Characterization of solitons in a pseudo-quasi-conformally flat and pseudo- $W_8$ flat Lorentzian Kähler space-time manifolds

1.   Introduction

The life we are leading today in a civilized society is created in such a way that the living space of each and every species is occupied by our human race. Being concentrated on this well-sophisticated life or at least a better life, people are working all the time without rest to fulfill the needs of the family like bringing education to children, after that, their employment, etc., This is the normal family life of a middle-class people who are especially from India. Due to the commitment all the time for the family well being, there is a lack of care for their own personal health. This is one of the reasons behind losing good health. The cultivation lands now became corporate land. So good health is not easy to expect by means of properly farmed organic foods. Though medically many new innovative inventions are progressing every day there is also a vacuum in the answer when there is a question about non-eradicated diseases. Many new diseases are slowly occupying us and we are all victims to any kind of such transferring diseases. Even not having symptoms every one of us is a victim and a host of such ailments. We must prepare ourselves to be strong to fight against daily emerging new diseases. Till yesterday, the whole world was busy with the hot topic called COVID-19. But today the trend has changed to Omicron. Omicron is nothing but the new variant of the coronavirus.

In the literature of biomathematics modeling, population dynamics and the spread of epidemics are the most prominent topics since they have a wide historical background. In this paper, we study an epidemic model which is the pioneer of all studies about the transmission of ailments. The old models in epidemiology science are bound with the assumptions that new cases in the count of birth, death is not all allowed. And one more interesting concept in the epidemic case study is that they will not take death as a parameter subject to change as susceptible or infected. Recently many new mathematicians started to break that rule. We people are a step ahead in making death a parameter. We are hereby considering two cases of death. i.e., one due to the infection and the other as a natural death.

The model we are framing in this innovative manuscript is new, say SIRD-susceptible-Infected-recovered-death-population epidemic model. Few people may observe that they are not physically well and after that, they will go to the hospital and diagnose themselves to prevent any infections by taking the precautions and prescriptions with proper medical advice. But many people will not be aware of the symptoms they are having. Those people are always in the mindset that the symptoms they are having will be because of climate change or may cause a fever and will recover soon. But medically every time is valuable. Every hour will be treated as an hour of the diamond moment. Think, about what would happen if the infections were not observed or diagnosed on time. Definitely, that disease may have a chance to lead to a severe stage of infections which might put the patients' life to death. Simply we can say a time delay in the study of or diagnosis of infections may make the infections sustain severely. In order to research them mathematically, we are using the concept of time delay while implementing the rate of infection population change. Instead of the time-shift property of Laplace transform we are using a simple approach called linear operation technique in both (LADM) and in (DTM). So we can hereby declare that the model we are about to construct here is of non-integer order with infectious population study of delay due to time. As there will always arise randomness and fluctuations in modeling natural ailment dynamics we are converting our model into the fuzzy model.

The manuscript was solely prepared with dedication by the authors but it was not created all of the sudden. There are numerous fore-paper studies that have been carried out before proceeding with our analysis. They are summarized below so that the readers may get fruitful knowledge while passing through them. They are Lofti Zadeh's theory of Fuzzy sets ^[1]. In 2000, the Buckley-Feuring proposal for FDE ^[2] created a great impact in the early 2000s. Abbasbandy's modified ADM in 2005 ^[3]. Allen's, mathematical biology ^[4]. Makinde ^[6], SIR-model for invariant vaccination procedure by ADM technique. Ongun's LADM for HIV infection of ^[7]. Arafa et al.'s fractional-order infant disease model of invariant vaccination procedure ^[8]. Atangana-Baleanu's novel fractional derivatives with nonlocal and non-singular kernel ^[9]. In ^[10], Aliyu provided an HIV-I cure model under ABC derivative. Farman presented the SEIR-measles model for the fractional-order derivatives using the technique known as LADM ^[13]. Authors in ^[14,15] solved influenza models. Authors like ^[17,18] presented a stochastic epidemic ODE model with perturbations. Authors like Prasantha Bharathi et al., frequently solved many types of FDEs ^[5,12,16]. Recently, Authors like P. Singh et al. contributed their innovative ideas about the dynamics of epidemic spread in ^[21,22].

The paper highlights if death is used also as a variable, what can be the model? How to analyze such a model?. In this manuscript, the retarded delay is introduced and studied extensively with the necessary analysis in the upcoming sections.

Other than the introductory section, the paper was arranged as follows. Under the Sections 2–7, SIRD-with delay-model creation, the fore-studies, qualitative analysis, analytical solutions, numerical estimations, and the respective graphical illustrations are presented. Finally, in Section 8, the manuscript was concluded in detail.

2.   SIRD delay epidemic model-formulation

We shall construct the fuzzy epidemic compartmental model under ABC fractional derivative $^{ABC}\Delta^{\alpha}_{0, t}$. Usually, S(t), I(t), R(t) denote Susceptible, Infected and Recovered hosts respectively and D(t) is given to the Death population where $t$ here is taken in days (see Figure 1). The main assumption behind the model formulations is that there arises a new strange disease among the society. Due to this, there arises 'S(t), I(t), R(t) and D(t)' here the death is not only because of the severity of infections but also due to some other health-related issues like heart attack, etc. Both are separately defined with the aid of separate parameters. Also, no additional birth or death exceeding the past count. The models that we are framing here are presented to bring the changes of approach to the classical Kermack-Mckendrick models ^[19,20]. Mathematics definitions for both death and recovery are the same under which there is no chance of infection ratio can be observed. All these assumptions and strategies are put together to form the following model. Throughout the entire manuscript $D$ is given for death population and $\Delta$ is given for $d/dt$.

The important thing we want to share to the readers is that the notations S(t), I(t), R(t) and D(t) are representing the number of susceptible, infected, recovered and death cases varying in time (in days) and they are dimensionless. S(t), I(t), R(t) and D(t) are initial populations in numbers at time $t = 0$. The rates $\beta, \delta, \psi, \gamma$ are representing the rates also. So our equations seem to be dimensionless. Also one can see when $D(t) = 0$, our entire study will match with the properties of SIR model ^[19,20] as in SIR model ^[19,20] assumes rates but dimensionless, i.e., S(t), I(t) and R(t) are representing the numbers.

SIRD with the initial function defined as $S(t_0) = m_1$, $I(t_0) = m_2$, $R(t_0) = m_3$ and $D(t_0) = m_4$ for all $-\tau\leq t_0\leq 0.$ Such that the initial functions are always constant as they are representing the population (numbers) at the time of study. All the symbols and parameters used in the above Eq (2.1) are completely described at the end of Section 3.

3.   Fore-studies

The operators with non-integer order seem in such a way that the many mathematical-physical-chemical and biological theories can be turned into a simplified model. Different types of fractional operators are available to easily analyze those theories. singular kernels are notably inconvenient while considering Riemann-Liouville and Caputo-fractional operators since the solutions are not fine and smooth. To get rid of this, Atangana-Baleanu(AB) introduced a very novel advanced operator which consists of Mittag-Leffler kernel which is ^[9] non-local as well as non-singular. The merit of using this operator is that it can be very useful in modeling the biological dynamical systems since AB-derivative eliminates the difficulty to model any before said problems with singularity. Various types of problems on physical phenomena have been constructed and studied with proper analyses by using the required operator (ABC). The below section lists out of the set of necessary results in the FFDE.

Definition 3.1. ^[5] The fuzzy non-integral single retarded DDE can be described in ABC way as

where $\tilde{f}:[0, \infty)\times R \times R\rightarrow E^{n}$ and $\tilde{\phi}\in R$ is a continuous fuzzy mapping and the initial condition $y_{0}\in\phi$ then $y_{0}(s) = y(s) = \phi(s), \ \ -\tau\leq s \leq 0$. Also $y_{0}$ is fuzzy valued with $r$-level intervals, $[y_{0}]^{r} = [\underline y_{0}^{r}, \overline y_{0}^{r}], \ \ \ \ 0\leq r \leq 1$.

Definition 3.2. ^[9] The Mitag-Leffler function can be defined as the result of the following fractional DE $\Delta^{\alpha} = ay\, 0 < \alpha < 1$ where generalized Mittag-Leffler function $E_{\alpha}(-t^{\alpha}) = \sum_{k = 0}^{\infty}{\frac{(-t)^{\alpha k}}{\Gamma(\alpha k+1)}}$ is considered as non local function.

Definition 3.3. ^[9] The fuzzy ABC derivative of $f$ over $[t_0, t_n]$ is

where $F(\alpha)$ is a normalized function which obviously satisfying $F(0) = F(1) = 1$ and $F(\alpha)$ will satisfy the properties in ABC in a same way as that in Caputo and Fabrizio case ^[11].

Definition 3.4. ^[9] We can make fuzzy ABC derivative to undergo Laplace transform implying, ${L\{^{ABC}\Delta^{\alpha}_{0, t}\tilde{f}(t)\}(s)} = \frac{F(\alpha)}{1-\alpha} \frac{s^{\alpha}{L\{\tilde{f}(t)\}(s)-s^{\alpha-1}}}{s^{\alpha}+{ \frac{\alpha}{1-\alpha}}}$ where $0 < \alpha\leq 1$.

Definition 3.5. ^[9] The AB integral of the function $\tilde{f}(t)$ having order $\alpha > 0$ is defined as

Also when $\alpha\rightarrow 1$ then the classical integral is obtained.

The DE referring (2.1) can be overwritten as the fractional DE which is given as

which in turn implies fuzzy fractional DE (FFDE) by the concepts that are given in the fore-studies. The primary functions are the same as that of (2.1).

where $\tilde{f}(t) = (0.75+0.25r, 1.125-0.125r)f(t)$ is the fuzzy function with $r\in[0, 1]$.

The primary conditions are satisfied and also implies that the total population is initially constant with size N. i.e., $S(t_0)+I(t_0)+R(t_0)+D(t_0) = N$.

Here we assume the following initial populations.

The parameters their descriptions and their respective assumed rates from the (2.1)–(3.2) are shown here under.

$\beta\rightarrow$ rate of susceptible becoming infectious = 0.00012;

$\delta\rightarrow$ the rate of infectious becoming recovered = 0.06;

$\psi\rightarrow$ rate of infectious becoming death due to severity of infections = 0.04;

$\gamma\rightarrow$ rate of infectious becoming death due to age or any other health issues = 0.02.

4.   Steady states and stability analysis

This section is devoted to analyzing the stability of the system. We are finding the eigenvalues of the steady states to study the stability. Take $(3.2)$, since the term $D(t)$ is not involved in $^{ABC}\Delta^{\alpha_1}_{0, t}\tilde{S}(t)$, $^{ABC}\Delta^{\alpha_1}_{0, t}\tilde{I}(t)$, $^{ABC}\Delta^{\alpha_1}_{0, t}\tilde{R}(t)$, $^{ABC}\Delta^{\alpha_1}_{0, t}\tilde{D}(t)$ considered for analyzing the stability. Also, $^{ABC}\Delta^{\alpha_1}_{0, t}\tilde{R}(t)$ is not considered because its population is not known since the total population is not known. It is now obvious to consider $^{ABC}\Delta^{\alpha_1}_{0, t}\tilde{S}(t$) and $^{ABC}\Delta^{\alpha_1}_{0, t}\tilde{I}(t)$ for which the steady states and the stability analysis can be done.

Take

and

Now set $^{ABC}\Delta^{\alpha_1}_{0, t}\tilde{I}(t) = 0$ to find disease free steady state $E_1$ along with basic reproduction number of the delayed $I(t)$ and put $^{ABC}\Delta^{\alpha_1}_{0, t}\tilde{S}(t) = 0$ to find disease depending steady state $E_2$. In previous existing models, when $R_0 = 1$ or $R^{Delay}_0 = 1$, $E_1 = E_2.$ In this model, We obtain the steady states as $E_1 = (\frac{\delta+\gamma}{\beta}, 0)$ and $E_2 = (0, \frac{-\gamma}{\beta})$. These states states infer that the growth of susceptible rate cannot turn out be infective whereas the decrease of infection can nullify the susceptible rate. If the inverse of these existing conditions prevail, these steady states remain to be steady always. In order to claim this, we shall prove the next theorem.

Theorem 4.1. The steady states remain to be steady always when the inverse of the product of steady-state matrices exists.

Proof. First let us confirm from the list of parameters given above that $\beta\neq\gamma\neq\delta\neq\psi$ Let us consider $E_1$ and $E_2$ in the algebraic form as $E_1(S, I) = (\frac{\delta+\gamma}{\beta}S)$ and $E_2(S, I) = (\frac{-\gamma}{\beta}I)$. Now assume, the determinant of the product of the steady state matrices to be zero so that the inverse of the product of steady state matrices does not exist.

It was found from the above determinant, that $|E_{1}E_{2}| = 0$ only when $\delta = \gamma$. But this is a contradiction to our parametric values. so the determinant of the product of steady-state matrices exists and obviously, the inverse of the product of steady-state matrices exists. This implies that these steady states always remain to be steady.

4.1.   Basic reproduction number

The basic reproduction number is the calculation of the minimum susceptible rate when there is no change in the rate of infection. Though the model is the system of delay differential equations, the basic reproduction number will be the same as that of the system of ordinary differential equations. Because for $-\tau\leq t_0\leq 0$, we have $I(0) = I(-\tau)$. At t = 0 the observations will be made on the rate at which the infectious cases progress and the estimation of this number will be found. When $^{ABC}\Delta^{\alpha_2}_{0, t}\tilde{I}(t_0) = 0$,

$\beta \tilde{S}(t_0)\tilde{I}(t_0)- (\delta+\gamma)\tilde{I}(t_0)) = 0$, $\beta S_{0}I_{0}-(\delta+\gamma)I_{0} = 0$, $I_0[\beta S_0-(\delta+\gamma)] = 0$,

$\beta S_{0} = \delta+\gamma$, $S_{0} = \frac{\delta+\gamma}{\beta}$,

${{R}_0} = {{R}^{delay}_0} = S_{0}\frac{\beta}{\delta+\gamma}$ is the Basic Reproduction number, Where $\frac{\gamma+\delta}{\beta} = S_{c}$.

When $S_{0} < S_{c}$, the epidemic will come to an end but if $S_0 > S_{c}$, the disease may propagate again and again until it comes to an end. i.e., there is an epidemic.

4.2.   Characteristic equations

Let us find the characteristic equations of the reduced system (4.1). The Jacobian matrix of the system is given by $J = [A_1+A_2.e^{-\lambda\tau}]$. Where,

Then the Jacobian matrix $J$ is given by

4.3.   Jacobian matrix at disease free steady state

Now Substitute $E_1$ at J, We get $J{E_1}$ as

The characteristic equation at the steady state $E_1$ is $CE_1 = |JE_1-\lambda I|$ is given by

i.e., $CE_1 = (-\lambda-\gamma)(\lambda\delta+\lambda\gamma-\lambda\delta e^{-\lambda\tau}-\lambda\gamma e^{-\lambda \tau})$

We can conclude that one of the eigenvalue, $\lambda_1(CE_1) = -\gamma$. If other eigenvalues also have negative real parts, then we can confirm that there is no Hopf bifurcation for $E_1$.

Consider,

i.e., $\lambda_2(CE_1)$ may lead to the infinite number of eigenvalues. Now take, $\frac{\lambda}{\delta+\gamma} = e^{-\lambda\tau}-1$. Now we have to establish that $(e^{-\lambda\tau}-1)$ lies in the left half of the complex plane. So that we can prove that $E_1$ is asymptotically stable. Now assume that $\lambda = a+ib$ with $a$ as zero or a positive real number and $b$ as a real number. Then the magnitude of $e^{-\lambda\tau}$ is

Since $a$ is a zero or positive real number, and $\tau > 0$ the following cases will arrive.

● When $a$ is zero, $|e^{-\lambda\tau}|$ is independent of $\tau$ will produce $|e^-\lambda\tau| = 1$.

● When $a$ is positive, and since $\tau$ is always positive, $|e^{-\lambda\tau}| < 1$.

Hence $|e^{-\lambda\tau}| = e^{-a\tau}\leq 1$. Then definitely, $(e^{-\lambda\tau}-1)$ is the complex number that lies in the left half of the complex plane. i.e., $\frac{(a+ib)}{\delta+\gamma} = e^{-a\tau}-1$. The right-hand side of the equation is already proved to lie in the left half of the complex plane irrespective of $a$ is zero or positive real number. Suppose if $a$ is the positive real number. i.e., if $a > 0$ then the left-hand side of the equation will be the complex number in the right half of the complex plane. This contradicts the right-hand side. So $a$ cannot be the positive real number. So $\lambda_2(CE_1) = (\delta+\gamma)(e^{-\lambda\tau}-1)$ leading to infinite eigenvalues are all negative which will never cross from left to right half of the complex plane. Then there is not a Hopf bifurcation. The disease-free steady state $E_1$ is asymptotically stable regardless of time delay $\tau$.

4.4.   Jacobian matrix at disease depending steady state

Now Substitute $E_2$ at J, We get $J{E_2}$ as

The characteristic equation at the steady state $E_2$ is $CE_2 = |JE_2-\lambda I|$ is given by

$CE_2 = \lambda^2+\lambda\delta+\lambda\gamma = 0$, here also, one of the eigenvalues $\lambda_1(CE_2)$ is negative as $\lambda_1(CE_2) = -(\delta+\gamma)$. Since there is no $e^{-\lambda\tau}$ term $\lambda_2(CE_2)$ will not lead to the infinite eigenvalue. The other eigenvalue $\lambda_2(CE_2)$ is given by $\lambda_2(CE_2) = 0$. So the eigenvalues of disease depending on steady-state $E_2$ is negative semi-definite. This is called a stable line of equilibrium. In order to prove that the steady-state is stable, even if one of the eigenvalues is zero, we have to prove the following theorem.

Theorem 4.2. The steady state is stable but not asymptotically stable when $\Delta > 0$, $Det(E_2) = 0$ and $tr(E_2) < 0$.

Proof. From $(4.8)$, It was found that $tr(E_2) = -(\delta+\gamma) < 0$, $Det(E_2) = 0$ and $\Delta\equiv[tr(E_2)]^2-4Det(E_2)\equiv[-(\delta+\gamma)]^2\equiv (\delta+\gamma)^2 > 0$. Hence the steady state $E_2$ is stable but not asymptotically stable. Since one of the eigenvalues is zero and the other is negative the solutions on the eigenspace are time independent, so obviously independent of time delay. The steady state $E_2$ is thus having an attractive line of equilibria. Hence the system is thus stable at each steady state independent of time delay.

5.   Analytical solution-SIRD-delay model

5.1.   Application of LADM technique

The analytical estimations can be addressed through the application of LADM of order 4 as that of ^[13]. Though the system is delay-dependent in $I$ we are not using the time shift property of the Laplace transform. Instead of that, we are treating the delay term $I(t-\tau)$ by using the concept of linear operations. We provide the entire study by taking $\tau = 1$. Because the initial function is unvaried constant for the period of delayed time from $-\tau$ to $0$. This method is very direct. i.e., We can take $L[I(t-\tau)]$ with $I(t_0) = c_2$ for $-\tau\leq t_0\leq 0.$ as $L[I(t)+I(-\tau)]$ which becomes $L[I(t)]+L[I(-\tau)]$ which in turn gives $L[I(t)]+L[I(t_0)]$. i.e., $L[I(t)]+L[I_{0}]$. Suppose the initial functions are the functions of $t$ then this direct linear operation method may not get worked out. Then we must go for the time-shifting property of Laplace transform or else we have to linearize the system and then we have to go for Laplace transform. All the cases can be solved by following procedures.

Where $(A_{k})$ is an Adomian polynomial defined by $A_{k} = \frac{1}{k!}\frac{d^{k}}{d\lambda^k}(\sum_{l = 0}^{k}(\lambda^l.S_{l}\lambda^l.I_{l})|_{\lambda = 0}$, i.e.,

$A_{0} = S_{0}I_{0}$,

$A_{1} = S_{0}I_{1}+S_{1}I_{0}$,

$A_{2} = S_{0}I_{2}+S_{1}I_{1}+S_{2}I_{0}$ and so on.

$S(t) = \sum_{k = 0}^{\infty}(S(k))$,

$I(t) = \sum_{k = 0}^{\infty}(I(k))$,

$R(t) = \sum_{k = 0}^{\infty}(R(k))$,

$D(t) = \sum_{k = 0}^{\infty}(D(k))$.

As it involves, For the model ($3.2$), the LADM-4 solutions for ($\alpha_1$, $\alpha_2$, $\alpha_3$, and $\alpha_4$) = 1 are obtained by neglecting the terms $t^5$ and above.

where $0 \leq r \leq 1.$ Note that the powers of $t$ do not mean the order of the system. Since we are using (LADM of order 4) we had computed the system up to $S_4$, $I_4$, $R_4$ and $D_4$. Their expansion were given in $(5.2)$.

5.2.   Application of DTM

The alternate method called DTM-4 is also taken in reference to ^[13]. The DTM was manifested with the aid of Taylor's expansion for the series. The complete DTM-4 solutions series is about $t = 0$. For $I(t-\tau)$, The same idea of linear operation as applied in (LADM-4) is applied here which was given in detail below. The differential transformation of the function $f(x)$ when $k/\alpha\in Z^{+}$ can be defined as $D_{t}(f(x)) = 1/(k/\alpha)![\frac{d^{k/\alpha}f(x)}{{dx}^{k/\alpha}}]_{x = 0}$. Also If $f(x) = g(x)h(x)$, $F(k) = \sum_{n = 0}{k}G(k)H(k-n)$. Now for ($\alpha_1$, $\alpha_2$, $\alpha_3$, and $\alpha_4$) = 1, the system of system of Eq $(3.2)$ is defined as

When the initial time is considered as zero, i.e., $t_{0} = 0.$ The inverse DT of $S(k)$, $I(k)$, $R(k)$ and $D(k)$ are obtained as $S(t) = \sum_{k = 0}^{\infty}S(k)t^k$, $I(t) = \sum_{k = 0}^{\infty}I(k)t^k$, $R(t) = \sum_{k = 0}^{\infty}R(k)t^k$ and $D(t) = \sum_{k = 0}^{\infty}D(k)t^k$. For the model ($3.2$) the DTM-4 solutions are given by

where $0 \leq r \leq 1.$ Here also we had computed the system up to $S_4$, $I_4$, $R_4$ and $D_4$ and had gotten the above expansion. But the highest power of $t$ here is 4 because of expansion of $(5.3)$. In general, $\tilde{S(t), I(t), R(t), D(t)} = (0.75+0.25r, 1.125-0.125r)(S(t), I(t), R(t), D(t))$.

6.   Numerical solution of SIRD-Delay model

6.1.   Fourth order Runge-Kutta method

In this section the RKM-4 for ($\alpha_1$, $\alpha_2$, $\alpha_3$, and $\alpha_4$) = 1 is applied to get the desired solution. Estimating $SIRD$ with delay at $h = 0.1$ is considered to get the better result over $0 \leq r \leq 1.$

Evaluation:

To estimate (6.1), consider the following.

$L_m = L_{m-1}$ for $m = 1$ and $I(t) = I(t-1)$ for $t = 0$.

For $p\in[1, 4]$ and $r\in[0, 4]$,

$\tilde{K}_{p} = \tilde{K}_{p}(t; r) = [\underline{K}_{p}(t; r), \overline{K}_{p}(t; r)],$

$\tilde{L}_{p} = \tilde{K}_{p}(t; r) = [\underline{L}_{p}(t; r), \overline{L}_{p}(t; r)],$

$\tilde{M}_{p} = \tilde{K}_{p}(t; r) = [\underline{M}_{p}(t; r), \overline{M}_{p}(t; r)],$

$\tilde{N}_{p} = \tilde{K}_{p}(t; r) = [\underline{N}_{p}(t; r), \overline{N}_{p}(t; r)].$

For $t\in[0, n]$, $n = 1, 2, 3, ...,$

and for $\underline{q} = t$, $\overline{q} = t+1$, $t = 0, 1, 2, 3, ...,$

$\tilde{S}(q) = \tilde{S}{(q)}(t; r) = [\underline{S}_{q}(t; r), \overline{S}_{q}(t; r)],$

$\tilde{I}(q) = \tilde{I}{(q)}(t; r) = [\underline{I}_{q}(t; r), \overline{I}_{q}(t; r)],$

$\tilde{R}(q) = \tilde{R}{(q)}(t; r) = [\underline{R}_{q}(t; r), \overline{R}_{q}(t; r)],$

$\tilde{D}(q) = \tilde{D}{(q)}(t; r) = [\underline{D}_{q}(t; r), \overline{D}_{q}(t; r)].$

Where $[\underline{f}(t; r), \overline{f}(t; r)] = [0.75+0.25r, 1.125-0.125r]f(t).$

7.   Graphical representations

The associateship among $SIRD$-delay at $\alpha_i = 1, i = 1, 2, 3, 4$ for $t\in[0, 1000]$ for the fuzzy valued model $(3.2)$ is given in figure.

For the various values of $t$, $\alpha_1$, $\alpha_2$, $\alpha_3$, $\alpha_4$ and $r\in[0, 1]$ the following tables are given. In Tables 1–4 the susceptible, infected, recovered and dead population solutions are calculated using two various methods, say, LADM-4, DTM-4 and after that by RKM-4, all these methods are compared. Figure 2 is plotted by taking $S(t)$, $I(t)$, $R(t)$ and $D(t)$ for $\alpha_i = 1, i = 1, 2, 3, 4$, $t\in[0,300]$ and $r = 1$. Figure 3 is plotted by taking $S(t)$, $I(t)$, $R(t)$ and $D(t)$ for $\alpha_i = 1, i = 1, 2, 3, 4$, $t\in[0, 3]$ and $r = 1$. In Tables 5 and 6, susceptible population values for $\alpha\in[0, 1]$ and $t\in[0, 1]$ are given and the respective plot is given in Figure 4. In Tables 7 and 8, infected population values for $\alpha\in[0, 1]$ and $t\in[0, 1]$ are given and the respective plot is given in Figure 5. In Tables 9 and 10, recovered population values for $\alpha\in[0, 1]$ and $t\in[0, 1]$ are given and the respective plot is given in Figure 6. In Tables 11 and 12, dead population values for $\alpha\in[0, 1]$ and $t\in[0, 1]$ are given and their respective plot is given in Figure 7. Figure 8 is given for $S(t)$, $I(t)$, $R(t)$, $D(t)$ for $t\in [0, 1]$, $\alpha\in[0, 1]$ and $r = 1$. After solving $(3.1)$ for $\alpha\in[0, 1]$ the values are calculated for $t\in[0, 1]$ and $r\in[0, 1]$ in (3.2). As a sample, fixing $t = 1$, $\alpha_i = 1, \, i = 1, 2, 3, 4$ and considering $r\in[0, 1]$ fuzzy valued solutions of susceptible, infected, recovered and dead populations are given in Table 13. Similarly we found the remaining values in $t\in[0, 1]$ with $\alpha_i, i = 1, 2, 3, 4$ for $r\in[0, 1]$ and other plots are obtained. Figures 9–12 are given for $S(t)$, $I(t)$, $R(t)$, $D(t)$ for $t\in [0, 1]$, $\alpha\in[0, 1]$, and $r\in[0, 1]$ which are the fuzzy valued plots.

From Tables 1–4, we can confirm the solutions are correct at least to one decimal place of accuracy by comparison of LADM-4, DTM-4, and RKM-4 methods.

Implications of Figures 2, 3 and 7

(1) The model we have shown here is not the cumulative case study. It is the daily change in the SIRD cases due to the epidemic's spread.

(2) If it is cumulative the graph of death will be fixed at a particular count and will not change, especially decreased with time.

(3) If it is cumulative, the curve of $D$ will not decrease since rebirth is physically impossible and it will increase or otherwise it is stable at a particular count if there is no new death.

(4) The model we considered cannot be taken as closed since there is a change in death.

(5) The model does not imply the death of people who died naturally due to age, accident, disaster, etc., it is only implying the death due to the epidemic spread. So decrease of curve $D$ implies only that when time increases the daily new death count decreases.

(6) The curve of $S$, $I$, $D$ decreases but $R$ increases for some $t$, and after that it also gets reduced. The curve $I$ is not hiking and continuously reducing because the model we considered is a delay model having $I(t-\tau)$. At initial time $I(t)$ is the same as the $I(t-\tau)$. When there is a decrease in daily new susceptible, it not only means the existing susceptible went to the compartment $I$, being susceptible the people might be gone out of $S$ due to recovering from the symptoms which is also the reason for the decrease of $I$, since $S$ is decreasing and by which $I$ is also decreasing, initially there is a hike in the curve of $R$. After a certain time, $R$ is also decreased since there is a decrease in $I$.

(7) In addition, the curves of $D$ are more dominated by $\psi$ than $\gamma$.

(8) All the curves decrease as time increases because it is a daily case study and not a cumulative case study. After the time $t\geq300$ $S, I, R, D$ become zero means at time $t\geq300^{th} day$, there is no new susceptible, new infected, new recovered and new dead since we are studying only daily case study and not cumulative study.

(9) Since the solutions are derived using the LADM technique they are not cumulative. Because in cumulative only the increase of time will increase at least one of the compartment cases mostly, $R$. But here we are addressing only a daily case study in which an increase of time need not imply an increase of at least a compartment case.

(10) Due to the above implications, the reason for the decrease of $D$ curves as shown in Figure 6 is also justified.

8.   Conclusions

The system of the single retarded delay fuzzy fractional epidemic model under ABC derivative was presented. To study its stability the system was reduced to two steady states such as disease-free and disease dependent since if the steady states are stable then obviously the complete system is stable. The basic reproduction number was found and stability analysis is done for both the steady states. We proved that the steady states always remain to be steady, by a theorem. For a negative semi-definite eigenvalue, we proved the theorem to claim that the system is stable. From our study steady-state 1 is asymptotically stable and steady-state 2 is stable and both are not affected by the delay. so no Hopf bifurcation will occur. For $\alpha_i\in [0, 1], i = 1, 2, 3, 4$, Fuzzy valued $S(t)$, $I(t)$, $R(t)$ and $D(t)$ at $t\in[0, 1]$, $r\in[0, 1]$ are shown in the tables and plots. Traditionally two numerical solutions will be compared with one analytical solution but we differently compared two analytical solutions with that of one numerical solution. Because the analytical solutions seem to coincide by only one decimal place of accuracy. In order to confirm the accuracy of the solution, we considered RKM-4 which is very direct. All these three methods were compared at $\alpha_i = 1$. The solutions by LADM, DTM and RKM are equally well-matched in accuracy up to one decimal place. But RKM-4 is considered as a quick as well as a direct solution in case of the delayed epidemic model. We are solving the delay term by means of linear operation instead of the time-shift property of Laplace transform in both LADM and in DTM. So one can get a doubt about the correctness of the solutions. That's why we are comparing LADM and DTM solutions with that of RKM-4 solutions which is very direct involving neither a transformation nor a linearization to confirm the correctness of the solutions. Also, we have to keep in mind that to solve the fuzzy ordinary epidemic model with a single retarded delay one can use RKM-4, and to solve the fuzzy fractional-order epidemic model with a single retarded delay one can use LADM-4 as well as DTM-4. The limitation of the model is the value of the delay term that could not be extended beyond the value used in the initial condition. In the future the authors would like to do the research on this retard delay epidemic model to predict the daily new cases.

Conflict of interest

The authors declare no conflict of interest.