# Fuzzy Analysis of SVIRS Disease System with Holling Type-II Saturated Incidence Rate and Saturated Treatment

Authors: Hu Zhang; V. Madhusudanan; B. S. N. Murthy; M. N. Srinivas; Biruk Ambachew Adugna
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This article presents a fuzzy SVIRS disease system with Holling type-II saturated incidence rate and saturated treatment, in which all parameters related to population dynamics have been considered as fuzzy numbers. Then, the existence condition and permanence of the SVIRS model have been discussed and we derived disease-free and endemic equilibrium points of the proposed fuzzy system. The local stability conditions of the fuzzy system around these equilibrium points using Routh–Hurwitz criteria are discussed. We also verified global stability around the interior steady state using Lozinskii measure. Computer simulations are provided to understand the dynamics of the proposed system. Parameter analysis is carried out with the help of computer simulation. Fuzzy provides better solution for any disease modeling in many ways like disease detection and transmission, different stages of disease, risk analysis (through parameter analysis), and optimal recovery solutions. Earlier literature acts as background to startup this disease modeling and optimal solutions provided by fuzzy logic is one of the motivational key element behind this fuzzy SVIRS model with Holling type-II and its analysis by both analytical and computer simulation.


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## Body

## 1. Introduction

Epidemiology, which studies the stealing and determinants of contamination hazard in individual communities, is sometimes referred to as the nucleus discipline of community wellbeing. Mathematical epidemic models aid in the understanding of infectious disease transmission and spread, the recognition of factors leading the transmission procedure in order to discover flourishing organize tactics, and the evaluation of observation strategies and interference procedures. Kermack and McKendrick [1] systematically introduced deterministic models for communicable diseases. In their concept, three epidemiological classes are regarded the fundamental aspects describing infectious diseases [2]: the vulnerable group S(t), the infective class I(t) [3], and the recovered class R(t) [4]. In order to better recognize the process of communicable disease transmission [5], some authors have investigated several types of epidemic systems by taking into account dissimilar compartment models such as SI [6], SIS [7], SIR [8], SIRS [9], SEIR [10], SVEIR [11], and others [12].In recent years, controlling infectious diseases has become a more difficult task [13]. Vaccination is one approach for controlling infectious diseases [14]. Vaccination is an important component of health treatments aiming at avoiding the spread of transferable diseases because of its wellbeing and rate efficiency [15]. Certainly, high vaccination rates have resulted in dramatic reductions or even eradication of various vaccine-preventable transferable diseases, as seen in the case of smallpox [16]. Nonetheless, one of the most important aspects of vaccination is its level of safety, both in terms of its capacity to prevent sickness and the longevity of the generated immunity. Some vaccines, such as measles [17], are quite successful, whereas others, such as varicella, are not [18]. Due to medical circumstances, as well as the variation and progress of communicable diseases, the efficacy and levels of fortification offered by a vaccine may gradually reduce with time since the flu virus [19] can evolve quickly. Last year’s influenza vaccine is improbable to protect people from this year’s viral strains. The measles syndrome [20], which is protected by the measles-mumps-rubella vaccine, does not change substantially from time to time, representative that it is just as probable to defend people now as it was 10 years ago. A quantity of vaccines reduces the risk of infection [21], but they do not prevent the disease from forming and spreading in someone who has been vaccinated [22]. Although these flawed vaccines may not completely avoid contagion [23], they may reduce the likelihood of infection or the severity of infection, hence reducing the burden of infectious disease [24]. Many researchers in the mathematical epidemiology literature have looked at epidemic models with poor immunization [25].The incidence rate plays an imperative function in determining the dynamics of epidemic models in mathematical modeling of communicable diseases. Kermack and Mckendrick proposed the prevalence rate in its traditional mass action version in 1927. The interaction term in this incidence rate is a linearly rising function of the numeral of infective, which is not optimal for large inhabitants. Anupama et al. [26] postulated a nonlinear prevalence rate for various psychological impacts as a result of this. Anupama et al. were motivated by behavioural changes: during periods of high frequency the apparent threat of disease might become quite high, leading to drastic changes in people’s behaviour and, as a result, lowering the real risk of infection. Only a few authors have emphasised the importance of nonlinear frequency rates in the study of infectious disease spread dynamics [27]: Anderson and May [28], Wei and Chen [29], Zhang et al. [30], Li et al. [31], Kumar [32, 33], and Goel [34, 35]. A nonlinear incidence rate SIR model was proposed by Zhou and Fan [36]. At this frequency rate, the quantity of effective interactions between infective and susceptible individuals may oversupply due to an overpopulation of infective individuals. Despite the fact that the dynamics of SIR or SIS diseases models with the saturated incidence rate have been extensively studied in the literature [37], few investigations on the saturated treatment function, even in SEIR epidemic models, have been published [38]. In this article, we will investigate the SEIR model with the saturated incidence rate and saturated treatment function in order to better understand the effects of these points on the spread of infectious diseases [39]. We believe that virus-infected hosts are unable to infect other hosts during the incubation period, and that recovered people and vaccinated-treated people have established eternal resistance and are no longer susceptible to infection [36].In recent literature, disease models with fuzzy analysis are trending and inspiring many scientists; likewise, we also studied disease models with risk analysis using fuzzy logic, which is the motivation for our work also [40]. Literature on disease modeling with fuzzy system is also little less in number, so we proposed the fuzzy SVIRS disease system with Holling type-II functional response. The current model is studied under fuzzy system and studied analytically and numerically using mathematical and computer software tools, respectively [41]. This work focused on parameter analysis also to analyze all parameters numerically to identify, which parameter influences and effects the system strongly [42]. The current work focused on the fuzzy system for the SVIRS disease model with Holling type-II functional response. All disease models are driven from the basic disease model SIR and SIER [43]. These models are very basic in nature. To capture disease detection, disease transmission, and risk analysis, we need to study the model with little complex constraints which may be interactions (like functional responses) which are complex in nature [44]. To serve an optimal solution for disease detection, disease transmission and its stages and recovery status and its stages can be achieved greatly by fuzzy systems. The fuzzy system works greatly when the scenario is complex in nature, uncertainty in scenario, and vague data are involved.The parameters used to mathematically express biological events are typically assumed to be accurate. Researchers have only made a few attempts to include environmental uncertainty into their research. The application of diverse mathematical approaches and concepts aids in the better understanding of biological and physical processes. In biological modeling, interval parameters or fuzzy parameters should be employed more frequently than recent attempts due to the realistic scenario. The fuzzy set theory, proposed by Zadeh [40], can be used to account for uncertainty in biological data. The application of fuzzy logic and fuzzy sets in biological systems offers a lot of potential, but there are not many of them. Reference [45] has several examples of fuzzy mathematics applications. Angalaeswari et al. describe some epidemic models that take into account parameter uncertainty and population heterogeneity [45]. We investigated a fuzzy SVIRS diseases system with Holling type-II incidence rate and saturated therapy based on the motivation of [30–36, 46], in which all parameters linked to population dynamics were considered as fuzzy numbers. The current work is definitely different previous works in view of parameter analysis, which says the control parameters for disease transmission and recovery greatly. Fuzzy analysis definitely added strength to the current work which shapes it as a novel innovative and informative study. Certainly, this work can contribute qualitative information and notable conclusions to the literature.


## 2. Preliminaries of Fuzzy Analysis

A fuzzy set is a collection of things in which there is no clear distinction between those that belong to the group and those that do not. We will go over some basic notations for fuzzy sets in this section [46]. We recommend Natrayan [47] and Klir and Yuan [48] for further information on fuzzy sets.Definition 1.
ζ-cut of a fuzzy number: a nonempty set P, a fuzzy set in P is a map f:P⟶0,1. A ζ-cut of a fuzzy number f˜ in P is denoted by fζ=flζ,fuζ, where f1ζandfuζ are the lower and upper bounds of the closed interval in that order and is defined as the following fuzzy set [49]:(1)fζ=pΞf˜p≥ζ,p∈Pwhereζ∈0,1.
Figure1 shows the ζ-cut of a triangular fuzzy number. Figure 2 shows the triangular fuzzy number.Figure 1
ζ-cut of a triangular fuzzy number.Figure 2
Triangular fuzzy number.Definition 2.
Triangular fuzzy number.
A triangular fuzzy number is a continuous membership functionΞf˜p:P⟶0,1 and is denoted by f˜.Definition 3.
Utility function method.
The utility function is defined as the weighted sum of the objectives:(2)U=∑k=1nσifi,σi≥0,where σi is a scalar that represents the relative importance of the objectives fi and subject to the condition ∑k=1nfi=1.


## 3. Fuzzy Model

LetS˜t, V˜t, I˜t, and R˜t denote the population densities of fuzzy susceptible, vaccinated, infected, and recovered human in the environment at time t, respectively. By using the concept on the fuzzy initial value problem [1] and differentials of fuzzy functions [2] and considering a set of fuzzy differential equations regarding the following SVIRS model [50], Table 1 shows the physical description of the symbols. Figure 3 shows the schematic diagram of the SVIRS fuzzy model.(3)DS˜=Δ˜−Ω˜S−α˜SI1+β˜I−γ˜S=ϕ˜R,DV˜=Ω˜S−χ˜SI1+β˜I−γ˜V,DI˜=α˜SI1+β˜I+χ˜SI1+β˜I−γ˜+η˜+ψ˜I−m˜I1+n˜I,DR˜=ψ˜I+m˜I1+n˜I−ϕ˜R−γ˜R,where D≡d/dt.Table 1
Physical description of the symbols.
Fuzzy parametersDescription of the symbolsΔ˜The recruitment rate of susceptibleα˜The vigor of infectionβ˜The reticence actions taken by the infectedχ˜The rate at which vaccinated persons become infectedΩ˜The transmission rate from susceptible individuals to vaccinated onesϕ˜The transmission rate from recovered individuals to susceptible onesψ˜The transmission rate from infected individuals to recovered onesγ˜The natural fatality rate of all the individualsη˜The death rate due to diseasem˜The treatment raten˜The rate of limitation in medical resourceFigure 3
A schematic diagram of the SVIRS fuzzy model.Let the solution of the fuzzy system (1) can be defined as DZ˜σ=DZlσ,DZuσ, then the deterministic system of the fuzzy model (1) is given by(4)DSlσ=Δlσ−δuσ+γuσS−αuσSI1+βlσI+ϕlσ,DSuσ=Δuσ−δlσ+γlσS−αlσSI1+βuσI+ϕuσ,DVlσ=δlσS−χuσVI1+βlσI−γuσV,DVuσ=δuσS−χlσVI1+βuσI−γlσV,DIlσ=αlσSI1+βuσI+χlσVI1+βuσI−γuσ+ηuσ+ψuσI−muσI1+blσI,DIlσ=αuσSI1+βlσI+χuσVI1+βlσI−γlσ+ηlσ+ψlσI−mlσI1+buσI,DRlσ=ψlσI+mlσI1+nuσI−ϕuσ+γuσR,DRuσ=ψuσI+muσI1+nlσI−ϕlσ+γlσR.The above system of differential equations can be solved using the UFM principle as follows [51]:(5)DS=σ1DSlσ+σ2DSuσ,DV=σ1DSlσ+σ2DSuσ,DI=σ1DSlσ+σ2DSuσ,DR=σ1DSlσ+σ2DSuσ,where σ1 and σ2 are two weight functions such that σ1+σ2=1 and σ1+σ2≥0; then, (5) can be written as(6)DS=u11−u12SI−u13S+u14R,DV=u21S−u22VI−u23V,DI=u31SI−u32VI−u13I+u34I,DR=u41I−u42I−u43R,where(7)u11=σ1Δlσ+σ2Δuσ;u12=σ1αuσ1+βlσI+σ2αl1+βuσI;u14=σ1ϕlσ+σ2ϕuσ,u13=σ1δuσ+γuσ+σ2δlσ+γuσ;u21=σ1δlσ+σ2δuσ,u22=σ1χuσ1+βu+σ2χlσ1+βl;u23=σ2γuσ+σ2γlσ;u31=σ1αlσ1+βu+σ2αuσ1+βl,u32=σ1χuσ1+βu+σ2χlσ1+βl;u34=σ1muσ1+nlσI+σ2mlσ1+nuσI,u33=σ1γuσ+ηuσ+ψuσ+σ2γlσ+ηlσ+ψlσ;u41=σ1ψlσ+σ2ψuσ,u22=σ1mlσ1+nuσ+σ2muσ1+nlσ;u43=σ1ϕuσ+σ2ϕlσ;u44=σ1γuσ+σ2γlσ.


## 4. Positivity and Boundedness of the SVIRS Fuzzy Model

In this section, all solutions of the proposed system (3) have been bounded [52]. First, we use Lemma 1 to verify the positivity of system (3).Lemma 1.
Ifu13−u21>0,u43−u14>0,u34−u33−u41−u42>0,u31−u12>0,u32−u22>0 provided that δlσ+γlσ>δuσ,ϕlσ>ϕuσ,mlσ/1+nuσ+γlσ+ηlσ+φlσ>φuσ+muσ/1+nlσ,αuσ/1+βlσ>δuσ, then the solutions St,Vt,It, and Rt of system (3) are positives.Proof.
Consider(8)u13−u21=σ1δuσ+γuσ+σ2δlσ+γlσ−σ1δlσ−σ2δuσ=σ12δuσ+γuσ−2δlσ+γlσ+δlσ+γlσ−σ2δuσ.
Now,σ1=0, we have u31−u21=δlσ+γlσ−δuσ; then, u13−u21 will be positive for σ1=0 if(9)δlσ+γlσ>δuσ.
Therefore, from equation (8), it is obtained that(10)ddσ1u13−u21=2δuσ+γuσ−2δlσ+γlσ>0,∀σ1∈0,1.
So,u13−u21 is an increasing function with respect to σ1, and it will be positive if condition (9) holds. In the similar way, it can be proved that u43−u14,u33+u34−u41−u42 is an increasing function with respect to σ1, and it will be positive if the condition ϕlσ>ϕuσ+ψ1σ>mlσ/1+nuσ+γlσ+ηlσ+ψlσ>ψuσ+muσ/1+nlσ,αuσ/1+βlσ>δuσ,χuσ/1+βlσ>χlσ/1+βuσ holds.Theorem 1.
All solutions of system (3) are bounded in the region R+4 provided that(11)u31>u12,u32>u22,μ=minu23,u13−u21,u43−u14,u33+u34−u41−u42.Proof.
Let us define a functionG=S+V+I+R.
Now, by separating and simplifying with regard to timet, we have(12)DG=DS+DV+DI+DR,DG=u11−u23V+u21−u13S+u41+u42−u34−u33I+u14−u43R+u31−u12SI+u32−u22VI.
For a positive real numberμ,(13)DG+μG=u11+μ−u23V+μ−u13−u21S+μ−u34+u33−u41−u42I+μ−u43−u14R+u31−u21SI+u32−u22VI.
Now, according to Lemma1, if μ=minu23,u13−u21,u43−u14,u34+u33−u41−u42, then the above equation becomes DG+μG≤u11. Solving this, we get G≤u11/μ+ke−μt. As t⟶∞, we have G≤u11/μ. So, it can show St≤u11/μ,Vt≤u11/μ,It≤u11/μ,Rt≤u11/μ. This establishes that the system’s solution is bounded.
The solutions of system (3) are bounded in the region R+4 provided that a31>a12, a32>a22 and μ=minu23,u13−u21,u43−u14,u33+u34−u41−u42σ1=0.


## 5. Existence and Stability Analysis

In this section, we look at the presence and stability of the fuzzy system’s nonnegative equilibrium point (3). In a fuzzy system (3), there are two nonnegative equilibrium points. The following are the existence and stability conditions for them [53]:(i)
Disease free equilibrium pointP1S0,0,0,0=u11/u13,0,0,0(ii)
Endemic equilibrium pointP2S∗,V∗,I∗,R∗Here,S∗=u11u43+u14u11+u14u42/u43u13+u43u12I∗, V∗=u21u11u43+u21u14u41+u21u14u42/u43u12u22I∗+u43u13u22I∗+u43u12u23I∗+u43u13u23,(14)R∗=u41+u42u43I∗,Γ1I∗2+Γ2I∗+Γ3=0,where(15)Γ1=u12u22u33u43+u12u22u34u43,Γ2=u13u22u33u43+u13u22u34u43+u12u23u33u43+u12u23u34u43−u11u22u31u43−u22u31u14u41−u14u22u31u42,Γ3=u13u23u33u43+u13u23u34u43−u11u31u23u43−u14u31u23u41−u14u31u23u42−u11u21u32u43−u14u21u32u41−u14u21u32u42.Theorem 2.
Fuzzy system (3) is locally asymptotically stable at disease free equilibrium point P1S0,0,0,0 if u132−u11u13>0andu13u44−u11u31>0; otherwise, it is unstable.Proof.
The characteristic equation of the fuzzy system (3) at disease free equilibrium point P1S0,0,0,0 is(16)λ4+∑1λ3+∑2λ2+∑3λ+∑4=0,where(17)∑1=u13+u23+u33+u43+u44−u31S0,∑2=u13u43+u13u33+u13u44+u23u43+u23u33+u23u44+u13u23−u31u13S0−u23u31S0,∑3=u13u33u43+u13u43u44+u23u33u43+u23u43u44+u23u43u13+u23u33u13+u23u13u44−u23u31u43S0−u23u31u13S0,∑4=u13u33+u13u44−u31u13S0.
The required and sufficient requirements for local stability of a disease-free equilibrium pointp1S0,0,0,0 according to Routh–Hurwitz criteria if ∑1>0, ∑3>0, ∑4>0, ∑3∑1∑2−∑3>∑12∑4, and ∑4∑1∑2∑3−∑12∑4−∑32>0. It is evident that u132−u11u13>0 and u13u44−u11u31>0.Theorem 3.
Fuzzy system (3) is locally asymptotically stable at endemic equilibrium point p2S∗,V∗,I∗,R∗ifS∗<maxu13/u31,u33/u31,V∗<maxu23/u32,u34/u32,u33−u31>0,u22−u12>0,u12u23u33>u14u31u41+u42,u12u23u33u43>u14u22u31u41+u42; otherwise, it is unstable [54].Proof.
The characteristic equation of fuzzy system (3) at the endemic equilibrium point p2S∗,V∗,I∗,R∗ is(18)λ4+Θ1λ3+Θ2λ2+Θ3λ+Θ4=0,where(19)Θ1=u13+u23+u33+u34+u43+u12I∗+u22I∗−u31S∗−u32V∗,Θ2=u13u23+u13u33+u13u34+u23u43+u23u34+u23u43+u33u43+u34u43+u12u23I∗+u12u33I∗+u12u34I∗+u12u43I∗+u13u22I∗+u22u33I∗+u22u34I∗+u22u43I∗−u13u31S∗−u23u31S∗−u31u43S∗−u13u32V∗−u23u32V∗−u32u43V∗I∗−u22u31S∗I∗,Θ3=u13u23u33+u13u23u34+u13u23u43+u13u33u43+u13u43u34+u33u23u43+u43u23u34−u31u14u42I∗−u14u31u41I∗+u22u34u43I∗+u22u33u43I∗+u13u22u43I∗+u13u22u34I∗+u13u33u22I∗++u22u34u43I∗+u22u33u43I∗+u13u22u43I∗+u13u22u34I∗+u13u33u22I∗+u13u33u22I∗+u12u34u43I∗+u12u33u43I∗+u12u23u33I∗−u31u23u43S∗−u23u43u32V∗+u22u34u43I∗−u13u23u31S∗−u13u23u32V∗−u13u31u43S∗−u13u43u32V∗+u12u22u34I∗2+u12u22u43I∗2−u12u23u32V∗I∗−u12u31u43S∗I∗−u31u22u43S∗I∗+u12u23u32VI∗2+u21u12u32S∗I∗+u12u22u33S∗I∗2+u12u31u43S∗I∗−u12u22u31S∗I∗2,Θ4=u13u23u33u43+u13u23u33u43+u13u23u34u43+u12u23u34u43I∗−u14u23u31u41I∗−u14u23u31u42I∗+u13u22u33u43I∗+u13u22u34u43I∗−u14u21u32u41I∗−u14u23u32u42I∗−u13u23u31u43S∗−u13u23u32u43V∗+u13u23u33u43I∗−u13u23u31u43S∗−u13u23u32u43V∗+u12u22u32u43V∗I∗2−u13u22u31u43S∗I∗−u12u23u32u43V∗I∗+u12u22u33u43I∗2+u12u32u22u43V∗I∗2+u12u22u34u43I∗2+u12u21u32u43S∗I∗−u14u22u31u41I∗2−u14u22u31u42I∗2.
The required and sufficient requirements for local stability of a disease-free equilibrium pointp1S0,0,0,0 according to the Routh–Hurwitz criteria if Θ1>0,Θ3>0,Θ4>0, Θ3Θ1Θ2−Θ3>Θ12Θ4 and Θ4Θ1Θ2Θ3−Θ12Θ4−Θ32>0.
It is evident that(20)S∗<MAXu13u31,u33u31,V∗<MAXu23u32,u34u32,u33−u31>0,u22−u12>0,u12u23u33>u14u31u41+u42,u12u23u33>u14u31u41+u42.Theorem 4.
Fuzzy system (3) is globally asymptotically stable at the endemic equilibrium point p2S∗,V∗,I∗,R∗. If ℵ1>0, where(21)ℵ1=u31ℵ+u32ℵ−u33−u34+minu12ℵ−u13−u22ℵ−u23−maxu22ℵ,u12ℵ,−u21−u31ℵ+minu22ℵ+u23,−u32,ℵu22−u32+u33+u34.Proof.
Let us consider the subsystem of fuzzy system (3):(22)DS=u11−u12SI−u13S+u14R,DV=u21S−u22VI−u23V,DI=u31SI+u33VI−u33I−u34I.
Let N be the variation matrix of the system (8) is(23)N=−u12I∗−u130−u12S∗u21−u22I∗−u23−u22V∗u13I∗u32u31S∗+u32V∗−u33−u34.
Let N[2] be the second additive compound matrix of N, we have(24)N=−u12I∗−u13−u22I∗−u23−u22V∗−u12S∗u32−u22I∗−u23u31S∗+u32V∗−u33−u340u13I∗u21−u22I∗−u23u31S∗+u32V∗−u33−u34.
Let us consider the function∏x=∂∏/∂x=diagS•/I−SI•/I2,S•/I−SI•/I2,S•/I−SI•/I2.
Thus, we have∏x∏−1=diagS•/I−SI•/I,S•/S−I•/I2,S•/I−I•/I and ∏N2∏−1=N2, so that(25)Ξ=∏x∏1+∏N2∏−1=Ξ11Ξ12Ξ21Ξ22,where Ξ11=S•/S−I•/I−u12I∗−u13−u22I∗−u23;Ξ12=−u22V∗u12S∗;Ξ21=u21u13I∗T.(26)Ξ22=S•S−I•I−u22I∗−u23−u22V∗u32S•S−I•I+u31S∗−u32V∗−u33−u34.
We consideredℜ as the Lozinskii measure by Martin [7] with respect to the norm defined in three-dimensional space x,y,z=maxx,y+z, where x,y,z is any vector in R3. Hence, ℜP≤supP1,P1, where Pi=ℜΞii+Ξij for i = 1, 2 and i≠j and ℜ1 denotes the Lozinskii measure with respect to the Γ1 vector norm. The Lozinskii measure for the matrix Ξ is given by(27)ℜ1Ξ11=S•S−I•I−−u12I∗−u13−u22I∗−u23;Ξ12=maxu22V∗,u12S∗Ξ21=u21u13I∗T;ℜ1Ξ22=S•S−I•I−MINu22I∗+u23,V∗u22−u32+u33+u34.
Here,P1=ℜ1Ξ11+Ξ12=S•/S−I•/I−u12I∗−u13−u22I∗−u23+MAXu22V∗+u12S∗ and(28)P2=ℜ1Ξ22+Ξ21=S•S−I•I−u21I∗−minu22I&lowast;+u23,−u32,Vu22−u32+u33+u34.
Thus, we haveI•/I=u31S+u32V−u33−u34, so that(29)ℜ1Ξ≤S•S−u31ℵ−u32ℵ+u34−minu12ℵ−u13−u22ℵ−u23−maxu22ℵ,u12ℵ,−u21−u31ℵ+minu22ℵ+u12,−u32,ℵu22−u32+u33+u34.
Sinceℵ=mininfS∗t,infV∗t,infI∗t, and hence ℜ1Ξ≤S•/S−ℵ1, where(30)ℵ1=u31ℵ+u32ℵ−u33−u34+minu12ℵ−u13−u22ℵ−u23−maxu22ℵ,u12ℵ,−u21−u31ℵ+minu22ℵ+u23,−u23,ℵu22−u32+u33+u34.
Now, integratingℜ1Ξ w.r.t t’ on both sides between the limits 0 to t and on letting t⟶∞, we obtain ∫0tℜΞdt=logSt/S0−ℵt⇒limt⟶∞supsup1/t∫0tℜ1Ξdt<−ℵ1, if ℵ1>0. Therefore, the interior equilibrium point p2S∗,V∗,I∗,R∗ will be globally asymptotically stable.


## 6. Numerical Simulation

In this numerical section, the dynamical behavior of the fuzzy SVIRS system is investigated.Example 1.
The case of endemic equilibrium is illustrated for the following numerical data.(31)Δ∼=6,15;α∼=0.01,0.09;β∼=0.01,0.3;χ∼=0.005,0.02;Ω∼=0.3,0.6;Φ∼=0.1,0.5;Ψ∼=0.1,0.6;γ∼=0.05,0.15;η∼=0.01,0.03;m∼=1,8;n∼=1,12.
Figure4 shows solution curves of the SVIRS disease system with the attributes in Example 1. Figure 5 shows time series evaluation of susceptible population for the recruitment rates Δ=8,12,16. Figure 6 shows time series evaluation of susceptible population for α=0.01;0.03;0.05;0.07;0.09. Figure 7 shows time series evaluation of infected population for χ=0.007,0.009,0.01,0.02,0.03.
Figure8 shows time series evaluation of susceptible population for β=0.05;0.10;0.15;0.20;0.25. Figure 9 shows time series evaluation of vaccinated population for β=0.05;0.10;0.15;0.20;0.25. Figure 10 shows time series evaluation of infected population for β=0.05;0.10;0.15;0.20;0.25. Figure 11 shows time series evaluation of vaccinated population for χ=0.007,0.009,0.01,0.02,0.03. Figure 12 shows time series evaluation of infected population χ=0.007,0.009,0.01,0.02,0.03. Figure 13 shows time series evaluation of infected population for η=0.002,0.006,0.01,0.02,0.03.
Figure14 shows time series evaluation of recovered population for ψ=0.10,0.15,0.20,0.25,0.30. Figure 15 shows time series evaluation of susceptible population for Ω=0.3,0.5,0.7. Figure 16 shows time series evaluation of vaccinated population for Ω=0.3,0.5,0.7. Figure 17 shows time series evaluation of susceptible population for γ=0.05;0.08;0.1;0.3;0.5.
Figure18 shows time series evaluation of vaccinated population for γ=0.05;0.08;0.1;0.3;0.5. Figure 19 shows time series evaluation of infected population for γ=0.05;0.08;0.1;0.3;0.5. Figure 20 shows time series evaluation of recovered population for γ=0.05;0.08;0.1;0.3;0.5.
Figure21 shows time series evaluation of susceptible population for ϕ=0.04;0.08;0.2;0.4;0.6. Figure 22 shows time series evaluation of vaccinated population for ϕ=0.04;0.08;0.2;0.4;0.6. Figure 23 shows time series evaluation of recovered population ϕ=0.04;0.08;0.2;0.4;0.6. Figure 24 shows time series evaluation of infected population for m=0.5,3.5,6.5,9.5,12.5. Figure 25 shows time series evaluation of Recovered population for m=0.5,3.5,6.5,9.5,12.5. Figure 26 shows time series evaluation of recovered population for n=2,5,8,10,14. Figure 27 shows time series evaluation of recovered population for n=2,5,8,10,14.Figure 4
Solution curves of the SVIRS disease system with the attributes in Example1.Figure 5
Time series evaluation of susceptible population for the recruitment ratesΔ=8,12,16.Figure 6
Time series evaluation of susceptible population forα=0.01;0.03;0.05;0.07;0.09.Figure 7
Time series evaluation of infected population forα=0.01;0.03;0.05;0.07;0.09.Figure 8
Time series evaluation of susceptible population forβ=0.05;0.10;0.15;0.20;0.25.Figure 9
Time series evaluation of vaccinated population forβ=0.05;0.10;0.15;0.20;0.25.Figure 10
Time series evaluation of infected population forβ=0.05;0.10;0.15;0.20;0.25.Figure 11
Time series evaluation of vaccinated population forχ=0.007,0.009,0.01,0.02,0.03.Figure 12
Time series evaluation of infected population forχ=0.007,0.009,0.01,0.02,0.03.Figure 13
Time series evaluation of infected population forη=0.002,0.006,0.01,0.02,0.03.Figure 14
Time series evaluation of recovered population forψ=0.10,0.15,0.20,0.25,0.30.Figure 15
Time series evaluation of susceptible population forΩ=0.3,0.5,0.7.Figure 16
Time series evaluation of vaccinated population forΩ=0.3,0.5,0.7.Figure 17
Time series evaluation of susceptible population forγ=0.05;0.08;0.1;0.3;0.5.Figure 18
Time series evaluation of vaccinated population forγ=0.05;0.08;0.1;0.3;0.5.Figure 19
Time series evaluation of infected population forγ=0.05;0.08;0.1;0.3;0.5.Figure 20
Time series evaluation of recovered population forγ=0.05;0.08;0.1;0.3;0.5.Figure 21
Time series evaluation of susceptible population forϕ=0.04;0.08;0.2;0.4;0.6.Figure 22
Time series evaluation of vaccinated population forϕ=0.04;0.08;0.2;0.4;0.6.Figure 23
Time series evaluation of recovered population forϕ=0.04;0.08;0.2;0.4;0.6.Figure 24
Time series evaluation of infected population form=0.5,3.5,6.5,9.5,12.5.Figure 25
Time series evaluation of recovered population form=0.5,3.5,6.5,9.5,12.5.Figure 26
Time series evaluation of recovered population forn=2,5,8,10,14.Figure 27
Time series evaluation of recovered population forn=2,5,8,10,14.


## 7. Conclusions and Remarks

In this study, we provide a fuzzy SVIRS epidemic model with a Holling type-II incidence rate and saturating treatment, in which all population dynamics parameters are handled as fuzzy integers. The SVIRS model’s existence and durability were next investigated, and the disease-free and endemic equilibrium points of the proposed fuzzy system were computed. The Routh–Hurwitz criteria are used to talk about the fuzzy system’s local stability conditions around these equilibrium locations. Using the Lozinskii metric, global stability surrounding the internal steady state was also verified. To further understand the dynamics of the suggested model, numerical simulations are provided. The following interpretations are based on the numerical simulations of Section6:(i)
Figure4 shows how the population of solution curves (population) changes over time.(ii)
Figure5 depicts how vulnerable populations change over time at various recruitment rates.(iii)
Figures6 and 7 show the variations in susceptible and infected populations respectively for various levels of infection force.(iv)
Figures8, 9, and 10 show how sensitive, vaccinated populations behave over time in days for various inhibitory tactics implemented by the infected. Also, as the sick person’s inhibitory measures become more effective, the number of vulnerable and vaccinated people grows. However, as seen in Figure 10, the number of infection individuals is decreasing as the population grows.(v)
The number of people who have been vaccinated has decreased as the values ofχ have risen (Figure 11), whereas the number of people who have been infected has increased (Figure 12).(vi)
When the death rate from disease (η) rises, the number of sick people decreases (Figure 13).(vii)
Figure14 illustrates that the number of recovered individuals grows as the transmission rate from infected persons to recovered individuals (ψ) increases.(viii)
When the transmission rate from susceptible to vaccinated (Ω) is growing, the number of susceptible and vaccinated persons is falling and increasing, respectively (Figures 15 and 16).(ix)
The number of vulnerable (Figure17), vaccinated (Figure 18), infected (Figure 19), and recovered (Figure 20) persons decreases as the natural death rate (γ) rises.(x)
The number of susceptible (Figure21) and vaccinated (Figure 22) individuals increases when the transmission rate from recovered individuals to susceptible (ϕ) increases, while the number of recovered individuals (Figure 23) decreases.(xi)
When the treatment rate (m) rises, the number of infected people (Figure 24) falls while the number of recovered people (Figure 25) rises.(xii)
In addition, whenever the rate of medical resource scarcity (m) rises, the number of infected (Figure 26) and recovered (Figure 27) individuals rises and falls, respectively.(xiii)
All parameters play major role in the proposed fuzzy SVIRS, which means the current system can be treated as one of the better solutions for any type of disease models. Previous works focused on stability analysis analytically and numerically, and some are focused on delay dynamics. Delay-induced mathematical model and its analysis also greatly helps us in detecting disease controlling parameters and to detect parameters which allows us to reduce disease transmission.(xiv)
The current work is mainly focused on deterministic (without randomness) approach both analytically and numerically. We can extend this model to capture rich dynamics of system (stochastic model) under noise (randomness, environmental factors around us) by both analytically (using Fourier transform tool) and numerically (using MATLAB software tool). We can extend this model to study the spatio-temporal analysis by constructing a diffusive model using partial differential equations. These can be considered as one of our future projects [51].



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