Prediction and decision making in corona virus using fuzzy mathematical model

Abstract

During the period of 2019–20, forecasting was of utmost priority for health care planning and to combat COVID-19 pandemic. Almost everyone’s life has been greatly impacted by COVID-19. Understanding how the disease spreads is crucial to know how the disease behaves dynamically. The aim of the research is to construct an SEI Q₁Q₂ R model for COVID-19 with fuzzy parameters. The fuzzy parameters are the transmission rate, the infection rate, the recovery rate and the death rate. We compute the basic reproduction number, using next-generation matrix method, which will be used further to study the model’s prediction. The COVID-free and endemic equilibrium points attain local and global stability when R₀ < 1. A sensitivity analysis of the reproduction number against its internal parameter has been done. The results of this model showed that intervention measures. The numerical simulation along with graphical representations at COVID-free and endemic points are shown. The SEIQ₁Q₂R model is a successful model to analyse the spreading and controlling the epidemics like COVID-19.

1 Introduction

The universe was brought to its knees by the COVID-19 illness epidemic during its inversion. A coronavirus is a CORONAVIRIDAE member that affects several vertebrates by causing gastrointestinal or respiratory illness [1]. When the sickness began to spread globally in early 2020 after its origin in Wuhan, China, in December 2019 [2], the world went through difficult times. The disease was initially discovered in Wuhan, the capital of Hubei, China, in December 2019. Since then it has spread around the world, leading to the 2020 pandemic breakout [3]. Due to the thousands of confirmed illnesses and thousands of deaths worldwide, the COVID-19 pandemic was regarded as the greatest global threat. Several approaches to assess and predict the pandemic’s evolution have been made as a result of the interest shown by experts from various fields in the worldwide challenge of the outbreak [4, 5]. In a short period of time, the sickness sickened millions of people globally and claimed 100,000 lives, causing unrest in every country. The World Health Organization (WHO) established disease-fighting tactics at the time to battle the pandemic and preserve lives. The strategic tactics included quarantine, physical and social seclusion, mask usage, border closures, and economic lock-downs. The actions taken had a social and economic impact on the planet. The most effective method of combating COVID-19 is immunization, which also helps to restore lost economic activity [6–8]. Scientists have developed a variety of vaccinations to battle the illness, and the WHO has approved them [9]. The study conducted by [10] examined vaccination in age group allocation techniques with their SEIR model in India and came out with the results revealing that the elderly generation aged more than 60 years, which was the most affected group by COVID-19 disease, was the group considered for vaccination. Moreover, [11] discussed a model with an age structure and time-varying contact rates.

As shown in the study [12], mathematical models are highly important tools used to assess the likelihood and severity of the disease and assist in determining the intervention measures to be implemented to battle and control the disease intensity. The article [13] discussed the future situation, after the calculation of reproduction number. The results of the manuscript [14], were used as an early prevention of the spread of COVID-19 in Indonesia. Many mathematical models have been developed since the COVID-19 outbreak first started. A compartmental model of the progression of chronic diseases, human epidemiological state, and intervention strategies was put forth by Tang et al. [15] and was deterministic. The manuscript [36] Huanlai Xing et.al proposed a robust semisupervised model with self-distillation that simplified existing semisupervised learning techniques for extracting high level semantic information, which is also helpful for capturing low level semantic information. A novel robust temporal feature network which was responsible for extracting local features was proposed [37]. The authors discovered that there can be up to 6.47 reproductive controls, and that engagement strategies like streamlined traceability combined with isolation and quarantine may successfully reduce the number of reproductive controls and transmission risk. Iman et al. [16] performed calculational modelling of the potential epidemic tracks with a focus on human-to-human transmissions to figure out the extent of the illness outbreak in Wuhan. According to its findings, more than 60% of the transmission must be effectively regulated in order to control the disease. Authors like Ahmaed, Parvaiz, Ihtisham, Fatma and Anne investigated the novel corona virus using various methods [38–40, 48]. In the manuscript [41], authors have suggested a new model that represents the interplay of two diseases (corona and bacterial pneumonia) by taking into account the two different ways that pneumonia can be acquired: in the community and in a hospital. They’ve demonstrated through numerical simulations that both diseases can endure if COVID’s basic reproduction number is higher than one. The model depicted MERS-CoV transmission between individuals and reservoirs like camels. The model was created with optimal control in view to reduce the number of infected people while keeping intervention costs low [44]. Using Caputo-Fabrizio advention reaction diffusion equation Hardik analyzed the effect of disturbance in intracellular calcium dynamic on fibroplast cell [45]. The majority of the models showed how important a direct human-to-human transmission mechanism was in the outbreak, which was supported by the observation that many of the infected people in the Wuhan region do not interact with one another and that the number of infections has been rising quickly and spreading to other Chinese provinces, affecting more than 20 people [17]. Hardik along with various co-authors have investigated several models for the spread of COVID-19 using Caputo fractional derivative, Adam-type predictor corrector, ABC derivative [42, 47]. To obtain an approximate solution, the Laplace. The suitable strategies for quarantine during the COVID-19 pandemic in a specific age group were first presented by Gondim and Machado [18]. Wang et. al used the mathematical model to deduce the earliest start date of COVID-19 infection in Wuhan [19]. The SEIR model was updated by Youssef et al. [20] and applied to the actual data of COVID-19 spreading in Saudi Arabia. Feng and Thieme [21, 22] evaluated the model dynamics of SEIQR (Susceptible-Exposed-Infected-Quarantined-Recovered) models with arbitrarily spaced periods of sickness, together with quarantine, and a general occurrence stated that all affected patients passed through the quarantine stage. A pandemic influenza SEIQR model for obtaining the numerical values of the model’s parameters was proposed by Jumpen et al. [23] who also examined the model’s characteristics. For paediatric illnesses, Gerberry and Milner [24] employed the SEIQR model. The authors of the articles [13, 18–22] evaluated the mathematical model with different compartments. They have computed basic reproduction number, analysed stability of the disease and some of them concluded with control strategies. This provided the motivation to add fuzzy analysis in this manuscript.

Deterministic models with fixed model parameters have been adopted by the majority of researchers. In general, they made the assumptions that each person could spread the disease and recover from it at a constant rate. The assumptions, however, be contrary to the actual epidemic. As a result, in the current study, we have taken a few parameters into account as fuzzy parameters. So, the purpose of this research is to develop a new COVID-19 dynamical model using mathematical analysis that is more applicable to scenarios in any country. Our model demonstrates how the corona virus spreads via interaction with an infected person who is carrying a variety of virus loads and depicts the transmission of the disease at both endemic and COVID-free equilibrium points. The outbreak is brought on by the new infections. In view of this, future predictions of the spread of any pandemic may be enhanced as a result of our research.

2 Transmission of COVID-19

The WHO Country Office in China initially received a report of a pneumonitis of unknown origin on December 31, 2019 [25]. Animal-to-human transmission was assumed to be the primary route because the first instances of the CoVID-19 disease were directly connected to exposure to the Huanan Seafood Wholesale Market in Wuhan, Republic of China. However, later occurrences were not connected to this exposure mechanism. Therefore, it was determined that the virus could potentially transfer from person to person and that symptomatic individuals constitute the main channel for COVID-19 transmission. Transmission prior to the onset of symptoms appears to be uncommon, although it cannot be ruled out. Furthermore, there are theories that those who are asymptomatic could spread the infection. According to the research, isolation is the most effective method for controlling this outbreak [26]. The longest time from infection to symptoms was 12.5 days (95% confidence interval, 9.2 to 18), according to data from the first cases in Wuhan and investigations by the Chinese Centre for Disease Control and Prevention (CDC) and local CDCs [27]. The incubation period could typically be between 3 and 7 days and up to 2 weeks. Additionally, this information demonstrated that this unusual pandemic doubled roughly every seven days, although the fundamental rate of production (R₀) is 2.2. In other words, each patient typically spreads the sickness to an additional 2.2 others. The reproduction number of the SARS-CoV epidemic in 2002–2003 was estimated to be around 3 [28]. According to a study, humans can develop coronavirus by inhaling it or by touching contaminated materials. Researchers found that the virus can be detected in aerosols for up to three hours, on copper for up to four hours, on cardboard for up to 24 hours, and on plastic and stainless steel for up to two to three days [29]. According to a recent study led by researchers at the Johns Hopkins Bloomberg School of Public Health [30], an analysis of publicly available data on infections from the new coronavirus, SARS-CoV-2, which causes the respiratory illness, COVID-19, produced an estimate of 5.1 days for the median disease incubation period. According to the data, around 97.5 percent of those who experience SARS-CoV-2 infection symptoms do so within 11.5 days after exposure. For every 10,000 people isolated for 14 days, Lauer et al. estimated that only approximately 101 would exhibit symptoms after being let out [30].

3 Basic terminologies

Susceptible: A susceptible person is a member of a population who is at risk of acquiring a disease, and is often referred to as a susceptible population in epidemiology.

Exposed: The group of population who are sick but non-infectious are the exposed population.

Infected: It refers to the population that has obtained a communicable disease (in our situation, COVID-19), which may have been spread locally by an individual or by a community.

Home Quarantined: It deals with the isolation or separation of healthy individuals who are susceptible to or may have been exposed to a contagious disease (in this case, COVID-19), prior to knowing whether they will become ill. Isolation typically occurs at home and might apply to a person, a group, or a community of susceptible or exposed people.

Medical Quarantine: In order to stop the spread of a contagious disease (in our example, COVID-19), it refers to the separation and restricted movement of sick people. Normally, it takes place in a hospital setting, but in extremely rare circumstances, it could be done at home with specific equipment.

Recovered: It refers to the group of population who have been recovered from COVID-19.

4 Assumptions

The population is considered as a heterogenous population.

Birth rates are assumed to be a susceptible population.

The following takes place at a constant rate:

The birth rate and natural deaths.

Death due to COVID-19 of the medically quarantined population.

Exposed population becoming susceptible under home quarantine.

Contact rate

The rate of susceptible population who were in their home becoming susceptible whose reports are negative.

Rate of infected population who undergo medical quarantine

Susceptible population under home quarantine

Medical quarantine population of confirmed cases.

5 Fuzzy mathematical model

The following is the system of fuzzy mathematical model for COVID-19, followed by overview of the parameters in Table 1.

Table 1

Overview of the parameters

S	Susceptible population
E	Exposed population
Q 1	Susceptible population under home quarantine
I	Infected population
Q 2	Medical quarantined population of confirmed cases
R	Recovered population
∧	Birth rate
δ 1	Natural death rate
δ 2	Rate of death due to COVID-19
δ 3	Rate of death of medical quarantined population due to COVID-19
β1	Rate of exposed population becoming susceptible under home quarantine
β2	Rate at which exposed population becoming infected population
ρ	Contact rate
β3	Transmission rate of susceptible population to infected population
β4	Rate of susceptible population who were in their home becoming susceptibles whose reports were negative
β5	Rate of infected population who undergo medical quarantine
γ1	Recovery rate
γ2	Recovery rate from medical quarantined population

$\frac{\mathit{dS}}{\mathit{dt}}=\wedge -\rho \mathit{SI}-{\delta }_{1}S+{\beta }_4{Q}_1$ ,

$\frac{\mathit{dE}}{\mathit{dt}}=\rho \mathit{SI}-\left({\beta }_1+{\beta }_2\left(\vartheta \right)+{\delta }_1\right)E$ ,

$\frac{{\mathit{dQ}}_1}{\mathit{dt}}={\beta }_{1}E-\left({\beta }_3\left(\vartheta \right)+{\beta }_4+{\delta }_1\right)Q_1$ ,

$\frac{\mathit{dI}}{\mathit{dt}}={\beta }_3\left(\vartheta \right)Q_1-\left({\beta }_5+{\gamma }_1\left(\vartheta \right)+{\delta }_1+{\delta }_2\left(\vartheta \right)\right)I+{\beta }_2\left(\vartheta \right)E$ ,

$\frac{{\mathit{dQ}}_2}{\mathit{dt}}={\beta }_{5}I-\left({\delta }_1+{\delta }_3+{\gamma }_2\left(\vartheta \right)\right)Q_2$ ,

$\frac{\mathit{dR}}{\mathit{dt}}={\gamma }_2\left(\vartheta \right)Q_2+{\gamma }_1\left(\vartheta \right)I-{\delta }_{1}R$ .

6 Schematic diagram

7 Fuzzy analysis

β₃ = β₃ (ϑ) be the chance of transmission rate from susceptible people to the infected. The virus load affects how quickly diseases spread. The following equation shows the fuzzy membership function for the transmission parameter. ${\beta }_3\left(\vartheta \right)=\{\begin{array}{c}0,\vartheta \le {\vartheta }_{\mathit{min}},\hfill \\ \frac{\vartheta -{\vartheta }_{\mathit{min}}}{{\vartheta }_M-{\vartheta }_{\mathit{min}}},{\vartheta }_{\mathit{min}}<\vartheta <{\vartheta }_M,\hfill \\ 1,{\vartheta }_M\le \vartheta \le {\vartheta }_{\mathit{max}.}\hfill \\ \hfill \end{array}$

When the virus load is very low (ϑ _min ), as shown by the equation, there is almost little chance of virus transmission. For transmission to take place, there must be a specific quantity of virus present (ϑ _M ). Each disease’s virus burden is always limited by ϑ _max .

The Fig. 1 shows the membership function of β₃ (ϑ).

Fig. 1

Membership function of β₃.

Let β₂ = β₂ (ϑ) be the rate at which the exposed population becomes infected. The Figs. 2 and 3 are the representation of membership function of β₂ (ϑ) and γ₁ (ϑ).

Fig. 2

Membership function of β₂.

Fig. 3

Membership function of γ₁.

${\beta }_2\left(\vartheta \right)=\frac{1–{{\beta }_2}_0}{{\vartheta }_{\mathit{max}}}\vartheta +{{\beta }_2}_0.$

Let γ₁ = γ₁ (ϑ) be the recovery rate from infectious population.

${\gamma }_1\left(\vartheta \right)=\{\frac{\left({\gamma }_{10}-1\right)}{{\vartheta }_{\mathit{max}}}\vartheta +1$ , where γ₁₀ > 0 is the lowest recovery rate from the infectious population.

Let γ₂ = γ₂ (ϑ) be the recovery rate from medical quarantined population.

${\gamma }_2\left(\vartheta \right)=\frac{1-{\gamma }_{20}}{{\vartheta }_{\mathit{max}}}\vartheta +{\gamma }_{20}$ , where γ₂₀ > 0 is the lowest recovery rate from medical quarantined population. The Figs. 4 and 5 are the membership functions of γ₂ (ϑ) and δ₂ (ϑ) respectively.

Fig. 4

Membership function of γ₂.

Fig. 5

Membership function of δ₂.

δ₂ = δ₂ (ϑ) Be the rate of death due to COVID-19

${\delta }_2\left(\vartheta \right)=\{\frac{\left(1-\eta \right)-{\delta }_{20}}{{\vartheta }_M}\vartheta +{\delta }_{20}$ , the maximum value odeath due to COVID-19 is (1 - η), (η ≤ 0). When the virus load is low, there won’t be any disease transmission, but when the virus load rises, the death rate will rise.

8 Equilibrium points

8.1 COVID-free equilibrium

$\begin{array}{cc}\hfill C_1& =\left(S_1,E_1,Q_{11},I_1,Q_{21},R_1\right)\hfill \\ \hfill & =\left(\frac{\wedge }{{\delta }_1},0,0,0,0,0\right).\hfill \end{array}$

8.2 Endemic equilibrium

$C_2=\left(S_2,E_2,Q_{12},I_2,Q_{22},R_2\right).$ $S_2=\frac{\wedge R_0{\beta }_3-\left(\rho {\beta }_3+R_0\left({\delta }_1+{\delta }_2+{\beta }_5+{\gamma }_1\right)\right)I_1}{R_0{\delta }_1{\beta }_3}$ $E_2=\frac{\rho I_1}{R_0\left({\beta }_1+{\beta }_2+{\delta }_1\right)},$ $Q_{12}=\frac{\left({\delta }_1+{\delta }_2+{\beta }_5+{\gamma }_1\right)I_1}{{\beta }_3},$ $I_2=\frac{{\beta }_3{Q}_{12}}{\left({\delta }_1+{\delta }_2+{\beta }_5+{\gamma }_1\right)},$ $Q_{22}=\frac{{\beta }_5{I}_2}{\left({\gamma }_2+{\delta }_1+{\delta }_3\right)},$ $R_2=\frac{({\gamma }_2{\beta }_5+{\gamma }_1({\gamma }_2+{\delta }_1+{\delta }_3\left)\right)I_2}{{\delta }_1({\gamma }_2+{\delta }_1+{\delta }_3)}.$

9 Basic reproduction number (BRN)

The basic reproduction is defined as the average numbe secondary infections when one infectious agent is introduced into a population that is completely susceptible. It is calculated using the next generation matrix method. $F=\left[\begin{array}{ccccc}0\hfill & 0\hfill & \rho S_1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \hfill \end{array}\right]$ $V=\left[\begin{array}{ccccc}{\beta }_1+{\beta }_2+{\delta }_1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ -{\beta }_1\hfill & \left({\beta }_3+{\beta }_4+{\delta }_1\right)\hfill & 0\hfill & 0\hfill & 0\hfill \\ -{\beta }_2\hfill & -{\beta }_3\hfill & \left({\beta }_5+{\gamma }_1+{\delta }_1+{\delta }_2\right)\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & -{\beta }_5\hfill & \left({\gamma }_2+{\delta }_1+{\delta }_3\right)\hfill & 0\hfill \\ 0\hfill & 0\hfill & -{\gamma }_1\hfill & -{\gamma }_2\hfill & {\delta }_1\hfill \\ \hfill \end{array}\right]$ $\begin{array}{c}{R}_0=\rho \left({\mathit{FV}}^{-1}\right)=\hfill \\ \frac{\rho \wedge \left({\beta }_1{\beta }_3+{\beta }_2\left({\beta }_3+{\beta }_4+{\delta }_1\right)\right)}{{\delta }_1\left({\beta }_1+{\beta }_2+{\delta }_1\right)\left({\beta }_3+{\beta }_4+{\delta }_1\right)\left({\beta }_5+{\gamma }_1+{\delta }_1+{\delta }_2\right)}\hfill \\ \hfill \end{array}.$

Remark: At COVID-free equilibrium, the reproduction number was found to be 0.9055, which is less than one. This makes clear that the infection spreading to the secondary person is very low and this situation can be brought under control. While at endemic equilibrium the reproduction number was found to be 3.622 which is greater than one, so this is the dangerous situation where one infected person is spreading the disease to three other persons. To bring this under control, we perform sensitive analysis and identify the parameters which leads to increase in reproduction number and provide the control measures.

10 Asymptotic stability analysis

Theorem 1. The COVID-free equilibrium is asymptotically stable locally if R₀ < ; 1, otherwise it is unstable.

Proof.

$J=\left[\begin{array}{ccccc}-{\delta }_1\hfill & 0\hfill & {\beta }_4\hfill & -\rho \hfill & 0\hfill \\ 0\hfill & -\left({\beta }_1+{\beta }_2+{\delta }_1\right)\hfill & 0\hfill & \rho \hfill & 0\hfill \\ 0\hfill & {\beta }_1\hfill & -\left({\beta }_3+{\beta }_4+{\delta }_1\right)\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & {\beta }_3\hfill & -\left({\beta }_5+{\gamma }_1+{\delta }_1+{\delta }_2\right)\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & {\beta }_5\hfill & -\left({\gamma }_2+{\delta }_1+{\delta }_3\right)\hfill \\ \hfill \end{array}\right]$

$\mathrm{J}-\lambda =\left[\begin{array}{ccccc}-{\delta }_1-\lambda \hfill & 0\hfill & {\beta }_4\hfill & -\rho \hfill & 0\hfill \\ 0\hfill & -\left({\beta }_1+{\beta }_2+{\delta }_1\right)-\lambda \hfill & 0\hfill & \rho \hfill & 0\hfill \\ 0\hfill & {\beta }_1\hfill & -\left({\beta }_3+{\beta }_4+{\delta }_1\right)-\lambda \hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & {\beta }_3\hfill & -\left({\beta }_5+{\gamma }_1+{\delta }_1+{\delta }_2\right)-\lambda \hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & {\beta }_5\hfill & -\left({\gamma }_2+{\delta }_1+{\delta }_3\right)-\lambda \hfill \\ \hfill \end{array}\right]$

The characteristic equation is λ³ + a₁ λ² + a₂ λ + a₃ = 0 $a_1=({\beta }_1+{\beta }_2+3{\delta }_1+{\beta }_3+{\beta }_4+{\beta }_5+{\gamma }_1+{\delta }_2),$

$\begin{array}{c}{a}_2=\left({\beta }_1+{\beta }_2+{\delta }_1\right)\left({\beta }_3+{\beta }_4+{\delta }_1\right)\hfill \\ +\left({\beta }_1+{\beta }_2+{\delta }_1\right)\left({\beta }_5+{\gamma }_1+{\delta }_1+{\delta }_2\right)\hfill \\ +\left({\beta }_5+{\gamma }_1+{\delta }_1+{\delta }_2\right)\left({\beta }_3+{\beta }_4+{\delta }_1\right),\hfill \\ \hfill \end{array}$ $\begin{array}{cc}\hfill a_3=& \left({\beta }_1+{\beta }_2+{\delta }_1\right)\left({\beta }_3+{\beta }_4+{\delta }_1\right)\hfill \\ \hfill & \left({\beta }_5+{\gamma }_1+{\delta }_1+{\delta }_2\right)-\rho {\beta }_1{\beta }_3.\hfill \end{array}$

According to Routh-Hurwitz criteria a₁ > 0, a₂ > 0, a₃ > 0, a₁a₂ > a₃ (Refer Appendix: 1). Thus the system is locally asymptotically stable.

11 Global stability

Theorem 1. The stem is globally stable at COVID-free equillibrium for R₀ < 1.

Proof. Consider the Lyapunov function ${\mathrm{F}=C}_1{\mathrm{E}+C}_2{\mathrm{Q}}_1{+C}_3\mathrm{I}.$ $\dot{F}{=C}_1\dot{E}{+C}_2\dot{Q}{+C}_3\dot{I}$ $\begin{array}{c}\le {\mathrm{C}}_1[\rho \mathit{SI}-\left({\beta }_1+{\beta }_2+{\delta }_1\right)E]\hfill \\ {+C}_2\hfill \\ +{\mathrm{C}}_3[{\beta }_3{Q}_1-\left({\beta }_5+{\gamma }_1+{\delta }_1+{\delta }_2\right)I+{\beta }_{2}E].\hfill \\ \hfill \end{array}$

Consider, $\begin{array}{c}\dot{F}=[-{\mathrm{C}}_1\left({\beta }_1+{\beta }_2+{\delta }_1\right)+{\mathrm{C}}_2{\beta }_1+{\mathrm{C}}_3{\beta }_2]\mathrm{E}.\hfill \\ \dot{F}\le [{\mathrm{C}}_2{\beta }_1+{\mathrm{C}}_3{\beta }_2-{\mathrm{C}}_1\left({\beta }_1+{\beta }_2+{\delta }_1\right)]\hfill \\ \hfill \end{array}$

Let, C₁ = δ₁ (β₃ + β₄ + δ₁) (β₅ + γ₁ + δ₁ + δ₂),

C₂ = ρ ∧ (β₁ β₃ + β₂ β₃ + β₂ β₄ + β₂δ₁)/β₁,

C3 = - δ₁ (β₁ + β₂ + δ₁) (β₃ + β₄ + δ₁) (β₅  + γ₁   + δ₁ + δ₂)/β₂

≤δ₁ (β₁ + β₂ + δ₁) (β₃ + β₄ + δ₁) (β₅ + γ₁ + δ₁   + δ₂) [R₀ - 1]

$\dot{F}\le 0$ iff E = 0.

Consider,

$\dot{F}{=[C}_1\rho \mathrm{S}-{\mathrm{C}}_3\left({\beta }_5+{\gamma }_1+{\delta }_1+{\delta }_2\right)]I$

Let C₁ = (β₁ β₃ + β₂ β₃ + β₂ β₄ + β₂δ₁) - [δ₁   (β₁ + β₂ + δ₁) (β₃ + β₄ + δ₁) (β₅ + γ₁ + δ₁ + δ₂)],

C₃ = (β₁ + β₂ + δ₁) (β₃ + β₄ + δ₁)

≤ (β₁ + β₂ + δ₁) (β₃ + β₄ + δ₁) (β₅ + γ₁ + δ₁   + δ₂) [R₀ - 1].

$\dot{F}\le 0$ iff I = 0.

Consider,

$\dot{F}{=[C}_3{\beta }_3-{\mathrm{C}}_2\left({\beta }_3+{\beta }_4+{\delta }_1\right)]\mathrm{Q}.$

Let C₂ = δ₁ (β₁ + β₂ + δ₁) (β₅ + γ₁ + δ₁ + δ₂),

C₃ = {ρ ∧ (β₁ β₃ + β₂ β₃ + β₂ β₄ + β₂δ₁) – [δ₁ (β₁   + β₂ + δ₁) (β₅ + γ₁ + δ₁ + δ₂) (β₃ + β₄ + δ₁)}/β₃

≤δ₁ (β₁ + β₂ + δ₁) (β₅ + γ₁ + δ₁ + δ₂) (β₃ + β₄   + δ₁) [R₀ - 1].

$\dot{F}\le 0$ iff R = 0.

Hence by LaSalle’s invariance principle [34], hence the system is globally stable at COVID-free equilibrium.

12 Fuzzy basic reproduction number (FBRN)

The basic reproduction number is R₀ $\left(\vartheta \right)=\frac{\rho \wedge \left({\beta }_1{\beta }_3+{\beta }_2\left({\beta }_3+{\beta }_4+{\delta }_1\right)\right)}{{\delta }_1\left({\beta }_1+{\beta }_2+{\delta }_1\right)\left({\beta }_3+{\beta }_4+{\delta }_1\right)\left({\beta }_5+{\gamma }_1+{\delta }_1+{\delta }_2\right)}$ , increases with the amount of viruses increase and cannot be a fuzzy set because it can be greater than 1, Thus 0 ≤ γ₁₀ R₀ ((ϑ) ≤ 1, where γ₁₀ R₀ (ϑ) is a fuzzy set and hence FEV [γ₁₀ R₀ (ϑ)] is well defined.

(γ₁₀ R₀ (ϑ)) is FBRN [32].

Where FEV (γ₁₀ R₀ (ϑ)) = sup {inf (α, k (α))}, 0≤ α ≤ 1,

k (α) = , is a fuzzy measure [33]. To get FEV (γ₁₀ R₀ (ϑ)) we, define fuzzy measure μ,

μ (X) = supΓ (ϑ), ∀σ ∈X, X ⊂ R, which is a possibility measure.

From FEV (γ₁₀ R₀ (ϑ)), it is clear that R₀ (ϑ) is not decreasing with, where the set, X = [ $\overline{\vartheta },{\vartheta }_{\mathit{max}}]$ , and $\overline{\vartheta }$ is the solution of the equation given below,

${\gamma }_{10}\frac{\rho \wedge \left({\beta }_1{\beta }_3+{\beta }_2\left({\beta }_3+{\beta }_4+{\delta }_1\right)\right)}{{\delta }_1\left({\beta }_1+{\beta }_2+{\delta }_1\right)\left({\beta }_3+{\beta }_4+{\delta }_1\right)\left({\beta }_5+{\gamma }_1+{\delta }_1+{\delta }_2\right)}=\alpha .$

Thus, k (α) = μ [ϑ′, ϑ _max ] = sup Γ (ϑ) with ϑ′ ≤ ϑ ≤ ϑ _max , here k (0) = 1 and k (1) = Γ (ϑ _max ).

The virus loads which are linguistic variable can be classified as low virus load (ϑ _min ), medium virus load (ϑ) and strong virus load (ϑ _max ).

Case 1: Low virus load

(i.e) when $\overline{\vartheta }+$ δ <ϑ _min , we have

FEV (γ₁₀ R₀ (ϑ)) = 0 < ${\gamma }_{10}\iff R_0^f<1$ .

As a result, we may say that the disease will vanish.

Case 2: Medium virus load

(i.e) when $\overline{\vartheta }--$ δ > ϑ _min and $\overline{\vartheta }+\delta <\vartheta$ .

Therefore, $\mathrm{k}(\alpha )=\{\begin{array}{c}1,\mathit{if}0<\alpha \le {\gamma }_{10}{R}_0\left(\overline{\vartheta }\right),\hfill \\ \Gamma \left(v^{\prime }\right),\mathit{if}{\gamma }_{10}{R}_0\left(\overline{\vartheta }\right)<\alpha \le {\gamma }_{10}{R}_0\left(\overline{\vartheta }+\delta \right)\hfill \\ 0,\mathit{if}{\gamma }_{10}{R}_0\left(\overline{\vartheta }+\delta \right)<\alpha \le 1\hfill \\ \hfill \end{array}$

So, if δ > 0, k(α) is continuous and decreasing function with k(0) = 1 and k(1) = 0. Hence, FEV (γ₁₀ R₀ (ϑ)) is the fixed point of k and

${\gamma }_{10}{R}_0\left(\overline{\vartheta }\right)\le \mathit{FEV}$ (γ₁₀ R₀ (ϑ)) $\le {\gamma }_{10}{R}_0\left(\overline{\vartheta }+\delta \right)$ ,

$R_0\left(\overline{\vartheta }\right)\le R_0^f\le R_0(\overline{\vartheta }+\delta )$ .

As the function R₀ (ϑ) is increasing and continuous function then by the intermediate value theorem there exists ϑ with $\overline{\vartheta }<\vartheta <\overline{\vartheta }+\delta$ .

$R_0^f=$ > $R_0\left(\overline{\vartheta }\right).$

$R_0^f$ And R₀ (ϑ) are similar as there exists a virus load. The average number of secondary cases $R_0^f$ exceeds the overall number of secondary cases $R_0\left(\overline{\vartheta }\right)$ because there is a medium quantity of virus load present.

Case 3: Strong virus load

(i.e) $\overline{\vartheta }--$ δ >ϑ & $\overline{\vartheta }+$ δ <ϑ_max, (then

$\mathrm{k}(\alpha )=\{\begin{array}{c}1,\mathit{if}0<\alpha \le {\gamma }_{10}{R}_0\left(\overline{\vartheta }\right),\hfill \\ \Gamma \left({\vartheta }^{\prime }\right),\mathit{if}{\gamma }_{10}{R}_0\left(\overline{\vartheta }\right)<\alpha \le {\gamma }_{10}{R}_0\left(\overline{\vartheta }+\delta \right)\hfill \\ 0,\mathit{if}{\gamma }_{10}{R}_0\left(\overline{\vartheta }+\delta \right)<\alpha \le 1\hfill \\ \hfill \end{array}$

Similar to case 2, we have

${\gamma }_{10}{R}_0\left(\overline{\vartheta }\right)\le \mathit{FEV}$ (γ₁₀ R₀ (ϑ)) $\le {\gamma }_{10}{R}_0\left(\overline{\vartheta }+\delta \right)$ ,

$R_0\left(\overline{\vartheta }\right)\le R_0^f\le R_0\left(\overline{\vartheta }+\delta \right).$

Thus, $R_0^f>1$ ; we can infer that the disease will spread more widely.

13 Numerical simulation

Case Study:

Bhilwara is a district in Rajasthan, India which is referred to as the nation’s textile capital. The size of Bhilwara district is about 10,455 square kilometres and the population density of 230 persons per square kilometre. Bhilwara is one of the first Indian towns to become hub for positive infection of corona virus. During the initial stages of the COVID-19 outbreak, the Bhilwara dtrict was one of India’s most severely impacted areas. By March 22, 2020, the Health Department and the district administration in Bhilwara had organised over 850 teams, conducted house-to-house surveys at 56,025 homes and interviewed 280937 people, all within three days of the first positive case. Patients who were tested positive underwent intense contact tracing, and the Health Department created detailed documentation of every person they have come into touch with since being sick. Essentials were arranged by administrators. The majority of the infected cases were found and quarantined as the number of tests grew. The district was divided off by the local government, which also established a containment zone of 1km from the epicentre and a 3km long buffer zone. Continuous screening was done along with cluster mapping.

The initial conditions are S (0)=0.69, E (0)=0.27, Q₁ (0)=0, I (0)=0.03, Q₂ (0)=0.01, R (0)=0 [35].

Example 1. The COVID-free equilibrium point has been depicted numerically by the following figures as an unsuccessful attack. Table 2 shows the parameter values [35].

Table 2

Parametric values for COVID-free equilibrium

Parameter	Value
∧	0.02
δ 1	0.01
δ 2	0.018
δ 3	0.015
β1	0.3
β2	0.05
ρ	0.1
β3	0.3
β4	0.05
β5	0.07
γ1	0.1
γ2	0.02

In Fig. 6a, represents the susceptible population which increases with respect to time, b represents the number of exposed population, where the curve decreases which indicates that the number of individuals from exposed compartment are moving to infected compartment, c represents the number of infected population due to disease, d represents the number of susctible population under home quarantine, e represents the number of population under medical quarantine who were tested positive for the disease and f represents the number of recovered population.

Fig. 6

Solution of Homotopy pertubation method at t = 0 to 100. a. Susceptible population; b. Exposed population; c. Infected population; d. Susceptible population under home quarantine; e. Medical quarantined population of confirmed cases; f. Recovered population; g. dynamical behavior of the model at COVID-free equilibrium.

Example 2. The endemic equilibrium point has been depicted numerically by the following figures as a successful attack. Table 3 shows parameter values [35].

Table 3

Parametric values for endemic point

Parameter	Value
∧	0.02
δ 1	0.02
δ 2	0.03
δ 3	0.035
β1	0.3
β2	0.05
ρ	0.4
β3	0.3
β4	0.02
β5	0.07
γ1	0.1
γ2	0.02

In Fig. 7a, represents the susceptible population at endemic equilibrium point; b, represents the number of exposed population at endemic equilibrium point; c, represents the number of infected population at endemic equilibrium point; d, represents the number of susceptible population under home quarantine at endemic equilibrium point; e, represents the number of population under medical quarantine of confirmed cases at endemic equilibrium point and f represents the number of recovered population at endemic equilibrium point.

Fig. 7

Solution of Homotopy pertubation method at t = 0 to 100. a. Susceptible population; b. Exposed population; c. Infected population; d. Susceptible population under home quarantine; e. Medical quarantined population of confirmed cases; f. Recovered population; g. dynamical behavior of the model at endemic equilibrium.

14 Sensitivity analysis

Our main concern is to know about the pandemic’s potential to infect a population. Sensitivity analysis is used to examine the elements that contribute to the spread and persistence of this disease in the community. We focus on the variables that cause a greater variance in the value of the basic reproduction number.

The sensitivity indices can be defined as follows [31] ${\gamma }_p^x=\frac{\partial x}{\partial p}x\frac{p}{x}.$

The sensitivity index of R₀ can be obtained from the above definition. The Table 1 below shows the sensitive indices of each parameter. The reproduction rate will rise with an increase in the sensitivity index number, and the reproduction rate will fall with a decrease in value, which in turn results in a reduction in the spread of disease. The Table 4 shows the sensitive index values for the parametric values in Table 2 and Fig. 8 shows its graphical representation.

Table 4

Sensitive index for COVID-free equilibrium

Parameter	Sensitive index
ρ	1
∧	1
β1	0.0117
β2	0.0160
β3	0.0768
β4	–0.5967
β5	–0.3535
δ 2	–0.0909
γ1	–0.5050

Fig. 8

Sensitive index for COVID-free equilibrium.

The Table 5 shows the sensitive index values for the parametric values in Table 3 and the Fig. 9 shows it’s the graphical representation.

Table 5

Sensitive index for endemic equilibrium

Parameter	Sensitive index
ρ	1
∧	1
β1	0.0303
β2	0.02375
β3	0.09898
β4	–0.04949
β5	–0.31828
δ 2	–0.13640
γ1	–0.4546

Fig. 9

Sensitive Index for endemic equilibrium.

Fig. 10

Reproduction number at COVID-free equilibrium.

15 Control measures

The only measures to prevent further spread were the time-tested practices of quarantine, case isolation, travel restrictions, and barrier precautions. The first line of protection against new deadly diseases will undoubtedly be such control measures, which have been used for ages. Without the use of models, the qualitative effect of control measures is known: they tend to reduce transmission. Models, on the other hand, offer a strict framework for analysis that gives a sophisticated, quantitative knowledge of the influence of control measures. Instead of being dependent on control and epidemiologic characteristics in a straightforward additive manner, the effectiveness of numerous control measures may depend on them in a complex, nonlinear way. The efficacy of numerous control measures may be complex, nonlinear, and dependent on control and epidemiologic characteristics rather than simply additive. When control measures are too expensive, difficult, or burdensome, detailed modelling can assist in determing whether and when partial relaxation of particular control measures is beneficial.

Remark. The limitation of our research work is data availability. When the data is not exact, we cannot predict the spread of the disease and identify the control parameters. But we can overcome this situation by solving the numerical simulation using some statistical tools, from which we can approximately give conclusions. In our future work, we intend to study the model by applying the different virus loads to the equilibrium points. In addition to this, we are trying to develop a mathematical model with compartments which includes the difficulties faced by corona infected individuals who have undergone severe medication.

16 Conclusion

An SEQ₁ IQ₂ R model has been developed for which local and global asymptotic stability is well established. The estimation of possible extent of the transmission is given by the basic reproduction number. There is negligible disease transmission when the system’s basic reproduction number is less than one, and when it is larger than one, the disease begins to spread more quickly. According to reality, not all sick people spread the disease in the same way, and each person has a unique level of disease infectivity that varies depending on their virus loads. The virus load classification aids in understanding of the fuzzy reproduction rate. The sensitive index provides scientists and the government with a clear picture that they can use to formulate policies to stop the disease’s spread and to take preventative action against and manage future pandemics caused by other new viruses. To establish the analytical results, numerical simulations using actual parric values are carried out.