta -1)}}(1-5\zeta )-0.0435(5\zeta -1)\hfill \\ -5.189907(5\zeta -1)){(t^y/\mathsf{\Gamma }(y+1))}^2+\dots \hfill \end{array}\right.$

$\left\{\begin{array}{c}{\overline{E}}_h(t)=(1-5\zeta )-3.699938(1-5\zeta )(t^y/\mathsf{\Gamma }(y+1))\hfill \\ +({\displaystyle \frac{6.786\times {10}^{-6}+0.000117(1-5\zeta )}{0.029-3.4000558(1-5\zeta )}}(1-5\zeta )+13.6897706\hfill \\ (1-5\zeta )){(t^y/\mathsf{\Gamma }(y+1))}^2+\dots \hfill \\ {\underline{E}}_h(t)=(5\zeta -1)-3.699938(5\zeta -1)(t^y/\mathsf{\Gamma }(y+1))\hfill \\ +({\displaystyle \frac{6.786\times {10}^{-6}+0.000117(5\zeta -1)}{0.029-3.4000558(5\zeta -1)}}(5\zeta -1)+13.6897706\hfill \\ (5\zeta -1)){(t^y/\mathsf{\Gamma }(y+1))}^2+\dots \hfill \end{array}\right.$

$\left\{\begin{array}{c}{\overline{I}}_h(t)=(1-5\zeta )-2.33(1-5\zeta )(t^y/\mathsf{\Gamma }(y+1))+5.1549124(1-5\zeta )\hfill \\ {(t^y/\mathsf{\Gamma }(y+1))}^2+\dots \hfill \\ {\underline{I}}_h(t)=(5\zeta -1)-2.33(5\zeta -1)(t^y/\mathsf{\Gamma }(y+1))+5.1549124(5\zeta -1)\hfill \\ {(t^y/\mathsf{\Gamma }(y+1))}^2+\dots \hfill \end{array}\right.$

$\left\{\begin{array}{c}{\overline{Q}}_h(t)=(1-5\zeta )-2.22(1-5\zeta )(t^y/\mathsf{\Gamma }(y+1))+1.968524(1-5\zeta )\hfill \\ {(t^y/\mathsf{\Gamma }(y+1))}^2+\dots \hfill \\ {\underline{Q}}_h(t)=(5\zeta -1)-2.22(15\zeta -1)(t^y/\mathsf{\Gamma }(y+1))+1.968524(5\zeta -1)\hfill \\ {(t^y/\mathsf{\Gamma }(y+1))}^2+\dots \hfill \end{array}\right.$

$\left\{\begin{array}{c}{\overline{R}}_h(t)=(1-5\zeta )-0.15(1-5\zeta )(t^y/\mathsf{\Gamma }(y+1))-2.8633(1-5\zeta )\hfill \\ {(t^y/\mathsf{\Gamma }(y+1))}^2+\dots \hfill \\ {\underline{R}}_h(t)=(5\zeta -1)-0.15(5\zeta -1)(t^y/\mathsf{\Gamma }(y+1))-2.8633(5\zeta -1)\hfill \\ {(t^y/\mathsf{\Gamma }(y+1))}^2+\dots \hfill \end{array}\right.$

$\left\{\begin{array}{c}{\overline{S}}_r(t)=(1-5\zeta )+(0.2+0.0074(1-5\zeta )+0.2)(t^y/\mathsf{\Gamma }(y+1))\hfill \\ -({\displaystyle \frac{0.0080892-0.0002993(1-5\zeta )}{0.2+0.4912(1-5\zeta )}}(1-5\zeta )+0.0004\hfill \\ -0.0000148(1-5\zeta )){(t^y/\mathsf{\Gamma }(y+1))}^2+\dots \hfill \\ {\underline{S}}_r(t)=(5\zeta -1)+(0.2+0.0074(5\zeta -1)+0.2)(t^y/\mathsf{\Gamma }(y+1))\hfill \\ -({\displaystyle \frac{0.0080892-0.0002993(5\zeta -1)}{0.2+0.4912(1-5\zeta )}}(5\zeta -1)+0.0004\hfill \\ -0.0000148(5\zeta -1)){(t^y/\mathsf{\Gamma }(y+1))}^2+\dots \hfill \end{array}\right.$

$\left\{\begin{array}{c}{\overline{E}}_r(t)=(1-5\zeta )-1.9966(1-5\zeta )(t^y/\mathsf{\Gamma }(y+1))+({\displaystyle \frac{0.0080892-0.0002993(1-5\zeta )}{0.2+0.4912(1-5\zeta )}}\hfill \\ (1-5\zeta )+3.9971932(1-5\zeta )){(t^y/\mathsf{\Gamma }(y+1))}^2+\dots \hfill \\ {\underline{E}}_r(t)=(5\zeta -1)-1.9966(5\zeta -1)(t^y/\mathsf{\Gamma }(y+1))+({\displaystyle \frac{0.0080892-0.0002993(5\zeta -1)}{0.2+0.4912(5\zeta -1)}}\hfill \\ (5\zeta -1)+3.9971932(5\zeta -1)){(t^y/\mathsf{\Gamma }(y+1))}^2+\dots \hfill \end{array}\right.$

$\left\{\begin{array}{c}{\overline{I}}_r(t)=(1-5\zeta )+1.498(1-5\zeta )(t^y/\mathsf{\Gamma }(y+1))-4.745196(1-5\zeta )\hfill \\ {(t^y/\mathsf{\Gamma }(y+1))}^2+\dots \hfill \\ {\underline{I}}_r(t)=(5\zeta -1)+1.498(5\zeta -1)(t^y/\mathsf{\Gamma }(y+1))-4.745196(5\zeta -1)\hfill \\ {(t^y/\mathsf{\Gamma }(y+1))}^2+\dots \hfill \end{array}\right.$

The spread of transmission of monkeypox virus is shown in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 using the fuzzy fractional order at $\zeta =0.003$. It also finds the solution for the following Equations (25)–(32). The existence of solutions is guaranteed by w-monotunity conditions (refer [39,40]).

Generalized Compartmental Model

According to model (1), close interaction with infected rodents will result in the spread of the monkeypox virus to humans. The monkeypox virus will not spread to humans if the infected rodents $I_r$ do not infect any humans. As a result, if $I_r$ falls to zero in (1), there are no suspected, exposed, or infected human populations. As a result, this model will reduce to the SEIQR model.

If $I_r=0$, then the model (1) will split into two compartments, such as

$$\left\{\begin{array}{c}{\displaystyle \frac{d}{dt}}{S}_h={\theta }_h-{\displaystyle \frac{{\beta }_2{I}_h{S}_h}{N_h}}-{\mu }_h{S}_h+\varphi Q_h\hfill \\ {\displaystyle \frac{d}{dt}}{E}_h={\displaystyle \frac{{\beta }_2{I}_h{S}_h}{N_h}}-({\alpha }_1+{\alpha }_2+{\mu }_h)E_h\hfill \\ {\displaystyle \frac{d}{dt}}{I}_h={\alpha }_1{E}_h-({\mu }_h+{\delta }_h+\nu )I_h\hfill \\ {\displaystyle \frac{d}{dt}}{Q}_h={\alpha }_2{E}_h-(\varphi +\tau +{\delta }_h+{\mu }_h)Q_h\hfill \\ {\displaystyle \frac{d}{dt}}{R}_h=\nu I_h+\tau Q_h-{\mu }_h{R}_h\hfill \end{array}\right.$$

(49)

and

$$\left\{\begin{array}{c}{\displaystyle \frac{d}{dt}}{S}_r={\theta }_r-{\mu }_r{S}_r\hfill \\ {\displaystyle \frac{d}{dt}}{E}_r=-({\mu }_r+{\alpha }_3)E_r\hfill \\ {\displaystyle \frac{d}{dt}}{I}_r={\alpha }_3{E}_r\hfill \end{array}\right.$$

(50)

Equations (49) and (50) represent the SEIQR and SEI models for humans and rodents, respectively. There are numerous approaches used to solve the solution for these two models [50,51,52,53]. As a result, the monkeypox virus model can be considered to be the generalized compartmental model for all other compartmental models.

5. Conclusions

The goal of the current paper is to use a fuzzy fractional mathematical model to analyze the dynamics of monkeypox virus transmission. Here, we used the Caputo’s derivative method in fuzzy system to solve the uniqueness and existence for the considered model. The results pertaining to fractional order differential equation solutions are advantageous in applications to various epidemic models for the deduction of viral spread. The graphic representation of the SEIQRSEI system’s response to changes for different parameters shows that the number of suspected and exposed human cases is gradually rising. This has the effect of rapidly accelerating infection transmission. The spread of infection within the population can be reduced using control measures. The spread of the monkeypox virus will slow down due to the reduction in the transmission parameter, which includes contact rates between exposed individuals $E_h$ and infected individuals $I_h$, and contributions from different $\zeta$ values. It was observed that exposed and infected people exhibit a reasonable reduction. As a result, the infection can be somewhat controlled by lowering effective contact rates. Furthermore, we deduced that the compartmental monkeypox virus model will result in the generalization of additional models such as SEI, SEIR, SEIQR, etc. The generalized compartmental section indicates that if the infection rate of rodent is zero ($I_r=0$), there will be no spread of rodent individuals to the human population. Future research can be conducted using several monkeypox epidemic models with various fractional order derivatives.