Modeling, analysis and prediction of new variants of covid-19 and dengue co-infection on complex network

Abstract

Recently, four new strains of SARS-COV-2 were reported in different countries which are mutants and considered as 70$\%$ more dangerous than the existing covid-19 virus. In this paper, hybrid mathematical models of new strains and co-infection in Caputo, Caputo-Fabrizio, and Atangana-Baleanu are presented. The idea behind this co-infection modeling is that, as per medical reports, both dengue and covid-19 have similar symptoms at the early stages. Our aim is to evaluate and predict the transmission dynamics of both deadly viruses. The qualitative study via stability analysis is discussed at equilibria and reproduction number ${\mathcal{R}}_0$ is computed. For the numerical purpose, Adams-Bashforth-Moulton and Newton methods are employed to obtain the approximate solutions of the proposed model. Sensitivity analysis is carried out to assessed the effects of various biological parameters and rates of transmission on the dynamics of both viruses. We also compared our results with some reported data against infected, recovered, and death cases.

1. Introduction

The mathematical modeling of communicable and non-communicable diseases have been attracting the attention of many mathematical modelers [2], [5], [16], [31] since a long time. The first case of the novel Carona virus was identified in Wuhan city of China in December 2019. The rate of infection of the covid-19 was very high therefore, WHO has declared it as a pandemic [35]. Mathematical modeling has been a powerful tool to study the transmission dynamics of covid-19 and other diseases. Many models have been presented by different mathematicians from time to time to get an insight into the dynamics of these diseases [7], [10], [13], [23], [24]. Recently, four new strains of covid-19 have been reported which are considered as 70$\%$ more dangerous than the early existing virus. The covid-19 has not only affected the healths of people but has caused big damage to the financial system of many countries.

Like covid-19, dengue fever is another challenging and very old disease spreading in the tropical and subtropical areas all over the world. The dengue epidemic is a major problem. As per data available in the literature, approximately 50 million people die due to dengue [34]. This epidemic is a mosquito-borne disease transmitted by Aedes albopictus and Aedes aegypti mosquitoes. This fever is caused by four different serotypes, which are $DEN(I,II,III,andIV)$.It is an RNA virus of the family Flaviviridae. However, a human is infected by only one serotype among these four. A human population is recovering, gaining full immunity to this type of serotype and only minor and transient immunity concerning the other three serotypes. Dengue fever can change from severe to mild. The other severe forms of dengue fever include dengue hemorrhagic fever (DHF) and shock syndrome. The infected populations remain asymptomatic for about three-fourteen days before they begin to experience a sudden onset of dengue fever. There is no specific treatment for this dengue disease; however, hospitalization, bed rest, analgesics, and antipyretics can be obtained for supportive care. People with weak immune systems develop these more serious forms of dengue. As usual, they need to be hospitalized. The full life cycle of epidemic dengue fever involves the role of the hosts and vectors as transmitters as the main source of infection [8], [25]. To prevent dengue fever virus transmission that depends fully on the control of vectors or interruption of host mosquitoes contact with the host, strategies are required at an early stage [30], [34].

As per the data available in the literature, approximately 0.6 Million people died due to covid-19 and many more infected [35]. It is confirmed that early symptoms of covid-19 are lung infections, breathing problems, fatigue, and cough. Strangely, some cases of gastroenteritis and neurological disorder have come to notice which open new vistas of research in the direction of neurological science [21]. The covid-19 spreads by droplets spreading in air and surface over a susceptible person expose to the droplets gets infected due to covid-19. Through mathematics, we can’t make any kind of vaccine for these epidemic diseases, but we can tell them how to prevent these viral diseases through mathematical models [32], [33]. Further, we set different rates by which everyone understands the mathematical model easily. Thus, we use fractional-order derivative in the Caputo sense because it gives a better outcome than the integer-order. Various fractional derivatives operators were developed, but the Riemann-Liouville and the Caputo are mostly used due to their simplicity and sincerity to handle [17], [20], [26]. But at present the other fractional derivatives are in lines namely, Hadamard, Atangana-Baleanu, Caputo-Fabrizio, and many others, [3], [4], [28]. These models have the suitability and efficiency of the Caputo operator. Moreover, Caputo-Fabrizio is the second best since it gives an error rate value of $1.97\%$ for a fractional-order derivative. It is important to mention that fractional-order derivative equations are more fitting than integer order modeling in economic, biological, and social mathematical models where memory effects are important.

We are motivated to study fractional-order differential equations because exponential laws are very traditional approaches to studying the chaotic behavior of a complex dynamical system of population densities and epidemics, but there are certain dynamical systems where dynamical changes undergo faster or slower than exponential laws. In such cases, the Mittag-Leffler function can be used to describe the dynamic changes in such systems. Also, due to the effective memory function of fractional derivative, fractional-order differential equations have been widely used to describe the biological situation. Fractional-order derivatives are useful in studying the chaotic behavior of the dynamical system. Even though fractional-order is the generalization of an ordinary differential equation to a random order. They have attracted considerable attention due to their ability to deal with more complex systems.

In this paper, we extend the work of the author [1] wherein the authors presented a fractional-order mathematical model in which only dengue class is considered. But in our work, we incorporated the new variants of the covid-19 class in addition to the dengue class and assess the effects of their co-infection. This co-dynamics situation is realistic as some cases were reported in Brazil in which dengue and covid-19 attacked the human population simultaneously [14]. We study a novel hybrid mathematical (SI-SICR) model of co-infection of dengue and covid-19 and address the following questions:-

$\left(a\right)$ Does the dengue virus act as a launch pad for new SARS-COV-2 strains $?$

$\left(b\right)$ Has new invariants of covid-19 strain possess existing SARS-COV-2 $?$

$\left(c\right)$ What will it take to achieve herd immunity with SARS-COV-2 $?$

$\left(d\right)$ What will be the optimal solution for the mitigation of the dengue and SARS-COV-2 co-infection $?$

The rest of the paper is organized as follows. Some basic preliminaries on fractional calculus are presented in the Section 2. Section 3 is devoted to the formulation of mathematical modeling. The basic properties of the proposed model are given in the Section 4. The stability analysis is discussed in Section 5. Optimization analysis is presented in the Section 6. The numerical solutions are obtained in the Section 7. The results and discussion are provided in the Section 8 and finally, the conclusion is drawn in the Section 9.

2. Preliminaries on fractional calculus

Some basic preliminaries on fractional calculus are given as:

Definition 2.1

The Riemann-Liouville fractional integral of the function $f:{\mathbb{R}}^{+}\to \mathbb{R}$ exists for order $\alpha >0$ in two forms, upper and lower. Consider the closed interval $[a,b]$, the integrals are defined as [29];

$${\displaystyle {}_a^{RL}{D}_t^{-\alpha }{=}_a^{RL}{I}_t^{\alpha }=\frac{1}{\Gamma \left(\alpha \right)}{\int }_a^t{(t-\omega )}^{\alpha -1}f\left(\omega \right)d\omega ,fort>a,}$$

$${\displaystyle {}_t^{RL}{D}_b^{-\alpha }{=}_t^{RL}{I}_b^{\alpha }=\frac{1}{\Gamma \left(\alpha \right)}{\int }_t^b{(\omega -t)}^{\alpha -1}f\left(\omega \right)d\omega ,fort<b,}$$

where $\Gamma$ is the gamma function.

Definition 2.2

The Riemann-Liouville fractional derivative of the function $f:{\mathbb{R}}^{+}\to \mathbb{R}$ is also exists in two forms, upper and lower. This derivative is calculated by using the Lagrange’s rule for differential operators. To compute the nth order derivative over the integral of order $(n-\alpha )$, the $\alpha$ order derivative is obtained. It is important to remember $n>\alpha$, where, $n$ is the smallest integer. Thus the derivatives are defined as [29];

$${\displaystyle {}_a^{RL}{D}_t^{\alpha }f\left(t\right)={\frac{d^n}{d{t}^n}}_a^{RL}{D}_t^{-(n-\alpha )}f\left(t\right)={\frac{d^n}{d{t}^n}}_a^{RL}{I}_t^{n-\alpha }f\left(t\right),}$$

$${\displaystyle {}_t^{RL}{D}_b^{\alpha }f\left(t\right)={\frac{d^n}{d{t}^n}}_t^{RL}{D}_b^{-(n-\alpha )}f\left(t\right)={\frac{d^n}{d{t}^n}}_t^{RL}{I}_b^{n-\alpha }f\left(t\right).}$$

Definition 2.3

Due to somedrawback of Riemann-Liouvile derivative an alternatetive definition was given by Caupto [27], and is defined as below

$$\begin{array}{ccc}& & {\displaystyle {}_0^C{D}_t^{\alpha }f\left(t\right)=\frac{1}{\Gamma (n-\alpha )}{\int }_0^t\frac{f^{\left(n\right)}\left(\omega \right)}{{(t-\omega )}^{\alpha -n+1}}d\omega ,}\hfill \\ & & \text{where}\alpha \in (n-1,n),\text{in}\text{which}n\in \mathbb{N}.\hfill \end{array}$$

Obviously, ${}_0^C{D}_t^{\alpha }f\left(t\right)\to D_t^{\alpha }f\left(t\right)$ whenever $\alpha \to 1.$ Thus, ${}_0^C{D}_t^{\alpha }f\left(t\right)$ and ${}_0^C{D}_t^{\alpha }g\left(t\right)$ exist almost everywhere and let $s_1,s_2\in \mathbb{R}$, then ${}_0^C{D}_t^{\alpha }[s_{1}f\left(t\right)+s_{2}g\left(t\right)]$ exist almost everywhere with

$${}_0^C{D}_t^{\alpha }[s_{1}f\left(t\right)+s_{2}g\left(t\right)]=s_1\left[{}_0^C{D}_t^{\alpha }f\left(t\right)\right]+s_2\left[{}_0^C{D}_t^{\alpha }g\left(t\right)\right].$$ (2.1)

Definition 2.4

Let us consider a constant point, say $c^{*}$ for the Caupto system that is called its equilibrium point, and is defined as below

$${}_0^C{D}_t^{\alpha }{c}^{*}\left(t\right)=f(t,c^{*}\left(t\right))\iff f(t,c^{*}t)=0,where0<\alpha <1.$$

Definition 2.5

Let $f\in H^1(a,b),b>a,\alpha \in [0,1]$,where $H^1(a,b)$ is the Sobolev space, of order $\alpha =1$ and is defined as

$$H^1(a,b)=\{f\in L^2(a,b):Df\in L^2(a,b)\},$$

then the Caupto fractional derivative is defined as [18];

$${\displaystyle {}_a^C{D}_t^{\alpha }f\left(t\right)=\frac{M\left(\alpha \right)}{1-\alpha }{\int }_a^t{D}_t^{\alpha }f\left(x\right)exp[-\alpha \frac{t-x}{1-\alpha }]dx,}$$

in which $M\left(\alpha \right)$ is the normalization function such that $M\left(0\right)=M\left(1\right)=1$.

Definition 2.6

If the function does not belong to the Sobolev space then the new derivative that comes is known as Caupto-Fabrizo fractional derivative and is defined as

$${\displaystyle {}_a^{CF}{D}_t^{\alpha }f\left(t\right)=\frac{M\left(\alpha \right)}{1-\alpha }{\int }_a^t(f\left(t\right)-f\left(x\right))exp[-\alpha \frac{t-x}{1-\alpha }]dx.}$$

Definition 2.7

Let $f\in H^1(a,b),b>a,\alpha \in [0,1]$. If the function $f$ is differentiable then, the new fractional derivative known as Atangana-Baleanu derivative in the Caupto sense and is defined as

$${\displaystyle {}_a^{ABC}{D}_t^{\alpha }f\left(t\right)=\frac{B\left(\alpha \right)}{1-\alpha }{\int }_a^t{D}_t^{\alpha }f\left(x\right)E_{\alpha }[-\alpha \frac{{(t-x)}^{\alpha }}{1-\alpha }]dx,}$$

where $B\left(\alpha \right)$ is the normalization function such that $B\left(0\right)=B\left(1\right)=1,$ in which ${\displaystyle B\left(\alpha \right)=1-\alpha +\frac{\alpha }{\Gamma \left(\alpha \right)}.}$

Here it should be notice that we don’t revover the original function when order $\alpha =0$, except when at the origin point function is get vinished. To avoid this type of issue, we have the following definition.

Definition 2.8

Let $f\in H^1(a,b),b>a,\alpha \in [0,1]$. Here the function $f$ is not necessary differentiable then, the new fractional derivative known as Atangana-Baleanu derivative in the Riemann-Liouville sense and is defined as

$${\displaystyle {}_a^{ABR}{D}_t^{\alpha }f\left(t\right)=\frac{B\left(\alpha \right)}{1-\alpha }{\int }_a^{t}f\left(x\right)E_{\alpha }[-\alpha \frac{{(t-x)}^{\alpha }}{1-\alpha }]dx,}$$

where $B\left(\alpha \right)$ is also the normalization function such that $B\left(0\right)=B\left(1\right)=1,$ in which ${\displaystyle B\left(\alpha \right)=1-\alpha +\frac{\alpha }{\Gamma \left(\alpha \right)}.}$

Definition 2.9

The new fractional derivative associted to fractional integral with nonlocal kernel is known as Atangana-Baleunu fractional integral and is defined as [9];

$${\displaystyle {}_a^{AB}{I}_t^{\alpha }f\left(t\right)=\frac{1-\alpha }{B\left(\alpha \right)}f\left(t\right)+\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_a^{t}f\left(x\right){(t-x)}^{\alpha -1}dx.}$$

If $\alpha =0$ we recover the intial function and $\alpha =1$ we get the ordinary integral.

3. Mathematical model formulation

In this section, we formulate a deterministic mathematical model of dengue and covid-19 co-infection by dividing the total population into two classes, namely, the vector (mosquito) and host (human). To make the dynamical transmission co-infection model with covid-19 induced death rate ${\varphi }_h$ and the fraction of covid-19 patient that have already dengue epidemic, the two classes of vector population are considered, namely susceptible vector (mosquito) class, at time $t$, is denoted by $S_m\left(t\right)$; infectious vector class $\left(I_m\left(t\right)\right)$. Hence, the total vector population $Z_m\left(t\right)$ is given by

$$Z_m\left(t\right)=S_m\left(t\right)+I_m\left(t\right).$$

On the other hand, four classes of host (human) population; susceptible $\left(S_h\left(t\right)\right)$, infectious $\left(I_h\left(t\right)\right)$, covid-19 patients in host population $\left(C_h\left(t\right)\right)$ and recovered$\left(R_h\left(t\right)\right)$ are considered so that the total host population $Z_h\left(t\right)$ becomes

$$Z_h\left(t\right)=S_h\left(t\right)+I_h\left(t\right)+C_h\left(t\right)+R_h\left(t\right).$$

Notations and parametric values of the variables used in the formulation of dynamical transmission of model are given in the below Tables 1 and 2 .

Table 1.

Description of the state variable of co-infection model (3.1) with variables.

State Variables	Description
$S_m$	Susceptible vectors(mosquitoes) class
$I_m$	Infected vectors class
$S_h$	Susceptible hosts(humans) class
$I_h$	Infected hosts class
$C_h$	covid-19 infected hosts class
$R_h$	Recovered hosts class

Table 2.

The description of parameters along with parametric values of the model (3.1).

Parameter	Description	Parametric values	Source
${\Delta }_1$	Recruitment rate of adults	$(4500-10000)mont{h}^{-1}$	[8]
	female mosquito population
${\Delta }_2$	Recruitment rate of host population	$10yea{r}^{-1}$	[8]
${\eta }_m$	Rate of infection in	0.85-1	[8]
	mosquitoes population
${\eta }_h$	Rate of infection in	0.75	Computed
	humans population
$b$	mosquitoes bitting rate	0.5	Computed
$d_m$	Natural death rate of	0.25	[8]
	mosquitoes population
$d_h$	Natural death rate of	0.0000457-0.25	Computed
	humans population
${\gamma }_h$	Recovery rate of dengue	0.1428	Computed
	in human population
${\varphi }_h$	covid-19 induced death	(0.6)million	Computed
	rate in the human population
$p$	The fraction of covid-19 patients that	1.0-1.50	Computed
	have already dengue epidemic

Let ${\Delta }_1$ is the recruitment rate of the vector population, ${\Delta }_2$ is the recruitment rate of the host population, $b$ is the vector biting rate, ${\eta }_m$ is the transmission chance from host to vector, ${\eta }_h$ is the transmission chance from vector to host, $d_m$ is the natural death rate of vector population, $d_h$ is the natural death rate of the host population, ${\gamma }_h$ is the recovery rate of the host population, $p$ is the fraction of covid-19 patient that have already dengue epidemic, and ${\varphi }_h$ is the covid-19 induced death rate in the host population.

The flows from the susceptible to infected classes of mosquitoes and humans populations depend on the transmission probabilites ${\eta }_m,{\eta }_h$, the bitting rate of the mosquitoes $b$, and the number of infectious and susceptibles of each species. In $S_m$ class, the recruitment rate of mosquiotoes population is ${\Delta }_1$. ${\eta }_m$ is the rate of infection between $S_m$ to $I_h$. Similarly in $S_h$ class the recruitment rate of humans population is ${\Delta }_2$. ${\eta }_h$ is the rate of infection between $S_h$ to $I_m$. $p{I}_h{C}_h$ is the rate of flow between $I_h$ and $C_h$ and ${\gamma }_h+(1-p)I_h{C}_h$ is the rate of flow between $I_h$ and $R_h$ respectively.

The transmission dynamics of the co-infection mathematical model is portrayed in Figs. 1  & 2 ((a) &  (b)), the new four-strain of covid-19 are shown [35].

Fig. 1.

Transmission diagram of co-infection of dengue $\&$ covid-19 new strains.

Fig. 2.

Four strains of SARS-COV-2.

3.1. Classical integer model

The classical integer model of co-infection epidemic between mosquito-to-human and vice-versa is as follows:

$$\begin{array}{cc}\hfill \mathbf{Mosq}& \mathbf{uitoes}\mathbf{population}\hfill \\ \hfill D{S}_m=& {\displaystyle {\Delta }_1-\frac{b{\eta }_m}{Z_h}{S}_m{I}_h-d_m{S}_m,}\hfill \\ \hfill D{I}_m=& {\displaystyle \frac{b{\eta }_m}{Z_h}{S}_m{I}_h-d_m{I}_m,}\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathbf{Huma}& \mathbf{ns}\mathbf{population}\hfill \\ \hfill D{S}_h=& {\displaystyle {\Delta }_2-\frac{b{\eta }_h}{Z_h}{S}_h{I}_m-d_h{S}_h,}\hfill \\ \hfill D{I}_h=& {\displaystyle \frac{b{\eta }_h}{Z_h}{S}_h{I}_m-(d_h+{\gamma }_h)I_h-p{I}_h{C}_h-(1-p)I_h{C}_h,}\hfill \\ \hfill D{C}_h=& p{I}_h{C}_h-(d_h+{\varphi }_h)C_h,\hfill \\ \hfill D{R}_h=& {\gamma }_h{I}_h-d_h{R}_h+(1-p)I_h{C}_h.\hfill \end{array}$$ (3.1)

The intial conditions are as below.

$$\begin{array}{ccc}\hfill S_m\left(0\right)& =& S_{m0},I_m\left(0\right)=I_{m0},S_h\left(0\right)=S_{h0},I_h\left(0\right)=I_{h0},\hfill \\ \hfill C_h\left(0\right)& =& C_{h0}\&R_h\left(0\right)=R_{h0}.\hfill \end{array}$$ (3.2)

3.2. Fractional order mathematical model

In this subsection, we have modified the classical integer model (2.1) into fractional-order in the sense of Caupto. The co-infection model system in the form of coupled non-linear fractional-order differential equation is given as:

$$\begin{array}{cc}\hfill \mathbf{Mosq}& \mathbf{uitoes}\mathbf{population}\hfill \\ \hfill {}_0^C{D}_t^{\alpha }{S}_m=& {\displaystyle {\Delta }_1-\frac{b{\eta }_m}{Z_h}{S}_m{I}_h-d_m{S}_m,}\hfill \\ \hfill {}_0^C{D}_t^{\alpha }{I}_m=& {\displaystyle \frac{b{\eta }_m}{Z_h}{S}_m{I}_h-d_m{I}_m,}\hfill \\ \hfill \mathbf{Huma}& \mathbf{ns}\mathbf{population}\hfill \\ \hfill {}_0^C{D}_t^{\alpha }{S}_h=& {\displaystyle {\Delta }_2-\frac{b{\eta }_h}{Z_h}{S}_h{I}_m-d_h{S}_h,}\hfill \\ \hfill {}_0^C{D}_t^{\alpha }{I}_h=& {\displaystyle \frac{b{\eta }_h}{Z_h}{S}_h{I}_m-(d_h+{\gamma }_h)I_h-p{I}_h{C}_h-(1-p)I_h{C}_h,}\hfill \\ \hfill {}_0^C{D}_t^{\alpha }{C}_h=& p{I}_h{C}_h-(d_h+{\varphi }_h)C_h,\hfill \\ \hfill {}_0^C{D}_t^{\alpha }{R}_h=& {\gamma }_h{I}_h-d_h{R}_h+(1-p)I_h{C}_h.\hfill \end{array}$$ (3.3)

The Caputo fractional-order co-infection model (3.3) gives the dynamics of host populations, and all state variables and parameters are supposed to be non-negative.

4. Basic properties of the model

For the well-posedness of the proposed model, its existence and uniqueness theorem is presented below.

4.1. Existence and uniqueness

In this subsection, the well-posedness of the Caputo-Fabrizio fractional differential co-infection model (3.3), by the use of fixed point theorems is presented. Thus, we simplify in the following way:

$$\begin{array}{cc}\hfill {}_0^{CF}{D}_t^{\alpha }{S}_m\left(t\right)=& Z_1(t,S_m,I_m,S_h,I_h,C_h,R_h),\hfill \\ \hfill {}_0^{CF}{D}_t^{\alpha }{I}_m\left(t\right)=& Z_2(t,S_m,I_m,S_h,I_h,C_h,R_h),\hfill \\ \hfill {}_0^{CF}{D}_t^{\alpha }{S}_h\left(t\right)=& Z_3(t,S_m,I_m,S_h,I_h,C_h,R_h),\hfill \\ \hfill {}_0^{CF}{D}_t^{\alpha }{I}_h\left(t\right)=& Z_4(t,S_m,I_m,S_h,I_h,C_h,R_h),\hfill \\ \hfill {}_0^{CF}{D}_t^{\alpha }{C}_h\left(t\right)=& Z_5(t,S_m,I_m,S_h,I_h,C_h,R_h),\hfill \\ \hfill {}_0^{CF}{D}_t^{\alpha }{R}_h\left(t\right)=& Z_6(t,S_m,I_m,S_h,I_h,C_h,R_h),\hfill \end{array}$$ (4.1)

where,

$$\begin{array}{cc}\hfill Z_1(t,S_m,I_m,S_h,I_h,C_h,R_h)=& {\displaystyle {\Delta }_1-\frac{b{\eta }_m}{Z_h}{S}_m{I}_h-d_m{S}_m,}\hfill \\ \hfill Z_2(t,S_m,I_m,S_h,I_h,C_h,R_h)=& {\displaystyle \frac{b{\eta }_m}{Z_h}{S}_m{I}_h-d_m{I}_m,}\hfill \\ \hfill Z_3(t,S_m,I_m,S_h,I_h,C_h,R_h)=& {\displaystyle {\Delta }_2-\frac{b{\eta }_h}{Z_h}{S}_h{I}_m-d_h{S}_h,}\hfill \\ \hfill Z_4(t,S_m,I_m,S_h,I_h,C_h,R_h)=& {\displaystyle \frac{b{\eta }_h}{Z_h}{S}_h{I}_m-(d_h+{\gamma }_h)I_h-p{I}_h{C}_h-(1-p)I_h{C}_h,}\hfill \\ \hfill Z_5(t,S_m,I_m,S_h,I_h,C_h,R_h)=& p{I}_h{C}_h-(d_h+{\varphi }_h)C_h,\hfill \\ \hfill Z_6(t,S_m,I_m,S_h,I_h,C_h,R_h)=& {\gamma }_h{I}_h-d_h{R}_h+(1-p)I_h{C}_h.\hfill \end{array}$$ (4.2)

Thus, the co-infection model (3.3) is generalized, in the following.

$$\begin{array}{cc}\hfill {}_0^{CF}{D}_t^{\alpha }\phi \left(t\right)=& K(t,\phi (t\left)\right),\alpha \in (0,1],t\in L=[0,c],\hfill \\ \hfill \phi \left(0\right)=& {\phi }_0\ge 0,\hfill \end{array}$$ (4.3)

along with iniatal conditions

$$\begin{array}{cc}\hfill \phi \left(t\right)=& {(S_m,I_m,S_h,I_h,C_h,R_h)}^T,\hfill \\ \hfill \phi \left(0\right)=& {(S_{m0},I_{m0},S_{h0},I_{h0},C_{h0},R_{h0})}^T,\hfill \\ \hfill K(t,\phi (t\left)\right)=& {\left(Z_i(t,S_m,I_m,S_h,I_h,C_h,R_h)\right)}^T,i=1to6,\hfill \end{array}$$ (4.4)

where, ${(.)}^T$ shows the transpose operation. By Ahmed et al. [5], the integral representation of Eq. (4.3) is equiv. to model (3.3) and is given by

$$\phi \left(t\right)={\displaystyle {\phi }_0+{\Theta }^{\alpha }K(t,\phi \left(t\right))={\phi }_0+\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t{(t-\chi )}^{\alpha -1}K(\chi ,\phi \left(\chi \right))d\chi .}$$ (4.5)

Let $E:[0,c]\to \mathbb{R}$ with the norm defined by

$${\parallel \phi \parallel }_E={\displaystyle \underset{t\in L}{\sup }\left|\phi \left(t\right)\right|,}$$

where, $\left|\phi \left(t\right)\right|=|S_m\left(t\right)|+|I_m\left(t\right)|+|S_h\left(t\right)|+|I_h\left(t\right)|+|C_h\left(t\right)|+|R_h\left(t\right)|,\&S_m,I_m,S_h,I_h,C_h,R_h\in E=C([0,c],\mathbb{R}).$

Theorem 4.1

Let$K:L\times {\mathbb{R}}_{+}^6\to \mathbb{R},$where$K$is in$C[L,\mathbb{R}]$. Also,$\exists$a positive constant${\mathcal{F}}_K$s.t.$|K(t,{\phi }_1\left(t\right))-K(t,{\phi }_2\left(t\right))|\le \mathcal{F}|{\phi }_1\left(t\right)-{\phi }_2\left(t\right)|,\forall t\in L,{\phi }_1,{\phi }_2\in C[L,\mathbb{R}]$. Then the Eq. (4.3) which is equiv. to the model (3.3) has a unique solution if$\aleph {\mathcal{F}}_K<1,$in which$\aleph =\frac{c^{\alpha }}{\Gamma (\alpha +1)}$.

Proof

See Appendix I. □

Next, by the Schauder fixed point theorem, we prove the existence of the solutions of the Eq. (4.3) that is Equiv. to model (3.3). So, we need the following corollary.

Corollary 4.2

let$\exists a_1,a_2\in E$s.t.$\left|K(t,\phi \left(t\right))\right|\le a_1\left(t\right)+a_2\left|\phi \left(t\right)\right|,forany\phi \in E,t\in L,$and${\displaystyle a_1^{*}=\underset{t\in L}{\sup }|a_2\left(t\right)|,a_2^{*}=\underset{t\in L}{\sup }\left|a_2\left(t\right)\right|<1.}$

Lemma 4.3

The function$R:E\to E$defined by$\left(R\phi \right)\left(t\right)={\phi }_0+\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t{(t-\chi )}^{\alpha -1}K(\chi ,\phi \left(\chi \right))d\chi ,$is completely continuous.

Proof

See Appendix II. □

Theorem 4.4

Let$K:L\times {\mathbb{R}}^6\to \mathbb{R}$is continuous and satisfies the Corollary 4.2. The Eq. (4.3) that is equiv. to the model (3.3) has at least one solution.

Proof

See Appendix III. □

4.2. Invariant region and attractivity

The dynamical transmission of the Caupto fractional derivative of the co-infection model (3.3) will be analyzed in the following biologically feasible region, $\Pi \subset {\mathbb{R}}_{+}^6,$ where.

$${\displaystyle \Pi =\{(S_m,I_m,S_h,I_h,C_h,R_h)\in {\mathbb{R}}_{+}^6:S_m+I_m=\frac{{\Delta }_1}{d_m},S_h+I_h+C_h+R_h=\frac{{\Delta }_2-{\varphi }_h{C}_h}{d_h}\}.}$$

Lemma 4.5

The feasible region$\Pi \subset {\mathbb{R}}_{+}^6$is + vely invariant with respect to the intial conditions in${\mathbb{R}}_{+}^6$for the co-infection model (3.3).

Proof

See Appendix IV. □

In order to show that the model (3.3) has positive solution, we take ${\mathbb{R}}_{+}^6=\{z\in {\mathbb{R}}^6:z\ge 0\}\&z\left(t\right)=(S_m\left(t\right),I_m\left(t\right),S_h\left(t\right),I_h\left(t\right),C_h\left(t\right),R_h\left(t\right)){}^T.$

Corollary 4.6

Let$f\left(t\right)\in C[u,v]$and${}_0^C{D}_t^{\alpha }f\left(t\right)\in (u,v]$, where$\alpha \in (0,1].$If

$$\left(i\right){}_0^C{D}_t^{\alpha }f\left(t\right)\ge 0,forallf\in (u,v),thenf\left(t\right)isnon-decreasing,$$

$$\left(ii\right){}_0^C{D}_t^{\alpha }f\left(t\right)\le 0,forallf\in (u,v),thenf\left(t\right)isnon-increasing.$$

4.3. Positivity and boundedness

Proposition 4.7

The solution for the model (3.3) is non-negative, bounded for all $(S_m\left(0\right),I_m\left(0\right),S_h\left(0\right),I_h\left(0\right),C_h\left(0\right),R_h\left(0\right))\in {\mathbb{R}}_{+}^6,$ and also defined for positive value of time $t$ .

Proof

See Appendix V. □

Lemma 4.8

The region $\Pi$ is positively invariant for the co-infection model (3.3) with an initial condition in ${\mathbb{R}}_{+}^6$ . It is sufficient to show the dynamics of the aforesaid model in the region given in $\Pi$ . So this region can be considered for study epidemiologically and biologically.

5. The analysis of the model

The analysis of the co-infection of model (3.3) has been done by using the theory of stability by finding the equilibrium points at various possibilities.

5.1. Equilibrium analysis

In order to find the equilibrium points let us assume the left hand sides of all the equations of the model (3.3) equal to zero, so that we get the five possible equilibrium points are obtained as follows:

$\left(i\right)$ The disease-free equilibrium(DFE) point:-

$$E_0=(S_m^0,I_m^0,S_h^0,I_h^0,C_h^0,R_h^0)=({\Delta }_1/d_m,0,{\Delta }_2/d_h,0,0,0).$$

$\left(ii\right)$ Infected mosquito free equilibrium point $(I_m=0)$:-

$$E_1=(S_m^1,I_m^1,S_h^1,I_h^1,C_h^1,R_h^1)=({\Delta }_1/d_m,0,{\Delta }_2/d_h,0,0,0).$$

$\left(iii\right)$ Infected mosquito free equilibrium point $(I_h=0)$:-

$$E_2=(S_m^2,I_m^2,S_h^2,I_h^2,C_h^2,R_h^2)=({\Delta }_1/d_m,0,{\Delta }_2/d_h,0,0,0).$$

$\left(iv\right)$ Covid-19 infected human free equilibrium point $(C_h=0)$:-

$$E_3=(S_m^3,I_m^3,S_h^3,I_h^3,C_h^3,R_h^3),$$

satisfies,

${\displaystyle S_m^3=\frac{p{\Delta }_1{Z}_h}{b{\eta }_m(d_h+{\varphi }_h)+p{d}_m{Z}_h},I_m^3=\frac{b{\eta }_m{\Delta }_1(d_h+{\varphi }_h)}{d_m(b{\eta }_m(d_h+{\varphi }_h)+p{d}_m{Z}_h)},S_h^3=\frac{{\Delta }_2{d}_m{Z}_h(b{\eta }_m(d_h+{\varphi }_h)+p{d}_m{Z}_h)}{(d_h+{\gamma }_h)\left(p{b}^2{\eta }_m{\eta }_h{\Delta }_1{\Delta }_2\right)},I_h^3=\frac{(d_h+{\varphi }_h)}{p},C_h^3=0,R_h^3=\frac{{\gamma }_h(d_h+{\varphi }_h)}{p{d}_h}.}$

$\left(v\right)$ The endemic equilibrium(EE) point of the Caupto fractional-order model (3.3) is as follows:-

$$E_4=(S_m^4,I_m^4,S_h^4,I_h^4,C_h^4,R_h^4),$$

satisfies,${\displaystyle S_m^4=\frac{p{\Delta }_1{Z}_h}{b{\eta }_m(d_h+{\varphi }_h)+p{d}_m{Z}_h},I_m^4=\frac{b{\eta }_m{\Delta }_1(d_h+{\varphi }_h)}{d_m(b{\eta }_m(d_h+{\varphi }_h)+p{d}_m{Z}_h)},S_h^4=\frac{{\Delta }_2{d}_m{Z}_h(b{\eta }_m(d_h+{\varphi }_h)+p{d}_m{Z}_h)}{E},I_h^4=\frac{(d_h+{\varphi }_h)}{p},C_h^4=\frac{N}{D(d_h+{\varphi }_h)},R_h^4=\frac{D{\gamma }_h(d_h+{\varphi }_h)+N(1-p)}{pD{d}_h},}$ where,

$$\begin{array}{ccc}\hfill N& =& p{b}^2{\eta }_m{\eta }_h{\Delta }_1{\Delta }_2(d_h+{\varphi }_h)-(d_h+{\gamma }_h)(d_h+{\varphi }_h)\hfill \\ & & (b^2{\eta }_m{\eta }_h{\Delta }_1(d_h+{\varphi }_h)+d_m{d}_h{Z}_h(b{\eta }_m(d_h+{\varphi }_h)+p{d}_m{Z}_h)),\hfill \end{array}$$

$$D=(b^2{\eta }_m{\eta }_h{\Delta }_1(d_h+{\varphi }_h)+d_m{d}_h{Z}_h(b{Z}_m(d_h+{\varphi }_h)+p{d}_m{Z}_h\left)\right),$$

and

$$E=(b^2{\eta }_m{\eta }_h{\Delta }_1(d_h+{\varphi }_h)+d_m{d}_h{Z}_h(b{\eta }_m(d_h+{\varphi }_h)+p{d}_m{Z}_h).$$

5.2. Stability analysis

Here we present the stability of the five possible equilibrium points to study the behaviour of dynamical system.

5.2.1. Stability of disease free equilibrium point $E_0$

The Jacobian matrix $J$ of the model (3.3) at $E_0$ is as follows:

$$\begin{array}{c}\hfill J_{E_0}=\left[\begin{array}{cccccc}-d_m& 0& 0& {\displaystyle -\frac{b{\eta }_m{\Delta }_1}{Z_h{d}_m}}& 0& 0\\ 0& -d_m& 0& {\displaystyle \frac{b{\eta }_m{\Delta }_1}{Z_h{d}_m}}& 0& 0\\ 0& {\displaystyle -\frac{b{\eta }_h{\Delta }_2}{Z_h{d}_h}}& -d_h& 0& 0& 0\\ 0& {\displaystyle \frac{b{\eta }_h{\Delta }_2}{Z_h{d}_h}}& 0& -(d_h+{\gamma }_h)& 0& 0\\ 0& 0& 0& 0& -(d_h+{\varphi }_h)& 0\\ 0& 0& 0& {\gamma }_h& 0& -d_h\end{array}\right].\end{array}$$ (5.1)

Theorem 5.1

$E_0$is locally asymptotically stable if$b^2{\eta }_m{\eta }_h{\Delta }_1{\Delta }_2<Z_h^2{d}_m^2{d}_h(d_h+{\gamma }_h)$and is unstable if$b^2{\eta }_m{\eta }_h{\Delta }_1{\Delta }_2>Z_h^2{d}_m^2{d}_h(d_h+{\gamma }_h)$.

Proof

See Appendix VI. □

Biologically if ${\mathcal{R}}_0<1$, then the infection will be eliminate, but if ${\mathcal{R}}_0>1$, the infection persist in the system and if ${\mathcal{R}}_0=1$, the bifurcation occur. After some simplification, we can find the threshold quantity that is

$${\displaystyle {\mathcal{R}}_0^c=\sqrt{{\displaystyle \frac{1}{(d_h+{\gamma }_h)}+\frac{1}{d_m}}}.}$$

On the base of threshold quantity we’ve the following proposition.

Proposition 5.2

$\left(i\right)$If${\mathcal{R}}_0<{\mathcal{R}}_0^c,$then the DFE$E_0$of the model (3.3) is locally asymptotically stable;

$\left(ii\right)$If${\mathcal{R}}_0>{\mathcal{R}}_0^c,$then the DFE$E_0$of the model (3.3) is unstable; and

$\left(iii\right)$If${\mathcal{R}}_0={\mathcal{R}}_0^c,$then the DFE$E_0$of the model (3.3) is stable.

5.2.2. Stability of mosquito free equilibrium point $E_1$$(I_m=0)$.

In case $E_0$ is replace by $E_1$, the rest of the analysis is similar.

5.2.3. Stability of human free equilibrium point $E_2$$(I_h=0)$.

In case $E_0$ is replace by $E_2$, the rest of the analysis is similar.

5.2.4. Stability of covid-19 free equilibrium point $E_3$$(C_m=0)$.

The Jacobian matrix $J_{E_3}$ evaluated at the covid-19 free equilibrium point and is given as below:

$$\begin{array}{cc}\hfill J_{E_3}=& \left[\begin{array}{cccccc}{\displaystyle -\frac{b{\eta }_m}{Z_h}{I}_h^3-d_m}& 0& 0& {\displaystyle -\frac{b{\eta }_m}{Z_h}{S}_m^3}& 0& 0\\ {\displaystyle \frac{b{\eta }_m}{Z_h}{I}_h^3}& -d_m& 0& {\displaystyle \frac{b{\eta }_m}{Z_h}{S}_m^3}& 0& 0\\ 0& {\displaystyle -\frac{b{\eta }_h}{Z_h}{S}_h^3}& {\displaystyle -\frac{b{\eta }_h}{Z_h}{I}_m^3-d_h}& 0& 0& 0\\ 0& {\displaystyle \frac{b{\eta }_h}{Z_h}{S}_h^3}& {\displaystyle \frac{b{\eta }_h}{Z_h}{I}_m^3}& -(d_h+{\gamma }_h)& -I_h^3& 0\\ 0& 0& 0& 0& p{I}_h^3-(d_h+{\varphi }_h)& 0\\ 0& 0& 0& {\gamma }_h& (1-p)I_h^3& -d_h\end{array}\right].\hfill \end{array}$$ (5.2)

The characterstic Eq. of the Jacobian matrix $J\left(E_3\right)$ is

$$\lambda (\lambda +d_m)(\lambda +d_h)({\lambda }^3+a{\lambda }^2+b\lambda +c)=0,$$ (5.3)

where,

$$\begin{array}{cc}\hfill a_1=& (2d_m+d_h+{\gamma }_h+u_1),\hfill \\ \hfill b_1=& (d_m(u_1+d_m)-u_2+(d_h+{\gamma }_h)(2d_m+u_1)),\hfill \\ \hfill c_1=& d_m(d_h+{\gamma }_h)(u_1+d_m)-u_2{d}_h\hfill \end{array}$$ (5.4)

in which ${\displaystyle u_1=\frac{b{\eta }_h}{Z_h}{I}_m^3,and{u}_2=\frac{b^2{\eta }_m{\eta }_h}{Z_h^2}{S}_m^3{S}_h^3.}$

If $f\left(z\right)=z^3+a_1{z}^2+b_{1}z+c_1.$ Let $D\left(f\right)$ denotes the disc. of a poly. $f\left(z\right)$; then we’ve

$$\begin{array}{cc}\hfill D\left(f\right)=& -\left|\begin{array}{ccccc}1& a_1& b_1& c_1& 0\\ 0& 1& a_1& b_1& c_1\\ 3& 2a_1& b_1& 0& 0\\ 0& 3& a_1& b_1& 0\\ 0& 0& 3& 2a_1& b_1\end{array}\right|=18a_1{b}_1{c}_1+{\left(a_1{b}_1\right)}^2-4c_1{a}_1^3-4b_1^3-27c_1^2.\hfill \end{array}$$ (5.5)

To check the stability of the $E_3$, we have the following proposition:

Proposition 5.3

$\left(i\right)$If$D\left(f\right)$, is + ve$\&$Routh-Hurwitz Criterian are satisfied, i.e.,$D\left(f\right)>0,a_1>0,c_1>0,\&a_1{b}_1>c_1$, then$E_3$is locally asymptotically stable.

$\left(ii\right)$If$D\left(f\right)<0,a_1>0,b_1>0,a_1{b}_1=c_1,\&0\le \alpha <1,$then$E_3$is locally asymptotically stable.

$\left(iii\right)$If$D\left(f\right)<0,a_1<0,b_1<0,\&\alpha >2/3,$then$E_3$is unstable.

$\left(iv\right)$The necessary condition for$E_3$, to be locally asymptotically stable, is$c_1>0.$

Theorem 5.4

Let$0<\alpha <1$, the covid-19 infection free equilibrium point of the Caupto fractional-order dengue fever and covid-19 model (3.3), is globally asymptotically stable in the closed set$\Pi$, whenever${\mathcal{R}}_0<1.$

Proof

See Appendix VII. □

5.2.5. Stability of endemic equilibrium point $E_4$

Now we discuss the stability of the endemic equilibrium point of the model (3.3) for this the Jacobian matrix $J_{E_4}$ is given as below:

$$\begin{array}{cc}\hfill J_{E_4}=& \left[\begin{array}{cccccc}{\displaystyle -\frac{b{\eta }_m}{Z_h}{I}_h^4-d_m}& 0& 0& {\displaystyle -\frac{b{\eta }_m}{Z_h}{S}_m^4}& 0& 0\\ {\displaystyle \frac{b{\eta }_m}{Z_h}{I}_h^4}& -d_m& 0& {\displaystyle \frac{b{\eta }_m}{Z_h}{S}_m^4}& 0& 0\\ 0& {\displaystyle -\frac{b{\eta }_h}{Z_h}{S}_h^4}& {\displaystyle -\frac{b{\eta }_h}{Z_h}{I}_m^4-d_h}& 0& 0& 0\\ 0& {\displaystyle \frac{b{\eta }_h}{Z_h}{S}_h^4}& {\displaystyle \frac{b{\eta }_h}{Z_h}{I}_m^4}& -(d_h+{\gamma }_h)-C_h^4& -I_h^4& 0\\ 0& 0& 0& p{C}_h^4& p{I}_h^4-(d_h+{\varphi }_h)& 0\\ 0& 0& 0& {\gamma }_h+(1-p)C_h^4& (1-p)I_h^4& -d_h\end{array}\right].\hfill \end{array}$$ (5.6)

The characterstic Eq. of the Jacobian matrix $J\left(E_4\right)$ is

$$\lambda (\lambda +d_m)(\lambda +d_h)({\lambda }^3+a_2{\lambda }^2+b_2\lambda +c_2)=0,$$ (5.7)

where

$$\begin{array}{cc}\hfill a_2=& (2d_m+d_h+{\gamma }_h+C_h^4+v_1),\hfill \\ \hfill b_2=& (d_m(v_1+d_m)-v_2+(d_h+{\gamma }_h+C_h^4)(2d_m+v_1)),\hfill \\ \hfill c_2=& d_m(d_h+{\gamma }_h+C_h^4)(v_1+d_m)-v_2{d}_h\hfill \end{array}$$ (5.8)

in which ${\displaystyle v_1=\frac{b{\eta }_h}{Z_h}{I}_m^4,and{v}_2=\frac{b^2{\eta }_m{\eta }_h}{Z_h^2}{S}_m^4{S}_h^4.}$

If $g\left(z\right)=z^3+a_2{z}^2+b_{2}z+c_2.$ Let $D\left(g\right)$ denotes the disc. of a poly. $g\left(z\right)$; then we’ve

$$\begin{array}{cc}\hfill D\left(g\right)=& -\left|\begin{array}{ccccc}1& a_2& b_2& c_2& 0\\ 0& 1& a_2& b_2& c_2\\ 3& 2a_2& b_2& 0& 0\\ 0& 3& a_2& b_2& 0\\ 0& 0& 3& 2a_2& b_2\end{array}\right|=18a_2{b}_2{c}_2+{\left(a_2{b}_2\right)}^2-4c_2{a}_2^3-4b_2^3-27c_2^2.\hfill \end{array}$$ (5.9)

To check the stability of the $E_3$, we have the following proposition:

Proposition 5.5

$\left(i\right)$If$D\left(g\right)$, is + ve$\&$Routh-Hurwitz Criterian are satisfied, i.e.,$D\left(g\right)>0,a_2>0,c_2>0,\&a_2{b}_2>c_2$, then$E_4$is locally asymptotically stable.

$\left(ii\right)$If$D\left(g\right)<0,a_2>0,b_2>0,a_2{b}_2=c_2,\&0\le \alpha <1,$then$E_4$is locally asymptotically stable.

$\left(iii\right)$If$D\left(g\right)<0,a_2<0,b_2<0,\&\alpha >2/3,$then$E_4$is unstable.

$\left(iv\right)$The necessary condition for$E_4$, to be locally asymptotically stable, is$c_2>0.$

6. Optimal control

Since covid-19 is spread via host contact with infected populations [19], [22]. But here in this paper, our aim is to inform people that those who have already been infected by dengue fever have more chances of covid-19 infection as compared to those who have not. Thus, we can put the control parameter on this serious problem to prevent its spreading. The following are the control assumptions:

$l_1:$ To control mosquitoes populations.

$l_2:$ Infectious mosquitoes should be killed.

$l_3:$ covid-19 patients should be quarantined.

On the basis of the above assumptions, the objective function is formulated as

$$F(l_i,\Omega )={\displaystyle {\int }_0^T(W_1{S}_m^2+W_2{I}_m^2+W_3{S}_h^2+W_4{I}_h^2+W_5{C}_h^2+W_6{R}_h^2+u_1{l}_1^2+u_2{l}_2^2+u_3{l}_3^2)dt}$$ (6.1)

where $\Omega$ denotes the set of all compartmental variables, $W_1,W_2,W_3,W_4,W_5,W_6$ denotes the positive weight contants for the variables $S_m,I_m,S_h,I_h,C_h,R_h$ respectively.

Now, we will find every value of control variables from $t=0toT$ s.t.,

$$F\left(l_i\left(t\right)\right)=\min \{\frac{F(l_i^{*},\Omega )}{l_i}\in M\},i=1,2,3$$ (6.2)

where, $M$ is the smooth function for the interval [0,1].

Therefore, the Langrangian function related to the objective function is given by,

$$\Theta (\Omega ,W_i)=W_1{S}_m^2+W_2{I}_m^2+W_3{S}_h^2+W_4{I}_h^2+W_5{C}_h^2+W_6{R}_h^2+u_1{l}_1^2+u_2{l}_2^2+u_3{l}_3^2+{\displaystyle {\zeta }_1({\Delta }_1-\frac{b{\eta }_m}{Z_h}{S}_m{I}_h-d_m{S}_m)+{\zeta }_2(\frac{b{\eta }_m}{Z_h}{S}_m{I}_h-(d_m+l_1+l_2)I_m)}+{\displaystyle {\zeta }_3({\Delta }_2-\frac{b{\eta }_h}{Z_h}{S}_h{I}_m-d_h{S}_h)+{\zeta }_4(\frac{b{\eta }_h}{Z_h}{S}_h{I}_m-(d_h+{\gamma }_h+l_3)I_h-I_h{C}_h+l_1{I}_m)}+{\zeta }_5\left(\right(p{I}_h-(d_h+{\varphi }_h))C_h+l_2{I}_m+l_3{I}_h)+{\zeta }_6({\gamma }_h{I}_h-d_h{R}_h+(1-p)I_h{C}_h).$$ (6.3)

The adjoint Eq. variables, ${\zeta }_i=({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4,{\zeta }_5,{\zeta }_6)$ for the system is calculated by taking the partial derivatives of $\Theta$ with respect to each variable.

$$\begin{array}{cc}\hfill \dot{{\zeta }_1}=& {\displaystyle -\frac{\partial \Theta }{\partial S_m}=-2W_1{S}_m+({\zeta }_1-{\zeta }_2)\frac{b{\eta }_m}{Z_h}{I}_h+{\zeta }_1{d}_m,}\hfill \\ \hfill \dot{{\zeta }_2}=& {\displaystyle -\frac{\partial \Theta }{\partial I_m}=-2W_2{I}_m+({\zeta }_3-{\zeta }_4)\frac{b{\eta }_h}{Z_h}{S}_h+({\zeta }_2-{\zeta }_4)l_1+({\zeta }_2-{\zeta }_5)l_2+{\zeta }_2{d}_m,}\hfill \\ \hfill \dot{{\zeta }_3}=& {\displaystyle -\frac{\partial \Theta }{\partial S_h}=-2W_3{S}_h+({\zeta }_3-{\zeta }_4)\frac{b{\eta }_h}{Z_h}{I}_m+{\zeta }_3{d}_h,}\hfill \\ \hfill \dot{{\zeta }_4}=& {\displaystyle -\frac{\partial \Theta }{\partial I_h}=-2W_4{I}_h+({\zeta }_1-{\zeta }_2)\frac{b{\eta }_m}{Z_h}{S}_m+({\zeta }_4-{\zeta }_5)l_3+{\zeta }_4(p{C}_h}\hfill \\ \hfill +& (1-p)C_h+{\gamma }_h+d_h)-{\zeta }_{5}p{C}_h-{\zeta }_6({\gamma }_h+(1-p)C_h),\hfill \\ \hfill \dot{{\zeta }_5}=& {\displaystyle -\frac{\partial \Theta }{\partial C_h}=-2W_5{C}_h+{\zeta }_4(p{I}_h+(1-p)I_h)-{\zeta }_5(p{I}_h-(d_h+{\varphi }_h))-{\zeta }_6\left((1-p)C_h\right),}\hfill \\ \hfill \dot{{\zeta }_6}=& {\displaystyle -\frac{\partial \Theta }{\partial R_h}=-2W_6{R}_h+{\zeta }_6{d}_h.}\hfill \end{array}$$ (6.4)

Hence, this calculation gives us,

$$\begin{array}{cc}\hfill l_1^{*}=& max(c_1,min(d_1,\frac{I_m({\zeta }_2-{\zeta }_4)}{2u_1}\left)\right),\hfill \\ \hfill l_2^{*}=& max(c_2,min(d_2,\frac{I_m({\zeta }_2-{\zeta }_5)}{2u_2}\left)\right),\hfill \\ \hfill l_3^{*}=& max(c_3,min(d_3,\frac{C_h({\zeta }_4-{\zeta }_5)}{2u_3}\left)\right).\hfill \end{array}$$ (6.5)

7. Numerical methods and simulations

Since most of the fractional-order differential Eq. don’t have exact analytic solutions, so numerical and approximate methods must obtain these solutions. So, we are constructing a numerical technique, for the fractional model based on the Caupto-fractional derivative, Caupto-Fabrizio and Atangana-Baleanu fractional derivative. For the use of this numerical technique we take some non-linear fractional ordinary differential Eq.:

7.1. Newton method for Caupto-fractional derivative

In this subsection, we deal with the following Cauchy problem

$${}_0^C{D}_t^{\alpha }g\left(t\right)=f(t,g\left(t\right)),g\left(0\right)=g_0,$$ (7.1)

where the derivative is in Caupto fractional derivative. Here, we aim to show the Newton method. For this, we firstly change the Eq. (7.1) into

$$g\left(t\right)-g\left(0\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{t}f(\chi ,g\left(\chi \right)){(t-\chi )}^{\alpha -1}d\chi .}$$ (7.2)

Since at the point $t_{i+1}=(i+1)\Delta t,$ we have

$$g\left(t_{i+1}\right)-g\left(0\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{t_{i+1}}f(\chi ,g\left(\chi \right)){(t_{i+1}-\chi )}^{\alpha -1}d\chi .}$$ (7.3)

Also, we have

$$g\left(t_{i+1}\right)={\displaystyle g\left(0\right)+\frac{1}{\Gamma \left(\alpha \right)}\sum _{n=2}^i{\int }_{t_n}^{t_{n+1}}f(\chi ,g\left(\chi \right)){(t_{i+1}-\chi )}^{\alpha -1}d\chi .}$$ (7.4)

Now, we replaced the Newton poly. with the Eq. (7.4), we have

$$g^{i+1}-g^0={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}\sum _{n=2}^i{\int }_{t_n}^{t_{n+1}}\{f(t_{n-2},g^{n-2})+(\frac{G_1}{\Delta t}+\frac{G_2}{2\Delta t^2}(\chi -t_{n-1})\left)(\chi -t_{n-2})\right\}}\times {(t_{i+1}-\chi )}^{\alpha -1}d\chi$$ (7.5)

where $G_1=f(t_{n-1},g^{n-1})-f(t_{n-2},g^{n-2}),G_2=f(t_n,g^n)-2f(t_{n-1},g^{n-1})+f(t_{n-2},g^{n-2}).$

This implies,

$$g^{i+1}={\displaystyle g^0+\frac{1}{\Gamma \left(\alpha \right)}\sum _{n=2}^i\left\{{\int }_{t_n}^{t_{n+1}}\right(f(t_{n-2},g^{n-2})+\frac{G_1}{\Delta t}(\chi -t_{n-2})+\frac{G_2}{2\Delta t^2}}\times (\chi -t_{n-2})(\chi -t_{n-1})\left){(t_{i+1}-\chi )}^{\alpha -1}d\chi \right\}.$$ (7.6)

This implies

$$g^{i+1}={\displaystyle g^0+\frac{1}{\Gamma \left(\alpha \right)}\sum _{n=2}^{i}f(t_{n-1},g^{n-2}){\int }_{t_n}^{t_{n+1}}{(t_{i+1}-\chi )}^{\alpha -1}d\chi +\frac{1}{\Gamma \left(\alpha \right)}\sum _{n=2}^i\frac{G_1}{\Delta t}{\int }_{t_n}^{t_{n+1}}(\chi -t_{n-2})}\times {\displaystyle {(t_{i+1}-\chi )}^{\alpha -1}d\chi +\frac{1}{\Gamma \left(\alpha \right)}\sum _{n=2}^i\frac{G_2}{2\Delta t^2}{\int }_{t_n}^{t_{n+1}}(\chi -t_{n-2})(\chi -t_{n-1}){(t_{i+1}-\chi )}^{\alpha -1}d\chi .}$$ (7.7)

The integrals in the Eq. (7.7) can be written as follows

$$\begin{array}{cc}\hfill {\int }_{t_n}^{t_{n+1}}& {\displaystyle {(t_{i+1}-\chi )}^{\alpha -1}d\chi =\frac{\Delta t^{\alpha }}{\alpha }[{(i-n+1)}^{\alpha }-{(i-n)}^{\alpha }],}\hfill \\ \hfill {\int }_{t_n}^{t_{n+1}}& {\displaystyle (\chi -t_{n-2}){(t_{i+1}-\chi )}^{\alpha -1}d\chi =\frac{\Delta t^{\alpha +1}}{\alpha (\alpha +1)}\left[\right({(i-n+1)}^{\alpha }-{(i-n)}^{\alpha }\left)(i-n+3+3\alpha )\right],}\hfill \\ \hfill {\int }_{t_n}^{t_{n+1}}& {\displaystyle (\chi -t_{n-2})(\chi -t_{n-1}){(t_{i+1}-\chi )}^{\alpha -1}d\chi =\frac{\Delta t^{\alpha +2}}{\alpha (\alpha +1)(\alpha +2)}\left[{(i-n+1)}^{\alpha }\right[2{(i-n)}^2+(3\alpha +10)}\hfill \\ \hfill \times (i-& n)+2{\alpha }^2+9\alpha 12]-{(i-n)}^{\alpha }[2{(i-n)}^2+(5\alpha +10)(i-n)+6{\alpha }^2+18\alpha +12\left]\right].\hfill \end{array}$$ (7.8)

Use Eq. (7.8) into (7.7). We have the following technique

$$g^{i+1}={\displaystyle g^0+\frac{\Delta t^v}{\Gamma (v+1)}\sum _{n=2}^{i}f(t_{n-1},g^{n-2})[{(i-n+1)}^v-{(i-n)}^v]+\frac{\Delta t^v}{\Gamma (v+2)}\sum _{n=2}^{i}f(t_{n-1},g^{n-1})}-{\displaystyle f(t_{n-2},g^{n-2})[{(i-n+1)}^v(i-n+3+2v)-{(i-n)}^v(i-n+3+3v)]+\frac{\Delta t^v}{2\Gamma (v+3)}}\times \sum _{n=2}^{i}f(t_n,g^n)-2f(t_{n-1},g^{n-1})+f(t_{n+2},g^{n-2})\left[{(i-n+1)}^v\right[2{(i-n)}^2+(3v+10)(i-n)+2v^2+9v+12]-{(i-n)}^v[2{(i-n)}^2+(5v+10)(i-j)+6v^2+18v+12\left]\right].$$ (7.9)

For simplicity, we write model (3.3) with C-F fractional derivative is already discussed in Eq. (4.1). Now the solution for the model (3.3), as followed.

$$S_m^{i+1}={\displaystyle S_m^0+\frac{\Delta t^{\alpha }}{\Gamma (\alpha +1)}\sum _{n=2}^j{\beta }_{13}[{(i-n+1)}^{\alpha }-{(i-n)}^{\alpha }]+\frac{\Delta t^{\alpha }}{\Gamma (\alpha +2)}\sum _{n=2}^j{\beta }_7-{\beta }_{13}[{(i-n+1)}^{\alpha }}\times {\displaystyle (i-n+3+2\alpha )-{(i-n)}^{\alpha }(i-n+3+3\alpha )]+\frac{\Delta t^{\alpha }}{\Gamma (\alpha +3)}\sum _{n=2}^j[{\beta }_1-2{\beta }_7+{\beta }_{13}]}\times \left[{(i-n+1)}^{\alpha }\right[2{(i-n)}^2+(3\alpha +10)(i-n)+2{\alpha }^2+9\alpha +12]-{(i-n)}^{\alpha }[2{(i-n)}^2+(5\alpha +10)(i-n)+5{\alpha }^2+18\alpha +12\left]\right]$$ (7.10)

$$I_m^{i+1}={\displaystyle I_m^0+\frac{\Delta t^{\alpha }}{\Gamma (\alpha +1)}\sum _{n=2}^j{\beta }_{14}[{(i-n+1)}^{\alpha }-{(i-n)}^{\alpha }]+\frac{\Delta t^{\alpha }}{\Gamma (\alpha +2)}\sum _{n=2}^j{\beta }_8-{\beta }_{14}[{(i-n+1)}^{\alpha }}\times {\displaystyle (i-n+3+2\alpha )-{(i-n)}^{\alpha }(i-n+3+3\alpha )]+\frac{\Delta t^{\alpha }}{\Gamma (\alpha +3)}\sum _{n=2}^j[{\beta }_2-2{\beta }_8+{\beta }_{14}]}\times \left[{(i-n+1)}^{\alpha }\right[2{(i-n)}^2+(3\alpha +10)(i-n)+2{\alpha }^2+9\alpha +12]-{(i-n)}^{\alpha }[2{(i-n)}^2+(5\alpha +10)(i-n)+5{\alpha }^2+18\alpha +12\left]\right]$$ (7.11)

$$S_h^{i+1}={\displaystyle S_h^0+\frac{\Delta t^{\alpha }}{\Gamma (\alpha +1)}\sum _{n=2}^j{\beta }_{15}[{(i-n+1)}^{\alpha }-{(i-n)}^{\alpha }]+\frac{\Delta t^{\alpha }}{\Gamma (\alpha +2)}\sum _{n=2}^j{\beta }_9-{\beta }_{15}[{(i-n+1)}^{\alpha }}\times {\displaystyle (i-n+3+2\alpha )-{(i-n)}^{\alpha }(i-n+3+3\alpha )]+\frac{\Delta t^{\alpha }}{\Gamma (\alpha +3)}\sum _{n=2}^j[{\beta }_3-2{\beta }_9+{\beta }_{15}]}\times \left[{(i-n+1)}^{\alpha }\right[2{(i-n)}^2+(3\alpha +10)(i-n)+2{\alpha }^2+9\alpha +12]-{(i-n)}^{\alpha }[2{(i-n)}^2+(5\alpha +10)(i-n)+5{\alpha }^2+18\alpha +12\left]\right]$$ (7.12)

$$I_h^{i+1}={\displaystyle I_h^0+\frac{\Delta t^{\alpha }}{\Gamma (\alpha +1)}\sum _{n=2}^j{\beta }_{16}[{(i-n+1)}^{\alpha }-{(i-n)}^{\alpha }]+\frac{\Delta t^{\alpha }}{\Gamma (\alpha +2)}\sum _{n=2}^j{\beta }_{10}-{\beta }_{16}[{(i-n+1)}^{\alpha }}\times {\displaystyle (i-n+3+2\alpha )-{(i-n)}^{\alpha }(i-n+3+3\alpha )]+\frac{\Delta t^{\alpha }}{\Gamma (\alpha +3)}\sum _{n=2}^j[{\beta }_4-2{\beta }_{10}+{\beta }_{16}]}\times \left[{(i-n+1)}^{\alpha }\right[2{(i-n)}^2+(3\alpha +10)(i-n)+2{\alpha }^2+9\alpha +12]-{(i-n)}^{\alpha }[2{(i-n)}^2+(5\alpha +10)(i-n)+5{\alpha }^2+18\alpha +12\left]\right]$$ (7.13)

$$C_h^{i+1}={\displaystyle C_h^0+\frac{\Delta t^{\alpha }}{\Gamma (\alpha +1)}\sum _{n=2}^j{\beta }_{17}[{(i-n+1)}^{\alpha }-{(i-n)}^{\alpha }]+\frac{\Delta t^{\alpha }}{\Gamma (\alpha +2)}\sum _{n=2}^j{\beta }_{11}-{\beta }_{17}[{(i-n+1)}^{\alpha }}\times {\displaystyle (i-n+3+2\alpha )-{(i-n)}^{\alpha }(i-n+3+3\alpha )]+\frac{{(t_{i+1}-t_i)}^{\alpha }}{\Gamma (\alpha +3)}\sum _{n=2}^j[{\beta }_5-2{\beta }_{11}+{\beta }_{17}]}\times \left[{(i-n+1)}^{\alpha }\right[2{(i-n)}^2+(3\alpha +10)(i-n)+2{\alpha }^2+9\alpha +12]-{(i-n)}^{\alpha }[2{(i-n)}^2+(5\alpha +10)(i-n)+5{\alpha }^2+18\alpha +12\left]\right]$$ (7.14)

$$R_h^{i+1}={\displaystyle R_h^0+\frac{\Delta t^{\alpha }}{\Gamma (\alpha +1)}\sum _{n=2}^j{\beta }_{18}[{(i-n+1)}^{\alpha }-{(i-n)}^{\alpha }]+\frac{\Delta t^{\alpha }}{\Gamma (\alpha +2)}\sum _{n=2}^j{\beta }_{12}-{\beta }_{18}[{(i-n+1)}^{\alpha }}\times {\displaystyle (i-n+3+2\alpha )-{(i-n)}^{\alpha }(i-n+3+3\alpha )]+\frac{\Delta t^{\alpha }}{\Gamma (\alpha +3)}\sum _{n=2}^j[{\beta }_6-2{\beta }_{12}+{\beta }_{18}]}\times \left[{(i-n+1)}^{\alpha }\right[2{(i-n)}^2+(3\alpha +10)(i-n)+2{\alpha }^2+9\alpha +12]-{(i-n)}^{\alpha }[2{(i-n)}^2+(5\alpha +10)(i-n)+5{\alpha }^2+18\alpha +12\left]\right]$$ (7.15)

where,

$$\begin{array}{cc}\hfill {\beta }_1=& S_{m1}(t_i,S_m^i,I_m^i,S_h^i,I_h^i,C_h^i,R_h^i),{\beta }_2=I_{m1}(t_i,S_m^i,I_m^i,S_h^i,I_h^i,C_h^i,R_h^i),\hfill \\ \hfill {\beta }_3=& S_{h1}(t_i,S_m^i,I_m^i,S_h^i,I_h^i,C_h^i,R_h^i),{\beta }_4=I_{h1}(t_i,S_m^i,I_m^i,S_h^i,I_h^i,C_h^i,R_h^i),\hfill \\ \hfill {\beta }_5=& C_{h1}(t_i,S_m^i,I_m^i,S_h^i,I_h^i,C_h^i,R_h^i),{\beta }_6=R_{h1}(t_i,S_m^i,I_m^i,S_h^i,I_h^i,C_h^i,R_h^i),\hfill \\ \hfill {\beta }_7=& S_{m1}(t_{i-1},S_m^{i-1},I_m^{i-1},S_h^{i-1},I_h^{i-1},C_h^{i-1},R_h^{i-1}),{\beta }_8=I_{m1}(t_{i-1},S_m^{i-1},I_m^{i-1},S_h^{i-1},I_h^{i-1},C_h^{i-1},R_h^{i-1}),\hfill \\ \hfill {\beta }_9=& S_{h1}(t_{i-1},S_m^{i-1},I_m^{i-1},S_h^{i-1},I_h^{i-1},C_h^{i-1},R_h^{i-1}),{\beta }_{10}=I_{h1}(t_{i-1},S_m^{i-1},I_m^{i-1},S_h^{i-1},I_h^{i-1},C_h^{i-1},R_h^{i-1}),\hfill \\ \hfill {\beta }_{11}=& C_{h1}(t_{i-1},S_m^{i-1},I_m^{i-1},S_h^{i-1},I_h^{i-1},C_h^{i-1},R_h^{i-1}),{\beta }_{12}=S_{h1}(t_{i-1},S_m^{i-1},I_m^{i-1},S_h^{i-1},I_h^{i-1},C_h^{i-1},R_h^{i-1}),\hfill \\ \hfill {\beta }_{13}=& S_{m1}(t_{i-2},S_m^{i-2},I_m^{i-2},S_h^{i-2},I_h^{i-2},C_h^{i-2},R_h^{i-2}),{\beta }_{14}=I_{m1}(t_{i-2},S_m^{i-2},I_m^{i-2},S_h^{i-2},I_h^{i-2},C_h^{i-2},R_h^{i-2}),\hfill \\ \hfill {\beta }_{15}=& S_{h1}(t_{i-2},S_m^{i-2},I_m^{i-2},S_h^{i-2},I_h^{i-2},C_h^{i-2},R_h^{i-2}),{\beta }_{16}=I_{h1}(t_{i-2},S_m^{i-2},I_m^{i-2},S_h^{i-2},I_h^{i-2},C_h^{i-2},R_h^{i-2}),\hfill \\ \hfill {\beta }_{17}=& C_{h1}(t_{i-2},S_m^{i-2},I_m^{i-2},S_h^{i-2},I_h^{i-2},C_h^{i-2},R_h^{i-2}),{\beta }_{18}=R_{h1}(t_{i-2},S_m^{i-2},I_m^{i-2},S_h^{i-2},I_h^{i-2},C_h^{i-2},R_h^{i-2}).\hfill \end{array}$$

7.2. Adams-Bashforth-Moulton method for Caupto fractional derivative

Now we impose many numerical and analytical techniques have been used to solve the fractional order differential Eq. For the numerical techniques of the fractional model (3.3) one can use the generalized ABM method. In order to find out the approximate solution by using this algorithm, we considered the following non-linear FDE [12]:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d^{\alpha }}{{}_C{dt}_t}{f}_h=}& y(h,f(h\left)\right),0\le h\le H,\hfill \\ \hfill f^{\left(\sigma \right)}\left(0\right)=& f_0^{\sigma },\sigma =0,1,2,3,...,l-1,wherel=\left[\alpha \right].\hfill \end{array}$$ (7.16)

Now Eq. (7.16) is equiv. to voltera integral equation:

$$f\left(h\right)={\displaystyle \sum _{\sigma =0}^{l-1}{f}_0^{\left(\sigma \right)}\frac{h^{\sigma }}{\sigma !}+\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^h{(h-s)}^{\alpha -1}y(s,f\left(s\right))ds.}$$ (7.17)

Based on ABM algorithm to integrate Eq. (7.17), [11] used the predictor-corrector scheme. On using predictor-corrector scheme to the fractional order dengue fever epidemic and covid-19 model (3.3) and letting $t=H/\mathbb{N},h_n=nt,\&n=0,1,2,3,...,\mathbb{N}\in {\mathbb{Z}}^{+}$. Therefore, Eq. (7.17) can be discretized as follows, we’ve:

$$\begin{array}{cc}\hfill S_{n+1}=& {\displaystyle S_0+\frac{t^{\alpha }}{\Gamma (\alpha +2)}({\Delta }_1-\frac{b{\eta }_m}{Z_h}{S}_{n+1}^r{X}_{n+1}^r-d_h{S}_{n+1}^r)}\hfill \\ \hfill +& {\displaystyle \frac{t^{\alpha }}{\Gamma (\alpha +2)}\sum _{k=0}^n{\phi }_{k,n+1}({\Delta }_1-\frac{b{\eta }_m}{Z_h}{S}_k{X}_k-d_m{S}_k),}\hfill \\ \hfill I_{n+1}=& {\displaystyle I_0+\frac{t^{\alpha }}{\Gamma (\alpha +2)}(\frac{b{\eta }_m}{Z_h}{S}_{n+1}^r{X}_{n+1}^r-d_h{I}_{n+1}^r)}\hfill \\ \hfill +& {\displaystyle \frac{t^{\alpha }}{\Gamma (\alpha +2)}\sum _{k=0}^n{\phi }_{k,n+1}(\frac{b{\eta }_m}{Z_h}{S}_k{X}_k-d_m{I}_k),}\hfill \\ \hfill W_{n+1}=& {\displaystyle W_0+\frac{t^{\alpha }}{\Gamma (\alpha +2)}({\Delta }_2-\frac{b{\eta }_h}{Z_h}{W}_{n+1}^r{I}_{n+1}^r-d_h{S}_{n+1}^r)}\hfill \\ \hfill +& {\displaystyle \frac{t^{\alpha }}{\Gamma (\alpha +2)}\sum _{k=0}^n{\phi }_{k,n+1}({\Delta }_2-\frac{b{\eta }_h}{Z_h}{W}_k{I}_k-d_m{W}_k),}\hfill \\ \hfill X_{n+1}=& {\displaystyle X_0+\frac{t^{\alpha }}{\Gamma (\alpha +2)}(\frac{b{\eta }_h}{Z_h}{W}_{n+1}^r{I}_{n+1}^r-(d_h+{\gamma }_h)X_{n+1}-p{X}_{n+1}{Y}_{n+1}-(1-p)X_{n+1}{Y}_{n+1})}\hfill \\ \hfill +& {\displaystyle \frac{t^{\alpha }}{\Gamma (\alpha +2)}\sum _{k=0}^n{\phi }_{k,n+1}(\frac{b{\eta }_h}{Z_h}{W}_k{I}_k-(d_h+{\gamma }_h)X_k-p{X}_k{Y}_k-(1-p)X_k{Y}_k),}\hfill \\ \hfill Y_{n+1}=& {\displaystyle Y_0+\frac{t^{\alpha }}{\Gamma (\alpha +2)}(p{X}_{n+1}^r{Y}_{n+1}^r-(d_h+{\varphi }_h)Y_{n+1}^r)}\hfill \\ \hfill +& {\displaystyle \frac{t^{\alpha }}{\Gamma (\alpha +2)}\sum _{k=0}^n{\phi }_{k,n+1}(p{X}_k{Y}_k-(d_h+{\varphi }_h)Y_k),}\hfill \\ \hfill Z_{n+1}=& {\displaystyle Y_0+\frac{t^{\alpha }}{\Gamma (\alpha +2)}({\gamma }_h{X}_{n+1}^r-d_h{Z}_{n+1}^r+(1-p)X_{n+1}^r{Y}_{n+1}^r)}\hfill \\ \hfill +& {\displaystyle \frac{t^{\alpha }}{\Gamma (\alpha +2)}\sum _{k=0}^n{\phi }_{k,n+1}({\gamma }_h{X}_k-d_h{Z}_k+(1-p)X_k{Y}_k),}\hfill \end{array}$$ (7.18)

where, ${\phi }_{k,r+1}=\{\begin{array}{c}{n}^{\alpha +1}-(r-\alpha )(r+1),k=0,\hfill \\ {(n-k+2)}^{\alpha +1}+{(r-k)}^{\alpha +1}-2{(r-k+1)}^{\alpha +1},1\le k\le n,\hfill \\ 1,k=n+1,\hfill \end{array}$ and the preliminary approxs. $S_{n+1}^r,I_{n+1}^r,W_{n+1}^r,X_{n+1}^r,Y_{n+1}^r\&Z_{n+1}^r$ are known as predictor and is given by

$$\begin{array}{cc}\hfill S_{n+1}^r=& {\displaystyle S_0+\frac{1}{\Gamma \left(\alpha \right)}\sum _{k=0}^n{\Psi }_{k,n+1}({\Delta }_1-\frac{b{\eta }_m}{Z_h}{S}_k{X}_k-d_m{S}_k),}\hfill \\ \hfill I_{n+1}^r=& {\displaystyle I_0+\frac{1}{\Gamma \left(\alpha \right)}\sum _{k=0}^n{\Psi }_{k,n+1}(\frac{b{\eta }_m}{Z_h}{S}_k{X}_k-d_m{I}_k),}\hfill \\ \hfill W_{n+1}^r=& {\displaystyle W_0+\frac{1}{\Gamma \left(\alpha \right)}\sum _{k=0}^n{\Psi }_{k,n+1}({\Delta }_2-\frac{b{\eta }_h}{Z_h}{W}_k{I}_k-d_m{W}_k),}\hfill \end{array}$$

$$\begin{array}{cc}\hfill X_{n+1}^r=& {\displaystyle X_0+\frac{1}{\Gamma \left(\alpha \right)}\sum _{k=0}^n{\Psi }_{k,n+1}(\frac{b{\eta }_h}{Z_h}{W}_k{I}_k-(d_h+{\gamma }_h)X_k-p{X}_k{Y}_k-(1-p)X_k{Y}_k),}\hfill \\ \hfill Y_{n+1}^r=& {\displaystyle Y_0+\frac{1}{\Gamma \left(\alpha \right)}\sum _{k=0}^n{\Psi }_{k,n+1}(p{X}_k{Y}_k-(d_h+{\varphi }_h)Y_k),}\hfill \\ \hfill Z_{n+1}^r=& {\displaystyle Z_0+\frac{1}{\Gamma \left(\alpha \right)}\sum _{k=0}^n{\Psi }_{k,n+1}({\gamma }_h{X}_k-d_h{Z}_k+(1-p)X_k{Y}_k),}\hfill \end{array}$$ (7.19)

where, ${\Psi }_{k,n+1}=\frac{h^{\alpha }}{\alpha }({(n-k+1)}^{\alpha }-{(n-k)}^{\alpha }),k\in [0,n].$

7.3. Newton method for Caupto-fabrizio fractional derivative

The Cauchy problem is given as

$${}_0^{CF}{D}_t^{\alpha }g\left(t\right)=f(t,g\left(t\right)),g\left(0\right)=g_0,$$ (7.20)

where the derivative is Caupto fabrizio derivative. Newton method is used to solve the Eq. (7.20). For this, we first transform the Eq. (7.20) into

$$g\left(t\right)-g\left(0\right)={\displaystyle \frac{1-\alpha }{M\left(\alpha \right)}f(t,g\left(t\right))+\frac{\alpha }{M\left(\alpha \right)}{\int }_0^{t}f(\chi ,g\left(\chi \right))d\chi .}$$ (7.21)

Since at $t_{i+1}=(i+1)\Delta t,$ we have

$$g\left(t_{i+1}\right)-g\left(0\right)={\displaystyle \frac{1-\alpha }{M\left(\alpha \right)}f(t_i,g\left(t_i\right))+\frac{\alpha }{M\left(\alpha \right)}{\int }_0^{t_{i+1}}f(\chi ,g\left(\chi \right))d\chi .}$$ (7.22)

At $t_i=i\Delta t,$ we have

$$g\left(t_i\right)-g\left(0\right)={\displaystyle \frac{1-\alpha }{M\left(\alpha \right)}f(t_{i-1},g\left(t_{i-1}\right))+\frac{\alpha }{M\left(\alpha \right)}{\int }_0^{t_i}f(\chi ,g\left(\chi \right))d\chi .}$$ (7.23)

From Eqs. (7.25) and (7.24), we get

$$g\left(t_{i+1}\right)-g\left(t_i\right)={\displaystyle \frac{1-\alpha }{M\left(\alpha \right)}{G}_3+\frac{\alpha }{M\left(\alpha \right)}\sum _{n=2}^i{\int }_{t_i}^{t_{i+1}}f(\chi ,g\left(\chi \right))d\chi ,}$$ (7.24)

and

$$g\left(t_{i+1}\right)-g\left(t_i\right)={\displaystyle \frac{1-\alpha }{M\left(\alpha \right)}{G}_3+\frac{\alpha }{M\left(\alpha \right)}\sum _{n=2}^i{\int }_{t_n}^{t_{n+1}}f(\chi ,g\left(\chi \right))d\chi .}$$ (7.25)

Using the Newton poly. we can write the approx. of the function $f(t,g(t\left)\right)$ as follows

$$Q_i\left(\chi \right)={\displaystyle f(t_{i-2},g\left(t_{i-2}\right))+\frac{G_4}{\Delta t}(\chi -t_{i-2})+\frac{G_5}{2\Delta t^2}(\chi -t_{i-2})(\chi -t_{i-1}),}$$ (7.26)

where $G_3=f(t_i,g\left(t_i\right))-f(t_{i-1},g\left(t_{i-1}\right)),G_4=f(t_{i-1},g\left(t_{i-1}\right))-f(t_{i-2},g\left(t_{i-2}\right)),G_5=f(t_i,g\left(t_i\right))-2f(t_{i-1},g\left(t_{i-1}\right))+f(t_{i-2},g\left(t_{i-2}\right)).$ Thus, using Eq. (7.26) into (7.25), we get

$$g^{i+1}-g^0={\displaystyle \frac{1-\alpha }{M\left(\alpha \right)}{G}_3+\frac{\alpha }{M\left(\alpha \right)}\sum _{n=2}^i{\int }_{t_n}^{t_{n+1}}\{f(t_{n-1},g^{n-2})+\frac{G_1}{\Delta t}\times (\chi -t_{n-2})+\frac{G_2}{2\Delta t^2}}\times (\chi -t_{n-2})(\chi -t_{n-1})\}d\chi ,$$ (7.27)

and more simplified as

$$g^{i+1}-g^0={\displaystyle \frac{1-\alpha }{M\left(\alpha \right)}{G}_3+\frac{\alpha }{M\left(\alpha \right)}\sum _{n=2}^i\{f(t_{n-2},g^{n-2}){\int }_{t_n}^{t_{n+1}}d\chi +\frac{1}{\Gamma \left(\alpha \right)}\sum _{n=2}^i\frac{G_1}{\Delta t}{\int }_{t_n}^{t_{n+1}}(\chi -t_{n-2})d\chi }+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}\sum _{n=2}^i\frac{G_2}{2\Delta t^2}{\int }_{t_n}^{t_{n+1}}(\chi -t_{n-2})(\chi -t_{n-1})d\chi \}.}$$ (7.28)

The calculations for the above integrals (7.28) is as

$${\displaystyle {\int }_{t_n}^{t_{n+1}}(\chi -t_{n-2})d\chi =}\frac{5}{2}\Delta t^2$$ (7.29)

$${\displaystyle {\int }_{t_n}^{t_{n+1}}(\chi -t_{n-2})(\chi -t_{n-1})d\chi =}\frac{23}{6}\Delta t^3.$$ (7.30)

Now by using Eqs. (7.29) and (7.30) into (7.28) we get

$$g^{i+1}={\displaystyle g^0+\frac{1-\alpha }{M\left(\alpha \right)}{G}_3+\frac{\alpha }{M\left(\alpha \right)}\sum _{n=2}^i{\int }_{t_n}^{t_{n+1}}\{f(t_{n-2},g^{n-2})+G_4\frac{5}{2}+G_5\frac{23}{6}\}\Delta t,}$$ (7.31)

and we can arrange as

$$g^{i+1}={\displaystyle g^0+\frac{1-\alpha }{M\left(\alpha \right)}{G}_3+\frac{\alpha }{M\left(\alpha \right)}\sum _{n=2}^i{\int }_{t_n}^{t_{n+1}}\left\{\right(\frac{32}{12}f(t_n,g^n)-\frac{4}{3}f(t_{n-1},g^{n-1})+\frac{5}{12}f(t_{n-2},g^{n-2}))}\times (t_{n+1}-t_n)\}.$$ (7.32)

After applying the C-F derivative, we have

$$S_m\left(t_{i+1}\right)={\displaystyle S_m\left(t_i\right)+\frac{1-\alpha }{M\left(\alpha \right)}[{\delta }_1-{\delta }_7]+\frac{\alpha }{M\left(\alpha \right)}\sum _{n=2}^j\left\{\right(\frac{23}{12}{\delta }_1-\frac{4}{3}{\delta }_7+\frac{5}{12}{\delta }_{13}\left)\Delta t\right\}}$$ (7.33)

$$I_m\left(t_{i+1}\right)={\displaystyle I_m\left(t_i\right)+\frac{1-\alpha }{M\left(\alpha \right)}[{\delta }_2-{\delta }_8]+\frac{\alpha }{M\left(\alpha \right)}\sum _{n=2}^j\left\{\right(\frac{23}{12}{\delta }_2-\frac{4}{3}{\delta }_8+\frac{5}{12}{\delta }_{14}\left)\Delta t\right\}}$$ (7.34)

$$S_h\left(t_{i+1}\right)={\displaystyle S_h\left(t_i\right)+\frac{1-\alpha }{M\left(\alpha \right)}[{\delta }_3-{\delta }_9]+\frac{\alpha }{M\left(\alpha \right)}\sum _{n=2}^j\left\{\right(\frac{23}{12}{\delta }_3-\frac{4}{3}{\delta }_9+\frac{5}{12}{\delta }_{15}\left)\Delta t\right\}}$$ (7.35)

$$I_h\left(t_{i+1}\right)={\displaystyle I_h\left(t_i\right)+\frac{1-\alpha }{M\left(\alpha \right)}[{\delta }_4-{\delta }_{10}]+\frac{\alpha }{M\left(\alpha \right)}\sum _{n=2}^j\left\{\right(\frac{23}{12}{\delta }_4-\frac{4}{3}{\delta }_{10}+\frac{5}{12}{\delta }_{16}\left)\Delta t\right\}}$$ (7.36)

$$C_h\left(t_{i+1}\right)={\displaystyle C_h\left(t_i\right)+\frac{1-\alpha }{M\left(\alpha \right)}[{\delta }_5-{\delta }_{11}]+\frac{\alpha }{M\left(\alpha \right)}\sum _{n=2}^j\left\{\right(\frac{23}{12}{\delta }_5-\frac{4}{3}{\delta }_{11}+\frac{5}{12}{\delta }_{17}\left)\Delta t\right\}}$$ (7.37)

$$R_h\left(t_{i+1}\right)={\displaystyle R_h\left(t_i\right)+\frac{1-\alpha }{M\left(\alpha \right)}[{\delta }_6-{\delta }_{12}]+\frac{\alpha }{M\left(\alpha \right)}\sum _{n=2}^j\left\{\right(\frac{23}{12}{\delta }_6-\frac{4}{3}{\delta }_{12}+\frac{5}{12}{\delta }_{18}\left)\Delta t\right\}}$$ (7.38)

where,

$$\begin{array}{cc}\hfill {\delta }_1=& S_{m1}(t_i,S_m\left(t_i\right),I_m\left(t_i\right),S_h\left(t_i\right),I_h\left(t_i\right),C_h\left(t_i\right),R_h\left(t_i\right)),\hfill \\ \hfill {\delta }_2=& I_{m1}(t_i,S_m\left(t_i\right),I_m\left(t_i\right),S_h\left(t_i\right),I_h\left(t_i\right),C_h\left(t_i\right),R_h\left(t_i\right)),\hfill \\ \hfill {\delta }_3=& S_{h1}(t_i,S_m\left(t_i\right),I_m\left(t_i\right),S_h\left(t_i\right),I_h\left(t_i\right),C_h\left(t_i\right),R_h\left(t_i\right)),\hfill \\ \hfill {\delta }_4=& I_{h1}(t_i,S_m\left(t_i\right),I_m\left(t_i\right),S_h\left(t_i\right),I_h\left(t_i\right),C_h\left(t_i\right),R_h\left(t_i\right)),\hfill \\ \hfill {\delta }_5=& C_{h1}(t_i,S_m\left(t_i\right),I_m\left(t_i\right),S_h\left(t_i\right),I_h\left(t_i\right),C_h\left(t_i\right),R_h\left(t_i\right)),\hfill \\ \hfill {\delta }_6=& R_{h1}(t_i,S_m\left(t_i\right),I_m\left(t_i\right),S_h\left(t_i\right),I_h\left(t_i\right),C_h\left(t_i\right),R_h\left(t_i\right)),\hfill \\ \hfill {\delta }_7=& S_{m1}(t_{i-1},S_m\left(t_{i-1}\right),I_m\left(t_{i-1}\right),S_h\left(t_{i-1}\right),I_h\left(t_{i-1}\right),C_h\left(t_{i-1}\right),R_h\left(t_{i-1}\right)),\hfill \\ \hfill {\delta }_8=& I_{m1}(t_{i-1},S_m\left(t_{i-1}\right),I_m\left(t_{i-1}\right),S_h\left(t_{i-1}\right),I_h\left(t_{i-1}\right),C_h\left(t_{i-1}\right),R_h\left(t_{i-1}\right)),\hfill \\ \hfill {\delta }_9=& S_{h1}(t_{i-1},S_m\left(t_{i-1}\right),I_m\left(t_{i-1}\right),S_h\left(t_{i-1}\right),I_h\left(t_{i-1}\right),C_h\left(t_{i-1}\right),R_h\left(t_{i-1}\right)),\hfill \\ \hfill {\delta }_{10}=& I_{h1}(t_{i-1},S_m\left(t_{i-1}\right),I_m\left(t_{i-1}\right),S_h\left(t_{i-1}\right),I_h\left(t_{i-1}\right),C_h\left(t_{i-1}\right),R_h\left(t_{i-1}\right)),\hfill \\ \hfill {\delta }_{11}=& C_{h1}(t_{i-1},S_m\left(t_{i-1}\right),I_m\left(t_{i-1}\right),S_h\left(t_{i-1}\right),I_h\left(t_{i-1}\right),C_h\left(t_{i-1}\right),R_h\left(t_{i-1}\right)),\hfill \\ \hfill {\delta }_{12}=& R_{h1}(t_{i-1},S_m\left(t_{i-1}\right),I_m\left(t_{i-1}\right),S_h\left(t_{i-1}\right),I_h\left(t_{i-1}\right),C_h\left(t_{i-1}\right),R_h\left(t_{i-1}\right)),\hfill \\ \hfill {\delta }_{13}=& S_{m1}(t_{i-2},S_m\left(t_{i-2}\right),I_m\left(t_{i-2}\right),S_h\left(t_{i-2}\right),I_h\left(t_{i-2}\right),C_h\left(t_{i-2}\right),R_h\left(t_{i-2}\right)),\hfill \\ \hfill {\delta }_{14}=& I_{m1}(t_{i-2},S_m\left(t_{i-2}\right),I_m\left(t_{i-2}\right),S_h\left(t_{i-2}\right),I_h\left(t_{i-2}\right),C_h\left(t_{i-2}\right),R_h\left(t_{i-2}\right)),\hfill \\ \hfill {\delta }_{15}=& S_{h1}(t_{i-2},S_m\left(t_{i-2}\right),I_m\left(t_{i-2}\right),S_h\left(t_{i-2}\right),I_h\left(t_{i-2}\right),C_h\left(t_{i-2}\right),R_h\left(t_{i-2}\right)),\hfill \\ \hfill {\delta }_{16}=& I_{h1}(t_{i-2},S_m\left(t_{i-2}\right),I_m\left(t_{i-2}\right),S_h\left(t_{i-2}\right),I_h\left(t_{i-2}\right),C_h\left(t_{i-2}\right),R_h\left(t_{i-2}\right)),\hfill \\ \hfill {\delta }_{17}=& C_{h1}(t_{i-2},S_m\left(t_{i-2}\right),I_m\left(t_{i-2}\right),S_h\left(t_{i-2}\right),I_h\left(t_{i-2}\right),C_h\left(t_{i-2}\right),R_h\left(t_{i-2}\right)),\hfill \\ \hfill {\delta }_{18}=& R_{h1}(t_{i-2},S_m\left(t_{i-2}\right),I_m\left(t_{i-2}\right),S_h\left(t_{i-2}\right),I_h\left(t_{i-2}\right),C_h\left(t_{i-2}\right),R_h\left(t_{i-2}\right)).\hfill \end{array}$$

7.4. Newton method for Atangana-Baleanu fractional derivative in Caupto sense

Here we take the following Cauchy problem.

$${}_0^{ABC}{D}_t^{\alpha }g\left(t\right)=f(t,g\left(t\right)),g\left(0\right)=g_0.$$ (7.39)

In this subsection, we provide a numerical scheme to solve the Eq. (7.39). Applying A-B integral, we convert the Eq. (7.39) into

$$g\left(t\right)-g\left(0\right)={\displaystyle \frac{1-\alpha }{AB\left(\alpha \right)}f(t,g\left(t\right))+\frac{\alpha }{AB\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_0^{t}f(\chi ,g\left(\chi \right)){(t-\chi )}^{\alpha -1}d\chi .}$$ (7.40)

At $t_{i+1}=(i+1)\Delta t,$ we have

$$g\left(t_{i+1}\right)-g\left(0\right)={\displaystyle \frac{1-\alpha }{AB\left(\alpha \right)}f(t_i,g\left(t_i\right))+\frac{\alpha }{AB\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_0^{t_{i+1}}f(\chi ,g\left(\chi \right)){(t_{i+1}-\chi )}^{\alpha -1}d\chi .}$$ (7.41)

Also, at $t_i=i\Delta t,$ we have

$$g\left(t_{i+1}\right)-g\left(0\right)={\displaystyle \frac{1-\alpha }{AB\left(\alpha \right)}f(t_i,g\left(t_{i-1}\right))+\frac{\alpha }{AB\left(\alpha \right)\Gamma \left(\alpha \right)}\sum _{n=2}^i{\int }_n^{t_{n+1}}f(\chi ,g\left(\chi \right))d\chi .}$$ (7.42)

Here, for the approx. of the function $f(t,g(t\left)\right)$ we apply the Newton poly. as

$$Q_i\left(\chi \right)={\displaystyle f(t_{i-2},g\left(t_{i-2}\right))+\frac{G_4}{\Delta t}(\chi -t_{i-2})+\frac{G_5}{2\Delta t^2}(\chi -t_{i-2})(\chi -t_{i-1}).}$$ (7.43)

Using Eq. (7.43) into (7.42) we get

$$g^{i+1}={\displaystyle g^0+\frac{1-\alpha }{AB\left(\alpha \right)}f(t_i,g\left(t_i\right))+\frac{\alpha }{AB\left(\alpha \right)\Gamma \left(\alpha \right)}\sum _{n=2}^i{\int }_{t_n}^{t_{n+1}}\{f(t_{n-2},g^{n-2})}+{\displaystyle (\frac{G_1}{\Delta t}+\frac{G_2}{2\Delta t^2}(\chi -t_{n-2}))(\chi -t_{n-1})\}{(t_{i+1}-\chi )}^{\alpha -1}d\chi .}$$ (7.44)

This implies,

$$g^{i+1}={\displaystyle g^0+\frac{1-\alpha }{M\left(\alpha \right)}f(t_i,g\left(t_i\right))+\frac{\alpha }{AB\left(\alpha \right)\Gamma \left(\alpha \right)}\sum _{n=2}^{i}f(t_{n-2},g^{n-2}){\int }_{t_n}^{t_{n+1}}{(t_{i+1}-\chi )}^{\alpha -1}d\chi }+{\displaystyle \frac{\alpha }{AB\left(\alpha \right)\Gamma \left(\alpha \right)}\sum _{n=2}^i\frac{G_1}{\Delta t}{\int }_{t_n}^{t_{n+1}}(\chi -t_{n-2}){(t_{i+1}-\chi )}^{\alpha -1}d\chi }+{\displaystyle \frac{\alpha }{AB\left(\alpha \right)\Gamma \left(\alpha \right)}\sum _{n=2}^i\frac{G_2}{2\Delta t^2}{\int }_{t_n}^{t_{n+1}}(\chi -t_{n-2})(\chi -t_{n-1}){(t_{i+1}-\chi )}^{\alpha -1}d\chi .}$$ (7.45)

Now the integral of Eq. (7.45) as

$$\begin{array}{cc}\hfill {\displaystyle {\int }_{t_n}^{t_{n+1}}}& (t_{n-1}-\chi )d\chi =\frac{\Delta t^{\alpha }}{\alpha }[{(i-n+1)}^{\alpha }-{(i-n)}^{\alpha }],\hfill \\ \hfill {\displaystyle {\int }_{t_n}^{t_{n+1}}}& (\chi -t_{n-2}){(t_{i+1}-\chi )}^{\alpha -1}d\chi =\frac{\Delta t^{\alpha +1}}{\alpha (\alpha +1)}\left[\right({(i-n+1)}^{\alpha }-{(i-n)}^{\alpha }\left)(i-n+3+3\alpha )\right],\hfill \\ \hfill {\displaystyle {\int }_{t_n}^{t_{n+1}}}& (\chi -t_{n-2})(\chi -t_{n-1}){(t_{n+1}-\chi )}^{\alpha -1}d\chi =\frac{\Delta t^{\alpha +2}}{\alpha (\alpha +1)(\alpha +2)}\left[{(i-n+1)}^{\alpha }\right[2{(i-n)}^2+(3\alpha +10)\hfill \\ \hfill \times (i-& n)+2{\alpha }^2+9\alpha +12]-{(i-n)}^{\alpha }[{(i-n+1)}^{\alpha }[2{(i-n)}^2+(5\alpha +10)(i-n)+6{\alpha }^2+18\alpha +12].\hfill \end{array}$$ (7.46)

We do the same process for Atangana-Baleanu fractional derivative for the Eq. (3.1) we have the following technique for numerical soltion of the Eq. (7.46)

$$S_m^{i+1}={\displaystyle S_m^0+\frac{1-\alpha }{AB\left(\alpha \right)}{\beta }_1+\frac{\alpha \Delta t^{\alpha }}{AB\left(\alpha \right)\Gamma (\alpha +1)}\sum _{n=2}^j{\beta }_{13}^2[{(i-n+1)}^{\alpha }(i-n+3+2\alpha )-{(i-n)}^{\alpha }}\times {\displaystyle (i-n+3+3\alpha )]+\frac{\alpha \Delta t^{\alpha }}{AB\left(\alpha \right)\Gamma (\alpha +2)}\sum _{n=2}^j[{\beta }_7-{\beta }_{13}\left]\right[{(i-n+1)}^{\alpha }(i-n+3+2\alpha )-{(i-n)}^{\alpha }}\times {\displaystyle (i-n+3+3\alpha )]+\frac{\alpha \Delta t^{\alpha }}{AB\left(\alpha \right)\Gamma (\alpha +3)}\sum _{n=2}^j[{\beta }_1-{\beta }_7-{\beta }_{13}\left]\right[{(i-n+1)}^{\alpha }[2{(i-n)}^2+(3\alpha +10)}\times (i-n)+2{\alpha }^2+9\alpha +12]-{(i-n)}^{\alpha }[2{(i-n)}^2+(5\alpha +10)(i-n)+6{\alpha }^2+18\alpha +12\left]\right]$$ (7.47)

$$I_m^{i+1}={\displaystyle I_m^0+\frac{1-\alpha }{AB\left(\alpha \right)}{\beta }_2+\frac{\alpha \Delta t^{\alpha }}{AB\left(\alpha \right)\Gamma (\alpha +1)}\sum _{n=2}^j{\beta }_{14}^2[{(i-n+1)}^{\alpha }(i-n+3+2\alpha )-{(i-n)}^{\alpha }}\times {\displaystyle (i-n+3+3\alpha )]+\frac{\alpha \Delta t^{\alpha }}{AB\left(\alpha \right)\Gamma (\alpha +2)}\sum _{n=2}^j[{\beta }_8-{\beta }_{14}\left]\right[{(i-n+1)}^{\alpha }(i-n+3+2\alpha )-{(i-n)}^{\alpha }}\times {\displaystyle (i-n+3+3\alpha )]+\frac{\alpha \Delta t^{\alpha }}{AB\left(\alpha \right)\Gamma (\alpha +3)}\sum _{n=2}^j[{\beta }_2-{\beta }_8-{\beta }_{14}\left]\right[{(i-n+1)}^{\alpha }[2{(i-n)}^2+(3\alpha +10)}\times (i-n)+2{\alpha }^2+9\alpha +12]-{(i-n)}^{\alpha }[2{(i-n)}^2+(5\alpha +10)(i-n)+6{\alpha }^2+18\alpha +12\left]\right]$$ (7.48)

$$S_h^{i+1}={\displaystyle S_h^0+\frac{1-\alpha }{AB\left(\alpha \right)}{\beta }_3+\frac{\alpha \Delta t^{\alpha }}{AB\left(\alpha \right)\Gamma (\alpha +1)}\sum _{n=2}^j{\beta }_{15}^2[{(i-n+1)}^{\alpha }(i-n+3+2\alpha )-{(i-n)}^{\alpha }}\times {\displaystyle (i-n+3+3\alpha )]+\frac{\alpha \Delta t^{\alpha }}{AB\left(\alpha \right)\Gamma (\alpha +2)}\sum _{n=2}^j[{\beta }_9-{\beta }_{15}\left]\right[{(i-n+1)}^{\alpha }(i-n+3+2\alpha )-{(i-n)}^{\alpha }}\times {\displaystyle (i-n+3+3\alpha )]+\frac{\alpha \Delta t^{\alpha }}{AB\left(\alpha \right)\Gamma (\alpha +3)}\sum _{n=2}^j[{\beta }_3-{\beta }_9-{\beta }_{15}\left]\right[{(i-n+1)}^{\alpha }[2{(i-n)}^2+(3\alpha +10)}\times (i-n)+2{\alpha }^2+9\alpha +12]-{(i-n)}^{\alpha }[2{(i-n)}^2+(5\alpha +10)(i-n)+6{\alpha }^2+18\alpha +12\left]\right]$$ (7.49)

$$I_h^{i+1}={\displaystyle I_h^0+\frac{1-\alpha }{AB\left(\alpha \right)}{\beta }_4+\frac{\alpha \Delta t^{\alpha }}{AB\left(\alpha \right)\Gamma (\alpha +1)}\sum _{n=2}^j{\beta }_{16}^2[{(i-n+1)}^{\alpha }(i-n+3+2\alpha )-{(i-n)}^{\alpha }}\times {\displaystyle (i-n+3+3\alpha )]+\frac{\alpha \Delta t^{\alpha }}{AB\left(\alpha \right)\Gamma (\alpha +2)}\sum _{n=2}^j[{\beta }_{10}-{\beta }_{16}\left]\right[{(i-n+1)}^{\alpha }(i-n+3+2\alpha )-{(i-n)}^{\alpha }}\times {\displaystyle (i-n+3+3\alpha )]+\frac{\alpha \Delta t^{\alpha }}{AB\left(\alpha \right)\Gamma (\alpha +3)}\sum _{n=2}^j[{\beta }_4-{\beta }_{10}-{\beta }_{16}\left]\right[{(i-n+1)}^{\alpha }[2{(i-n)}^2+(3\alpha +10)}\times (i-n)+2{\alpha }^2+9\alpha +12]-{(i-n)}^{\alpha }[2{(i-n)}^2+(5\alpha +10)(i-n)+6{\alpha }^2+18\alpha +12\left]\right]$$ (7.50)

$$C_h^{i+1}={\displaystyle C_h^0+\frac{1-\alpha }{AB\left(\alpha \right)}{\beta }_5+\frac{\alpha \Delta t^{\alpha }}{AB\left(\alpha \right)\Gamma (\alpha +1)}\sum _{n=2}^j{\beta }_{17}^2[{(i-n+1)}^{\alpha }(i-n+3+2\alpha )-{(i-n)}^{\alpha }}\times {\displaystyle (i-n+3+3\alpha )]+\frac{\alpha \Delta t^{\alpha }}{AB\left(\alpha \right)\Gamma (\alpha +2)}\sum _{n=2}^j[{\beta }_{11}-{\beta }_{17}\left]\right[{(i-n+1)}^{\alpha }(i-n+3+2\alpha )-{(i-n)}^{\alpha }}\times {\displaystyle (i-n+3+3\alpha )]+\frac{\alpha \Delta t^{\alpha }}{AB\left(\alpha \right)\Gamma (\alpha +3)}\sum _{n=2}^j[{\beta }_5-{\beta }_{11}-{\beta }_{17}\left]\right[{(i-n+1)}^{\alpha }[2{(i-n)}^2(3\alpha +10)}\times (i-n)+2{\alpha }^2+9\alpha +12]-{(i-n)}^{\alpha }[2{(i-n)}^2+(5\alpha +10)(i-n)+6{\alpha }^2+18\alpha +12\left]\right]$$ (7.51)

$$R_h^{i+1}={\displaystyle R_h^0+\frac{1-\alpha }{AB\left(\alpha \right)}{\beta }_6+\frac{\alpha \Delta t^{\alpha }}{AB\left(\alpha \right)\Gamma (\alpha +1)}\sum _{n=2}^j{\beta }_{18}^2[{(i-n+1)}^{\alpha }(i-n+3+2\alpha )-{(i-n)}^{\alpha }}\times {\displaystyle (i-n+3+3\alpha )]+\frac{\alpha \Delta t^{\alpha }}{AB\left(\alpha \right)\Gamma (\alpha +2)}\sum _{n=2}^j[{\beta }_{12}-{\beta }_{18}\left]\right[{(i-n+1)}^{\alpha }(i-n+3+2\alpha )-{(i-n)}^{\alpha }}\times {\displaystyle (i-n+3+3\alpha )]+\frac{\alpha \Delta t^{\alpha }}{AB\left(\alpha \right)\Gamma (\alpha +3)}\sum _{n=2}^j[{\beta }_6-{\beta }_{12}-{\beta }_{18}\left]\right[{(i-n+1)}^{\alpha }[2{(i-n)}^2+(3\alpha +10)}\times (i-n)+2{\alpha }^2+9\alpha +12]-{(i-n)}^{\alpha }[2{(i-n)}^2+(5\alpha +10)(i-n)+6{\alpha }^2+18\alpha +12\left]\right].$$ (7.52)

8. Results and discussion

In this section, we present results and discussion of co-infection mathematical model (3.3). We have used MATLAB software to compute all numerical computation. For the numerical purpose, we set all default parameters as: $d_m=0.25,d_h=0.0000457-0.25,b=0.5,{\gamma }_h=0.1428,{\eta }_m=0.85-1,{\eta }_h=0.75,Z_h=10000,{\varphi }_h=696147,{\Delta }_2=10,andp=1.0-1.5$. The dynamics of different population of host and vectors with different values of fractional-order $\alpha =0.97,0.96.0.95,0.93$ and different rate of transmission is studied. We have seen that the fractional modeling of biological systems can be advantage and applicable as memory effects are finding in those systems. Although $\alpha$ is not a biological parameter yet, it plays a very significant role in the dynamics of the model (3.3). In Fig. 3, Fig. 4, Fig. 5 , it is observed that the

Fig. 3.

Variations of the total population $I_m\left(t\right),S_h\left(t\right),I_h\left(t\right)\&C_h\left(t\right)$ w.r.t. time for $\alpha =0.97$.

Fig. 4.

Variations of the total population $I_m\left(t\right),S_h\left(t\right),I_h\left(t\right)\&C_h\left(t\right)$ w.r.t. time for $\alpha =0.96$.

Fig. 5.

Variations of the total population $I_m\left(t\right),S_h\left(t\right),I_h\left(t\right)\&C_h\left(t\right)$ w.r.t. time for $\alpha =0.95$.

total population increases beginning up to time $t=10$(Days) and it decreases afterward. It is also clear that the SARS-COV-2 virus takes at least 10-15 days to show symptoms and attain a peak and afterward, it declines to normal but is still prevalent in the system.

Next, the variations of the variables $I_m\left(t\right),S_h\left(t\right),\&I_h\left(t\right)$ are demonstrated in Figs. 6 -8 with $\alpha =0.97,0.96,0.95$. In each figure, we have considered different population variables against time. From these three figures, it is learned that if we take the covid-19 infection class is equal to zero, the fractional-order models approach the fixed point over a large period. Figs. 9 -10 exhibited the variation of the variables $I_m\left(t\right),I_h\left(t\right)\&C_h\left(t\right)$ for time $t$ and it is seen that as we increase the value of the biting rate of mosquitoes $b$ the line of the graph is slightly increasing. It means that we decrease the value of bitting rate $b$ mitigates the rate of infection which is possible in real life too. Hence, the infected mosquito class is fully dependent on the biting rate. Further, from Figs. 11 -12 , one thing we have seen as we increase the value of the fraction rate $p$ of covid-19 patients the graphs of the covid-19 class $C_h\left(t\right)$ increases.

Fig. 6.

Variations of the total population $I_m\left(t\right),S_h\left(t\right)\&I_h\left(t\right)$ w.r.t. time for $\alpha =0.97.$

Fig. 8.

Variations of the total population $I_m\left(t\right),S_h\left(t\right)\&I_h\left(t\right)$ w.r.t. time for $\alpha =0.95.$

Fig. 9.

Variations of the total population $I_m\left(t\right),I_h\left(t\right)\&C_h\left(t\right)$ w.r.t. time for $\alpha =0.97$.

Fig. 10.

Variations of the total population $I_m\left(t\right),I_h\left(t\right)\&C_h\left(t\right)$ w.r.t. time for $\alpha =0.96$.

Fig. 11.

Effect of biological parameter $p=1.14$ on $C_h\left(t\right)$ w.r.t. time.

Fig. 12.

Effect of biological parameter $p=1.15$ on $C_h\left(t\right)$ w.r.t. time.

9. Conclusion

Since dengue and covid-19 are very strange infectious diseases that cause havoc all over the world, their mode of transmission is not yet fully understood. In this paper, we investigate the dynamics of dengue and covid-19 co-infection. We computed the reproduction number and established a stability analysis based on some important theorems. It is observed that for the fractional values of $\alpha =0.96$, the curve of co-infection of dengue and covid-19 gets flattened well. It means that if we can control the values of fractional order $\alpha$ within suitable intervals (0,1), the curve of the dengue and covid-19 co-infection reducible to a certain control level as depicted in Fig. 7. The effects of fractional-order $\alpha$ and other transmission rates are also depicted graphically in figures. Our results are in good agreement with results which show that dengue fever acts as a launch pad for the SARS-COV-2 virus and causes death. We compared these different models in the sense of Caputo, Caputo-Fabrizio, and Atangana-Baleanue. It is inferred in the numerical simulation that Caputo shows still better results in the form of stability as compared to the other operators.

Fig. 7.

Variations of the total population $I_m\left(t\right),S_h\left(t\right)\&I_h\left(t\right)$ w.r.t. time for $\alpha =0.96$.

CRediT authorship contribution statement

Attiq ul Rehman: Conceptualization, Methodology, Writing - original draft, Software. Ram Singh: Conceptualization, Methodology, Formal analysis, Software, Supervision. Praveen Agarwal: Methodology, Formal analysis.

Declaration of Competing Interest

The author(s) declare(s) that there has been no conflict of interest.

Appendix A

Let $R:E\to E$ is a function defined by

$$\left(R\phi \right)\left(t\right)={\displaystyle {\phi }_0+\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t{(t-\chi )}^{\alpha -1}K(\chi ,\phi \left(\chi \right))d\chi .}$$ (11.1)

Since, the function $R$ is well defined and the unique of the model (3.3) is just the fixed point of the function. Now, let us take,

$${\displaystyle \underset{t\in L}{\sup }\parallel K(t,0)\parallel =\mathcal{M},\&k\ge \parallel {\phi }_0\parallel +\aleph \mathcal{M}.}$$

Therefore, it is enough to show that $R{\mathcal{B}}_k\subset {\mathcal{B}}_k,$ where ${\mathcal{B}}_k=\{\phi \in E:\parallel \phi \parallel \le k\}$ is convex and closed set. Now, for any $\phi$ in ${\mathcal{B}}_k$, it gives

$$\begin{array}{cc}\hfill \left|\right(R\phi \left)\right(t\left)\right|\le & |{\phi }_0|+\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t{(t-\chi )}^{\alpha -1}\left|K(\chi ,\phi \left(\chi \right))\right|d\chi \hfill \\ \hfill \le & {\phi }_0+\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t{(t-\chi )}^{\alpha -1}\left[\right|K(\chi ,\phi \left(\chi \right))-K(\chi ,0)|+|K(\chi ,0)\left|\right]d\chi \hfill \\ \hfill \le & {\phi }_0+\frac{{\mathcal{F}}_{K}k+\mathcal{M}}{\Gamma \left(\alpha \right)}{\int }_0^t{(t-\chi )}^{\alpha -1}d\chi \hfill \\ \hfill \le & {\phi }_0+\frac{{\mathcal{F}}_{K}k+\mathcal{M}}{\Gamma \left(\alpha \right)}{c}^{\alpha }\hfill \\ \hfill \le & {\phi }_0+\aleph ({\mathcal{F}}_{K}k+\mathcal{M})\hfill \\ \hfill \le & k.\hfill \end{array}$$ (11.2)

Thus, the target is followed. Also, for any ${\phi }_1,{\phi }_2\in E$, we have

$$\begin{array}{cc}\hfill |\left(R{\phi }_1\right)\left(t\right)-\left(R{\phi }_2\right)\left(t\right)|\le & \frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t{(t-\chi )}^{\alpha -1}|K(\chi ,{\phi }_1\left(\chi \right))-K(\chi ,{\phi }_2\left(\chi \right))|d\chi \hfill \\ \hfill \le & \frac{{\mathcal{F}}_K}{\Gamma \left(\alpha \right)}{\int }_0^t{(t-\chi )}^{\alpha -1}|{\phi }_1\left(\chi \right)-{\phi }_2\left(\chi \right)|d\chi \hfill \\ \hfill \le & \aleph {\mathcal{F}}_K|{\phi }_1\left(t\right)-{\phi }_2\left(t\right)|.\hfill \end{array}$$ (11.3)

This implies that $\parallel \left(R{\phi }_1\right)\left(t\right)-\left(R{\phi }_2\right)\left(t\right)\parallel \le \aleph {\mathcal{F}}_K|{\phi }_1-{\phi }_2|.$ Hence, by the consquence of the principle of Banach contraction. The model (3.3) has a unique solution.

Appendix B

Since, the continuity of $K\Rightarrow R.$ So, for any $\phi$ in ${\mathcal{B}}_k$, we have

$$\begin{array}{cc}\hfill \left|\right(R\phi \left)\right(t\left)\right|\le & |{\phi }_0+\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t{(t-\chi )}^{\alpha -1}K(\chi ,\phi \left(\chi \right))d\chi |\hfill \\ \hfill \le & \parallel {\phi }_0\parallel +\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t{(t-\chi )}^{\alpha -1}K(\chi ,\phi \left(\chi \right))d\chi \hfill \\ \hfill \le & \parallel {\phi }_0\parallel +\frac{a_1^{*}+a_2^{*}\parallel \phi \parallel }{\Gamma \left(\alpha \right)}{\int }_0^t{(t-\chi )}^{\alpha -1}d\chi \hfill \\ \hfill \le & \parallel {\phi }_0\parallel +\frac{a_1^{*}+a_2^{*}\parallel \phi \parallel }{\Gamma \left(\alpha \right)}{c}^{\alpha }\hfill \\ \hfill \le & \parallel {\phi }_0\parallel +\aleph (a_1^{*}+a_2^{*}\parallel \phi \parallel )\hfill \\ \hfill \le & \infty .\hfill \end{array}$$ (11.4)

Therefore, $R$ is uniformaly bounded. Next,we will prove that $R$ is equicountinuity. For this, we suppose ${\displaystyle \underset{(t,\phi )\in L\times {\mathcal{A}}_k}{\sup }\left|K(t,\phi \left(t\right))\right|=K^{*}.}$ Then, for any $t_1,t_2$ in $L$ s.t. $t_2\ge t_1$, we have

$$\begin{array}{ccc}& & |\left(R\phi \right)\left(t_2\right)-\left(R\phi \right)\left(t_1\right)|\le \frac{1}{\Gamma \left(\alpha \right)}\left|{\int }_0^{t_1}\right[{(t_2-\chi )}^{\alpha -1}-{(t_1-\chi )}^{\alpha -1}+{(t_2-\chi )}^{\alpha -1}]\hfill \\ & & K(\chi ,\phi (\chi \left)\right)d\chi |\hfill \\ & & \le \frac{K^{*}}{\Gamma \left(\alpha \right)}[2{(t_2-t_1)}^{\alpha }+(t_2^{\alpha }-t_1^{\alpha })]\hfill \end{array}$$ (11.5)

This implies, $|\left(R\phi \right)\left(t_2\right)-\left(R\phi \right)\left(t_1\right)|\to 0,as{t}_2\to t_1$.

Hence, $R$ is equicontinuous and so ${\mathcal{A}}_k$ is relatively compact. Therefore, $R$ is completely continuous by the consequence of Arzela-Ascoli theorem.

Appendix C

First of all, we define a set $\mathcal{J}=\{\phi \in E:\phi =\mathcal{U}(R\phi \left)\right(t),\mathcal{U}\in (0,1\left)\right\}.$ Clearly, by the corollary 4.2, the function $R:\mathcal{J}\to E$ as defined in Eq. (4.3) is completely continuous. Now, for any $\phi$ in $\mathcal{J}$ and corollary 4.2, it gives

$$\begin{array}{cc}\hfill \left|\right(\phi \left)\right(t\left)\right|=& \left|\mathcal{U}\right(R\phi \left)\right(t\left)\right|\hfill \\ \hfill \le & |{\phi }_0|+\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t{(t-\chi )}^{\alpha -1}K(\chi ,\phi \left(\chi \right))d\chi \hfill \\ \hfill \le & \parallel {\phi }_0\parallel +\frac{a_1^{*}+a_2^{*}\parallel \phi \parallel }{\Gamma (\alpha +1)}{c}^{\alpha }\hfill \\ \hfill \le & \parallel {\phi }_0\parallel +\aleph (a_1^{*}+a_2^{*}\parallel \phi \parallel )\hfill \\ \hfill \le & \infty .\hfill \end{array}$$ (11.6)

Therefore, $\mathcal{J}$ is bounded. So, $R$ has at least one fixed point which is only the solution of the model (3.3). Hence the result is obtained.

Appendix D

On adding the first two Eq. of the co-infection model (3.3), the total mosquitoes population, is given by

$${\displaystyle {}_0^C{D}_t^{\alpha }{Z}_h\left(t\right)=}{\Delta }_1-d_m{Z}_h\left(t\right).$$ (11.7)

The solution of the model (3.3) is given by

$$Z_h\left(t\right)=Z_h\left(0\right)E_{\alpha ,1}(-d_h,t^{\alpha })+{\Delta }_1{t}^{\alpha }{E}_{\alpha ,\alpha +1}(-d_h{t}^{\alpha }),$$

where $E_{\alpha ,\beta }$ is known as the Mittag-Leffler function. On concerning [15] the point that the behaviour of this function is an asymptotic, so we’ve

$$E_{\alpha ,\beta }\left(g\right){\displaystyle \sim -\sum _{j=1}^{\mu }\frac{g^{-j}}{\Gamma (\beta -\alpha j)}+{O\left(\right|j|}^{-1-\mu }),(\left|g\right|\to \infty ,\left|arg\left(g\right)\right|\in (\alpha \pi /2,\pi ]).}$$ (11.8)

Therefore, from the Eq. (11.7) we’ve

$$Z_h\left(t\right)\to {\Delta }_1/d_{m}ast\to \infty .$$

Further, the proof in the case of the human population is completely similar to the vector population and hence that’s why it is omitted. Thus, for all positive values of time, all the solutions of the Caputo fractional derivative the model (3.3) with initial conditions of $\Pi$ remain in $\Pi$. Therefore, the region $\Pi$ is + vely invariant fr the model (3.3) and attracts all solutions in ${\mathbb{R}}_{+}^6.$

Appendix E

In order to explore the non-negativity solution, it is require to show that on every hyperplane bounding of ${\mathbb{R}}_{+}^6$. From the model (3.3), we have:

$$\begin{array}{cc}\hfill {}_0^C{D}_t^{\alpha }{S}_m(at{S}_m=0)=& {\Delta }_1\ge 0,\hfill \\ \hfill {}_0^C{D}_t^{\alpha }{I}_m(at{I}_m=0)=& {\displaystyle \frac{b{\eta }_m{\Delta }_1}{Z_h{d}_m}{I}_h\ge 0,}\hfill \\ \hfill {}_0^C{D}_t^{\alpha }{S}_h(at{S}_h=0)=& {\Delta }_2\ge 0,\hfill \\ \hfill {}_0^C{D}_t^{\alpha }{I}_h(at{I}_h=0)=& {\displaystyle \frac{b{\eta }_h{\Delta }_2}{Z_h{d}_h}{I}_m\ge 0,}\hfill \\ \hfill {}_0^C{D}_t^{\alpha }{C}_h(at{C}_h=0)=& 0\ge 0,\hfill \\ \hfill {}_0^C{D}_t^{\alpha }{R}_h(at{R}_h=0)=& {\gamma }_h{I}_h+(1-p)I_h{C}_h\ge 0.\hfill \end{array}$$

Thus, by the corollary 4.6, the above target set has been achieved that the solution will stay in ${\mathbb{R}}_{+}^6$ and thus we have the following biologically feasible region:

$$\begin{array}{ccc}\hfill \Pi & =& \left\{\right(S_m\left(0\right),I_m\left(0\right),S_h\left(0\right),I_h\left(0\right),C_h\left(0\right),R_h\left(0\right))\hfill \\ & & \in {\mathbb{R}}_{+}^6:S_m\left(0\right),I_m\left(0\right),S_h\left(0\right),I_h\left(0\right),C_h\left(0\right),R_h\left(0\right)\ge 0\}.\hfill \end{array}$$

Therefore all the terms of the sum are positive, then the solution of the model (3.3) is bounded.

Appendix F

The disease-free equilibrium is locally asymptotically stable if all the characterstic values ${\lambda }_i$, where $i=1,2,3,4,5,6$ of the Jacobian matrix $J_{E_0}$ is satisfying the condition [6]:

$$\left|arg\right({\lambda }_i\left)\right|>\alpha \pi /2.$$ (11.9)

The characterstic values of the Jacobian matrix $J_{E_0}$ are ${\lambda }_1=-d_m,{\lambda }_2=-d_h,{\lambda }_3=-(d_h+{\varphi }_h),\&{\lambda }_4=-d_h,$ and the other two roots are find out from the quadratic equation

$${\lambda }^2+\lambda (d_m+d_h+{\gamma }_h)+d_m(d_h+{\gamma }_h)(1-{\mathcal{R}}_0)=0,$$ (11.10)

where

$${\displaystyle {\mathcal{R}}_0=\frac{b^2{\eta }_m{\eta }_h{\Delta }_1{\Delta }_2}{Z_h^2{d}_m^2{d}_h(d_h+{\gamma }_h)}.}$$

and is called a basic reproduction number. Hence $E_0$ is locally asymptotically stable if ${\mathcal{R}}_0$ is less than 1 and is unstable if ${\mathcal{R}}_0$ is greater than 1.

Appendix G

Let us consider the following Lyapnuov function:

$$V\left(t\right)=a_1{I}_m\left(t\right)+a_2{I}_h\left(t\right),$$

where, ${\displaystyle a_1=\frac{1}{d_m}and{a}_2=\frac{1}{d_h+{\gamma }_h}.}$ Therefore, the Lyapunov function $V$ is well defined, positive definite and continuous for all $I_m\left(t\right)\&I_h\left(t\right)>0.$ By the Eq. (2.1), we have

$${}_0^C{D}_t^{\alpha }V{=}_0^C{D}_t^{\alpha }{I}_m{+}_0^C{D}_t^{\alpha }{I}_h.$$

From the Caupto fractional order dengue fever and covid-19 model (3.3), we have

$${\displaystyle {}_0^C{D}_t^{\alpha }V=a_1(\frac{b{\eta }_m}{Z_h}{S}_m{I}_h-d_m{I}_m)+a_2(\frac{b{\eta }_h}{Z_h}{S}_h{I}_m-(d_h+{\gamma }_h)I_h).}$$

Therefore,

$${\displaystyle \frac{1}{d_m}>0\&\frac{1}{d_h+{\gamma }_h}>0.}$$

So, we have

$${}_0^C{D}_t^{\alpha }V={\displaystyle \frac{1}{d_m}(\frac{b{\eta }_m}{Z_h}{S}_m{I}_h-d_m{I}_m)+\frac{1}{d_h+{\gamma }_h}(\frac{b{\eta }_h}{Z_h}{S}_h{I}_m-(d_h+{\gamma }_h)I_h)}={\displaystyle \frac{b{\eta }_m}{d_m{Z}_h}{I}_h{S}_m-I_m+\frac{b{\eta }_h}{(d_h+{\gamma }_h)Z_h}{I}_m{S}_h-I_h.}$$

Since $S_m\le S_m^3\&S_h\le S_h^3,$

$${}_0^C{D}_t^{\alpha }V\le {\displaystyle \frac{b{\eta }_m}{Z_h}{I}_h{S}_m^3-I_m+\frac{b{\eta }_h}{Z_h}{I}_m{S}_h^3-I_h.}$$

As, ${\displaystyle I_h\le \frac{b{\eta }_h}{Z_h(d_h+{\gamma }_h)}{S}_h^3{I}_m\&I_m\le \frac{b{\eta }_m}{Z_h{d}_m}{S}_m^3{I}_h}$, we have

$$\begin{array}{cc}\hfill {}_0^C{D}_t^{\alpha }V\le & {\displaystyle \frac{b{\eta }_m}{Z_h}\times \frac{b{\eta }_h}{Z_h(d_h+{\gamma }_h)}{S}_m^3{S}_h^3{I}_m-I_m+\frac{b{\eta }_h}{Z_h}\times \frac{b{\eta }_m}{Z_h{d}_m}{S}_h^3{S}_m^3{I}_h-I_h}\hfill \\ \hfill =& {\displaystyle (\frac{b^2{\eta }_m{\eta }_h{\Delta }_1{\Delta }_2}{Z_h^2{d}_m^2{d}_h(d_h+{\gamma }_h)}-1)I_m+(\frac{b^2{\eta }_m{\eta }_h{\Delta }_1{\Delta }_2}{Z_h^2{d}_m^2{d}_h(d_h+{\gamma }_h)}-1)I_h}\hfill \\ \hfill =& ({\mathcal{R}}_0-1)I_m+({\mathcal{R}}_0-1)I_h\hfill \\ \hfill \le & 0for{\mathcal{R}}_0<1.\hfill \end{array}$$

Since all the biological parameters of the model (3.3) are nonnegative, it follows that ${}_0^C{D}_t^{\alpha }V\le 0for{\mathcal{R}}_0<1$ with ${}_0^C{D}_t^{\alpha }V=0$ iff $I_m=I_h=0.$ On substituting the value of $I_m\&I_h$ equal to zero into the model (3.3), we have ${\displaystyle S_m\to S_m^3=\frac{{\Delta }_1}{d_m}}$ as $t\to \infty$ and ${\displaystyle S_h\to S_h^3=\frac{{\Delta }_2}{d_h}}$ as $t\to \infty$.