The present study on modeling epidemics like infectious diseases is an interesting topic in the fields of mathematical biology. The medical world is still struggling a lot to provide medicines that can cure deadly diseases and prevent recurring infections. For example, the Human Immunodeficiency Virus (HIV) disease already found is still being treated instead of being cured. The epidemic models are the pioneers of all mathematical models in studying the growth of diseases. Epidemics are the root of various diseases. The existing models in epidemiology are close-ended where no new birth or death occurs. Basically, a patient can die with or without the symptoms of a disease. So death is not only a parameter to make changes in susceptible, and infected. It is also subject to change depending on immunity, the dosage of medicine, age, etc. So, we are interested to propose a new model in epidemiology called Susceptible-Infected-Recovered-Dead (SIRD). In this paper, such a model is suggested and we convert it into a fuzzy fractional model. These models are studied with three methods of the same order to find the best out of them and the used methods are all fourth-order in Laplace Adomian decomposition method (LADM), differential transform method (DTM), and fourth-order Runge-Kutta method (RKM-4). In 1965 Zadeh introduced the fuzzy sets [1]. Bukley et al. [2] proposed the fuzzy differential equations in the year 2000. Abbasbandy [3] has extended Newton’s method for a system of nonlinear equations by modified the ADM in 2005. Allen [4] in 2007 studied an introduction to mathematical biology. In the same year, Makinde [5] suggested a SIR epidemic model with constant vaccination within ADM. Ongun [6] has applied the Laplace Adomian decomposition method for solving a model for HIV infection of CD4+T cells in 2011. Arafa et al. [7] studied the solutions of the fractional order model of Childhood disease with constant vaccination strategy in 2012. Farman et al. [8] analyzed and numerically found the solution of SEIR epidemic model of measles with non-integer time- fractional derivatives by using LADM in 2018. Moustafa et al. [9] and Palese et al. [10] mathematically solved the influenza type problems. Recently, many authors showed interest to study fractional-order mathematical models in HIV, Tuberculosis (TB), and cancer [11–13]. After the classical epidemic model [14], many research papers arrived at the advanced epidemic models. Authors of manuscripts [15–19] have regularly treated fuzzy differential equations numerically. Numerical solutions of epidemic models are studied by many authors [20–25]. Recently, many researchers discussed the coronavirus, a pandemic disease [26–30].

This manuscript consists of 8 sections. In Section 2, we are forming a new epidemic model. In Section 3, we are discussing the basic definitions and describing the parameters involved in solving this new model. In Section 4, we are analyzing the equilibrium and stability of the presented model. In Section 5, we are analytically finding the solution of the model by fourth-order DTM and fourth-order LADM. In Section 6, we are using the RKM-4 to find the numerical solution of the model. Section 7 presents numerical simulations of our study. In Section 8, the study of the whole paper was concluded.

Let us consider a fuzzy fractional epidemic model-SIRD (See, Fig. 1) under Caputo derivative cΔα , where ‘S’ denotes Susceptible, ‘I’ denotes the Infected, ‘R’ denotes the recovered hosts, and ‘D’ denotes death hosts. The model is considered to be close-ended i.e., no new birth or death can occur. So, we shall assume the following information as the background of the model. Let us consider an unknown disease spread in a place where it holds few susceptible, infected, recovered, and dead cases in the overall population. Also, at the stage of observation, it is noted that few people died out because of the severity of the disease, due to lack of immunity, or due to lack of medication. We also assert that no new people further died or were further assumed to be susceptible. The model is also framed to improve on the classical epidemic model SIR framed by Kermack-Mckendrick [14]. The classical epidemic model does not change for susceptible, infected, and dead cases (SID). For any common flu or severe disease, it is quite natural that few people can recover, few people can die and few recovered people may become infected. Recovered is also a part of susceptibility. Both recovered and dead people are free from infection, but the recovered people may become infected again, or even die naturally, whereas the dead people would not become infected, susceptible or recovered. With these ideas in mind the following model is assumed:

(1) ΔS(t)=−βS(t)I(t)−γS(t)ΔI(t)=βS(t)I(t)−(δ+γ)I(t)ΔR(t)=(2δ−ψ)I(t)−γR(t)ΔD(t)=(ψ−δ)I(t)−γD(t)

Throughout the paper, D represents the death population and Δ represents d/dt.

This section consists of some basic results in fractional differential equation in terms of fuzzy.

Definition 3.1: The fuzzy Caputo fractional order derivative of a function f on the interval [0,t] is defined as cΔαf~(t)=1Γ(n−α)∫0t⁡(t−s)n−α−1f~n(s)ds , where n=[α]+1 and α represents the integer part of α .

Definition 3.2: The Laplace transform of fuzzy Caputo fractional derivative is given to be L{cΔαf~(t)}=kαg(k)−∑s=0n−1⁡kα−i−1f~(s)(0), where n−1<α≤n,n∈N . For arbitrary Ci∈R,i=0,1,2,…,n−1. where n=[α]+1 and [α] represents the integer part of α.

The system of differential equations (1) is rewritten as the fractional system of differential equations (FDE) which is given as

(2) cΔα1S(t)=−βS(t)I(t)−γS(t)cΔα2I(t)=βS(t)I(t)−(δ+γ)I(t)cΔα3R(t)=(2δ−ψ)I(t)−γR(t)cΔα4D(t)=(ψ−δ)I(t)−γD(t)

It becomes a fuzzy fractional-order system of differential equation using the ideas given in preliminaries

(3) cΔα1S~(t)=−βS~(t)I~(t)−γS~(t)cΔα2I~(t)=βS~(t)I~(t)−(δ+γ)I~(t)cΔα3R~(t)=(2δ−ψ)I~(t)−γR~(t)cΔα4D~(t)=(ψ−δ)I~(t)−γD~(t)

where f~(t)=(0.75+0.25r,1.125−0.125r)f(t) is the fuzzy number with r∈[0,1] .

The initial conditions are satisfied implying that the total population is constant with the size N. S(0)+I(0)+R(0)+D(0)=N . Here we shall take the initial populations as

The parameters are defined as follows:

β→ The rate at which the susceptible become infected = 0.0006

δ→ The rate at which the infectious become recovered = 0.03

ψ→ The rate at which the infectious become dead because of disease = 0.02

γ→ The rate at which natural death occurs in each population = 0.01

Calculation of Basic Reproduction Number (R₀):

Basic reproduction number or ratio is defined as the number of secondary infections produced by an infected single individual during the total epidemic period. This number is calculated from the rate of change in the population of infected people when the time t=0 . We found that, R0=S0βδ+γ . Suppose the critical value Sc=δ+γβ then if S0<Sc the disease will not survive and if {S_c}$]]>S0>Sc then there is an epidemic. Also, if 1$]]>R0>1 , the disease will spread and for R0<1, the disease will die out.

Equilibrium Points:

In Eq. (3), we take cΔα1S~(t)=0,cΔα2I~(t)=0,cΔα3R~(t)=0&cΔα4D~(t)=0 . i.e.,

(4) −βS~(t)I~(t)−γS~(t)=0

(5) βS~(t)I~(t)−(δ+γ)I~(t)=0

(6) (2δ−ψ)I~(t)−γR~(t)=0

(7) (ψ−δ)I~(t)−γD~(t)=0

From Eqs. (4)–(7) we get the disease-free equilibrium point as (0,0,0,0) and disease dependant equilibrium point as (δ+γβ,γβ,ψ−2δβ,δ−ψβ) . i.e., (66.667,−16.667,−66.667,16.667) is the disease dependant equilibrium point.

Theorem 1:

When all the real values of complex or non-complex eigenvalues of the linearized form of (3)<0 then the model is asymptotically stable.

Proof:

Let us first linearize the model (3) in the form of Jacobian matrix and name it as E .

(8) E=(−(βI(t)+γ)−βS(t)00βI(t)βS(t)−(δ+γ)0002δ−ψ−γ00ψ−δ0−γ)

After substituting all values of all parameters we get, the characteristic polynomial of a matrix is t4+0.053t3+0.000825t2−5.3×10−6t−9.25×10−8 . Equating the above polynomial to zero we have found the eigenvalues as (−0.0265+0.0163631i),(−0.0265−0.0163631i),−0.01,−0.01 . Since eigenvalues have the negative real parts we can conclude the system (3) is asymptotically stable.

In this section, we are using RKM-4 with α1=α2=α3=α4=1 . We are finding the values of S~(t) , I~(t) , R~(t) and D~(t) at h=0.1 for the best approximation. For 0≤r≤1 ,

We evaluate S~(t) , I~(t) , R~(t) and D~(t) :

K~1=h(−β(S~(t))(I~(t))−γ(S~(t)))L~1=h(β(S~(t))(I~(t))−(δ+γ)(I~(t)))M~1=h((2δ−ψ)(I~(t))−(γ)R~(t))N~1=h((ψ−δ)I(t))−((γ)D~(t))

K~2=h(−β(S~(t))+(K~12)(I~(t))+(L~12)−(γ(S~(t)+(K~12))))L~2=h(β(S~(t))+(K~12)(I~(t))+(L~12)−((δ+γ)(I~(t)+(L~12))))M~2=h((2δ−ψ)(I~(t)+(L~12))−(γ(R~(t)+(M~12))))N~2=h((ψ−δ)(I~(t)+(L~12))−(γ(D~(t)+(N~12))))

(13) K~3=h(−β(S~(t))+(K~22)(I~(t))+(L~22)−(γ(S~(t)+(K~22))))L~3=h(β(S~(t))+(K~22)(I~(t))+(L~22)−((δ+γ)(I~(t)+(L~22))))M~3=h((2δ−ψ)(I~(t)+(L~22))−(γ(R~(t)+(M~22))))N~3=h((ψ−δ)(I~(t)+(L~22))−(γ(D~(t)+(N~22))))

K~4=h(−β(S~(t))+(K~3)(I~(t))+(L~3)−(γ(S~(t)+(K~3))))L~4=h(β(S~(t))+(K~3)(I~(t))+(L~3)−((δ+γ)(I~(t)+(L~3))))M~4=h((2δ−ψ)(I~(t)+(L~3))−(γ(R~(t)+(M~3))))N~4=h((ψ−δ)(I~(t)+(L~3))+−(γ(D~(t)+(N~3))))

For1≤p≤4and0≤r≤1

K~p=K~p(t;r)=[K_p(t;r),K¯p(t;r)],L~p=L~p(t;r)=[L_p(t;r),L¯p(t;r)]

M~p=M~p(t;r)=[M_p(t;r),M¯p(t;r)],N~p=N~p(t;r)=[N_p(t;r),N¯p(t;r)]

For 0≤t≤n,n=1,2,3,…and for q_=t,q¯=t+1, t=0,1,2,3,…

Here,[f_(t;r),f¯(t;r)]=[0.75+0.25r,1.125−0.125r]f(t)with

(14) S~(t+1)=(S~(t)+16(K~1+2K~2+2K~3+K~4))I~(t+1)=(I~(t)+16(L~1+2L~2+2L~3+L~4))R~(t+1)=(R~(t)+16(M~1+2M~2+2M~3+M~4))D~(t+1)=(D~(t)+16(N~1+2N~2+2N~3+N~4))

The relationship between S~(t) , I~(t) , R~(t) and D~(t) at r=1,αi=1,i=1,2,3,4 for t∈[0,1000] for the fuzzy model Eq. (3) is given in Fig. 2.

In Tabs. 1–4, non-fuzzy, non-fractional valued susceptible infected, recovered and dead populations have been provided. In Tabs. 5–8, fractional valued susceptible infected, recovered, and dead populations have been provided and their respective plots have been given in Figs. 3–6. In Tab. 9 as a sample, fuzzy fractional SIRD populations are provided for t=0.5,α∈[0,1],r∈[0,1]. But in Figs. 7–10, the plots of fuzzy fractional SIRD populations for t∈[0,1],α∈[0,1],r∈[0,1] are provided.

The stability of the epidemic model SIRD was confirmed by studying the nature of eigenvalues. Since the system is nonlinear, we applied and compared three different methods LADM-4, DTM-4, and RKM-4. Fuzzy valued Sα1,Iα2,Rα3 and Dα4 at t∈[0,1] and at r∈[0,1] are shown in the tables and figures. The solutions obtained by LADM-4, DTM-4, and RKM-4 were compared and it was noticed that all the above three methods are equally good in accuracy. Also, we have to keep in mind that all these three methods are very direct since there is no linearization made. The table values were presented only by fixing α1=α2=α3=α4 . The graphical representations were plotted by considering all the values of αi,i=1,2,3,4 irrespective of whether they are equal or not. As future works, we would like to frame the model of COVID-19 by using the fuzzy fractional differential equations and their solutions will be found by using any of the above three equally good accuracy methods.