# Fuzzy Modeling for the Dynamics of Alcohol-Related Health Risks with Changing Behaviors via Cultural Beliefs

Authors: Maranya M. Mayengo; Moatlhodi Kgosimore; Snehashish Chakraverty
Journal: Journal of Applied Mathematics
 (2020)

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## Abstract

In this paper, we propose and analyze a fuzzy model for the health risk challenges associated with alcoholism. The fuzziness gets into the system by assuming uncertainty condition in the measure of influence of the risky individual and the additional death rate. Specifically, the fuzzy numbers are defined functions of the degree of peer influence of a susceptible individual into drinking behavior. The fuzzy basic risk reproduction numberR0f is computed by means of Next-Generation Matrix and analyzed. The analysis of R0f reveals that health risk associated with alcoholism can be effectively controlled by raising the resistance of susceptible individuals and consequently reducing their chances of initiation of drinking behavior. When perceived respectable individuals in the communities are involved in health education campaign, the public awareness about prevailing risks increases rapidly. Consequently, a large population proportion will gain protection from initiation of drinks which would accelerate their health condition into more risky states. In a situation where peer influence is low, the health risks are likely to be reduced by natural factors that provide virtual protection from alcoholism. However, when the perceived most influential people in the community engage in alcoholism behavior, it implies an increase in the force of influence, and as such, the system will be endemic.


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## Body

## 1. Introduction

Alcohol consumption that leads to health-related epidemics has been one of the leading causes of mortality of individuals worldwide. In Tanzania, just like in other developing countries, the situation gets worse as alcohol consumption becomes an important part of community social activities. However, available literature cite alcohol consumption behavior is one among the risk factors for various health challenges in Tanzanian population [1–9]. While communities identify themselves with alcohol consumption, a large part of the religious population forbid its uptake. Being religious therefore plays an important role in promoting health and molding an individual’s behaviors [10, 11]. Increased health defect cases such as malnutrition, chronic pancreatitis diseases, liver cirrhosis, and some types of cancer have been found to be associated to alcoholic behavior [12–19]. While in some parts of the world individuals believed that a low level of alcohol consumption prevents thrombosis [20], recent studies challenge the belief. For example, Griswold et al. [13] established that any amount of alcohol consumption leads to health risks.Mathematical modeling has been very useful in providing solutions to epidemiology-related problems for both autonomous and nonautonomous system models [12, 14, 15, 21–29]. While autonomous systems literally mean the use of constant controls, a nonautonomous system occurs when a continuous variable control is used in the system as a function of time [22]. However, an application of fuzzy set theory in solving dynamical systems has recently been an interesting research area [30–33].In this paper, we use an application of fuzzy set theory approach to modify the basicSPLMAR model developed in system (1) into a fuzzy model. Two main distinguishing aspects of the model are as follows: social cultural beliefs as an integral part of the society [34–36] and the staged process in which alcoholism behaviors take in the spread of health risks [12, 14, 15, 24, 26, 29] are considered.


## 2. Fuzzy Model Formulation

This paper presents crisp model (1) with six time-dependent state variables described as follows: susceptible population, St—comprising of individuals susceptible to alcohol drinking habits; protected population, Pt—comprising of individuals with strong religious beliefs that provide them with virtual protection from alcohol drinking; and low-risk population, Lt—individuals at low risk who drink alcoholic beverages once in a while. Others include medium/moderate-risk population, Mt—individuals at medium/moderate risk who drink alcoholic beverages regularly; alcoholics, At—high-risk drinkers or alcohol addicts who have developed physical or psychological dependence on alcoholic beverages; and recovered population, Rt—former drinkers who have voluntarily quit alcohol drinking due to various reasons. The model considers multirisk levels established under the drinking cultures of Tanzanian population with active religious beliefs by using constant model parameters defined in Table 1(1)Ṡ=−μ+γ1+λS+γ2P+ωR+1−ϕπN,Ṗ=ϕπN+γ1S−μ+γ2P+νL+τM+ψA,L̇=λρS−ν+μ+σL,Ṁ=λ1−ρS+σL−τ+μ+δ+ξM,Ȧ=δM−μ+α+η+ψA,Ṙ=ξM+ηA−ω+μR,with S0>0, P0>0, L0≥0, M0≥0, A0≥0, R0≥0, and N=S+P+L+M+A+R.Table 1
Model parameters and their description.
SymbolDescriptionsπPer capita recruitment rateφThe proportion of recruitment joining the protected populationμNatural mortality rateλForce of peer influence to induce drinkingβThe measure of influence of the risky individualscThe contact rate between a susceptible member and a drinker necessary to convince the susceptible member to drinkθ1The chances of becoming an alcoholic after successful influence of a moderate-risk drinkerθ2The chances of becoming an alcoholic after successful influence of a high-risk drinkerρThe proportion of susceptible individual recruited to the low-risk drinking populationωThe rate at which recovered individuals join susceptible compartmentγ1The rate at which susceptible population joins protected compartmentγ2Virtual protection wane rateνThe rate at which low-risk population joins protected compartmentτThe rate at which moderate-risk population joins protected compartmentψThe rate at which high-risk population joins protected compartmentσProgressive rate from low to moderate risk compartmentsδProgressive rate from moderate- to high-risk compartmentsξRecovery rate for moderate-risk populationαAlcohol-induced fatality rateηRecovery rate for high-risk populationThe model is developed on the fact that a nonalcoholic drinker acquires alcohol drinking habits through regular social contacts [12, 37] at a rate of λ defined as
(2)λ=cβL+θ1M+θ2AN.We propose a fuzzy model usingx as the variable describing the degree of peer influence of a susceptible individual to initiate drinking behavior. The model development is made under the assumption that both β and α are the functions of x. The physical meaning of parameters used in the model determined the choice of the two parameters. While β is attributed to the spread of health risks associated with alcoholism in the community, α translates the consequences of health risks by means of additional death rate. The fuzzy numbers β=βx and α=αx represent the likelihood that a susceptible individual will drink alcohol after prolonged contact with drinking individuals and the additional death rate induced by alcoholism, respectively. A fuzzy number may be defined as a generalization of a real number referring to a connected set of possible values, rather than referring to one single value, where each possible value is weighted between 0 and 1. Thus, the fuzzy system (3) of differential equations is established
(3)Ṡ=1−ϕπN−μ+γ1+λxS+γ2P+ωR,Ṗ=ϕπN+γ1S−μ+γ2P+νL+τM+ψA,L̇=λxρS−ν+μ+σL,Ṁ=λx1−ρS+σL−τ+μ+δ+ξM,Ȧ=δM−μ+αx+η+ψA,Ṙ=ξM+ηA−ω+μR,with nonnegative initial conditions of the state variables: S0>0, P0≥0, L0≥0, M0≥0, A0≥0 and R0≥0 where
(4)N=S+P+L+M+A+R,Ṅ=π−µN−αxA.It is further assumed that the population is homogeneous and, at the initial level, drinking behavior is acquired by choice. Further, it is assumed that individuals in the recovered population do not develop a permanent immunity system against alcohol drinking. Similarly, individuals in the protected compartment acquire a nonpermanent virtual protection from alcohol drinking for their entire life in the compartment. On the other hand, a susceptible drinker acquires alcohol drinking habits through prolonged social contacts with drinkers in shared drinking venues. The force of peer influence is redefined into(5)λx=cβxL+θ1M+θ2AN.


## 3. Analysis of the Fuzzy System

Following Verma et al. [38] and Nandi et al. [32], we study the fuzzy SPLMAR model (3) and provide the model analysis and interpretations. The parameters βx and αx can be described through two fuzzy membership functions. For example, considering a situation where a drinking individual interacts with a susceptible member, the minimum amount of the degree of peer influence, x=xmin, is required to have an impact on a susceptible member. That is to say, when x<xmin, the impact of behavioral influence is considered negligible. The quantity xmin is taken as a parameter whose exact value would depend upon both the attitude, public opinions towards the drinking behavior or the drinking individual, and willingness of a susceptible individual to conform with the peer pressures. As x increases, the behavior inducement rate increases to the maximum which is equal to unity at x≥x0. Further, it is also assumed that the degree of peer influence is bounded above where x=xmax marks an upper bound. Therefore, the values of x with an effect to the system lies in the interval of xmin≤x≤xmax. The fuzzy membership function for the fuzzy number βx is given by
(6)βx=0,ifx<xmin,x−xminx0−xmin,ifxmin≤x≤x0,1,ifx0<x<xmax.Similarly, we also assume that the addition death rate,αx, is a fuzzy number as it occurs due to the risk “transmission.” When x<xmin, we have a negligible amount of risk transmissions implying that the additional death rate, αx, gets to the minimum level, say αx<xmin=α0. As x increases, the additional death rate increases and gets the highest value at x=x0. Since the additional death rate may not reach αx=1 as its highest score due to several limitations, we let the maximum value αx=1−u for some real number u such that 0<u<1−α0. We therefore establish the fuzzy membership function of αx as follows:
(7)αx=α0,if0≤x<xmin,α0+1−u−α0x0−xminx,ifxmin≤x<x0,1−u,ifx0≤x≤xmax.The graphs of membership functionsβx and αx are presented in Figures 1(a) and 1(b), respectively. For the model to be more realistic, we assume that the degree of peer influence of the studied group Π differs from among drinking individuals depending on their social influence in the community. Consequently, Π can be considered a linguistic variable with varying classifications. If we use xc and d as, respectively, the central value and dispersion of each one of the fuzzy sets assumed by Π, we model each classification by using a triangular fuzzy number whose membership function is given in (8) and the graph of membership function Π is presented in Figure 2.
(8)∏x=0,ifx<xc−d,x−xc+dd,ifxc−d≤x≤xc,−x−xc−dd,ifxc<x≤xc+d,0,ifxc+d<x.Figure 1
The graph of membership functionsβ and α.
(a)(b)Figure 2
The graph of membership functionΠ.LetLx,t, Mx,t, and Ax,t be the family solutions of the fuzzy model system (3). These are the numbers of risky population proportions created as the result of social interactions between the susceptible members and risky individuals with social influence x at time t. Now, Lx,t, Mx,t, and Ax,t are fuzzy numbers which lie in the interval 0,1.


### 3.1. Existence of Equilibrium Points

The equilibrium points of the fuzzy model (3) can be obtained by setting the RHS to be equal to zero and solve the system simultaneously. Thus, we have
(9)0=1−ϕπN∗−κ1+λ∗xS∗+γ2P∗+ωR∗,0=ϕπN∗+γ1S∗−κ2P∗+νL∗+τM∗+ψA∗,0=λ∗xρS∗−κ3L∗,0=λ∗x1−ρS∗+σL∗−κ4M∗,0=δM∗−κ5+αxA∗,0=ξM∗+ηA∗−κ6R∗,where κ1=µ+γ1, κ2=µ+γ2, κ3=ν+µ+σ, κ4=τ+µ+δ+ξ, κ5=µ+η+ψ, and κ6=ω+μ are simplifying parameters.The risk-free equilibriumRFE point ε0 is obtained when the entire population is free from health risks associated with alcoholism, that is, L=M=A=0 implying that λx=0. Solving system (9) by maintaining risky classes at zero, we have
(10)ε0=S0∗,P0∗,L0∗,M0∗,A0∗,R0∗=1−κN0∗,κN0∗,0,0,0,0,with the positive constant κ=πϕμ+γ1/μμ+γ1+γ2.Similarly, whenL=0, M=0, A=0, and consequently λ∗x=0, the risk endemic equilibrium REE point ε1 (11) is established.
(11)ε1=S∗,P∗,L∗,M∗,A∗,R∗,where by P∗,L∗,M∗,A∗, and R∗ are defined in
(12)P∗=1γ2κ1+1−ωQ3λ∗xS∗−1−ϕπN∗,(13)L∗=Q0λ∗xS∗,(14)M∗=Q1λ∗xS∗,(15)A∗=Q2λ∗xS∗,(16)R∗=Q3λ∗xS∗,and Q0=ρ/κ3, Q1=κ3−ρν+μ/κ3κ4, Q2=κ3−ρν+μδ/κ3κ4κ5+αx, and Q3=ξκ5+αx+ηδκ3−ρν+μ/κ3κ4κ5+αxκ6 are the simplifying factors. Thus, the Theorem 1 is established.Theorem 1.
System (3) has a risk-free equilibrium ε0=1−κN0∗,κN0∗,0,0,0,0 and a unique risk endemic equilibrium ε1=S∗,P∗,L∗,M∗,A∗,R∗.


### 3.2. Bifurcation and Fuzzy Basic Risk Reproduction Number

In this section, we first compute the basic health risk reproduction number, denoted asR0 by using the Next-Generation Matrix method [39, 40]. The value of R0 serves as the determinant to indicate whether the health risk epidemic is possible or not. Based on Diekmann et al.’s study [39], the risk transmission model consists of nonnegative initial conditions together with the following system of equations:
(17)Ż=FZ−VZ=λxρSλx1−ρS0−κ3L−σL+κ4M−δM+κ5+αxA,where VZ=V−Z−V+Z, X=S,P,RT∈ℝ3, Z=L,M,AT∈ℝ3, and ⋅T denotes transpose.We then formulate the risk “transmissions” matrix F and risk “transitions” matrix V such that
(18)F=1−κcβxρcβxθ1ρcβxθ2ρcβx1−ρcβxθ11−ρcβxθ21−ρ000,V=κ300−σκ400−δκ5+αx.Thus, by direct computation, we have(19)V−1=1κ300σκ3κ41κ40σδκ3κ4κ5+αxδκ4κ5+αx1κ5+αx.Now, we have the following Next-Generation Matrix(20)FV−1=1−κcβxρκ31+θ1σκ4+θ2δσκ4κ5+αxcβxρκ4θ1+θ2δκ5+αxcβxθ2ρκ5+αxcβx1−ρκ31+θ1σκ4+θ2δσκ4κ5+αxcβx1−ρκ4θ1+θ2δκ5+αxcβxθ21−ρκ5+αx000.The basic risk reproduction number,R0, is given by the dominant eigenvalue of matrix FV−1 Therefore,
(21)R0x=cβxρκ4κ5+αx+θ1κ5+αx+θ2δ1−ρκ3+ρσ1−κκ3κ4κ5+αx.With appropriate choices ofρ,ϕ≤1, Theorem 2 is establishedTheorem 2.
The risk-free equilibriumε0 is locally asymptotic stable when R0<1and unstable otherwise.Proof.
Consider the Jacobian matrix evaluated at the risk free equilibrium point bellow(22)Jε0=J11J12J13J14J15J16J21J22J23J24J25ϕπ00J33J34J35000J43J44J450000δJ550000ξη−k6,where J11=−κ1+1−ϕπ, J12=γ2+1−ϕπ, J13=−cβx1−κ+1−ϕπ, J14=−cβxθ11−κ+1−ϕπ, J15=−cβxθ21−κ+1−ϕπ, J16=ω+1−ϕπ, J21=ϕπ+γ1, J22=ϕπ−κ2, J23=ϕπ+ν, J24=ϕπ+τ, J25=ϕπ+τ, J33=cβxρ1−κ−κ3, J34=cβxθ1ρ1−κ, J35=cβxθ1ρ1−κ, J43=1−ρcβx1−κ+σ, J44=cβxθ11−ρ1−κ−κ4, J45=cβxθ11−ρ1−κ−κ4, and J55=−κ5−αx
Using a trace determinant method, the risk-free equilibrium pointε0 is locally stable provided that TrJε0<0 and DetJε0>0 The trace and determinant of Jε0 are, respectively, given by
(23)TrJε0=π−αx+cβx1−ρθ1+ρ1−κ−∑i=16κi,DetJε0=1−R0π−μμ+γ1+γ2κ3κ4κ5+αxκ6.
Sinceμ<π, it follows that ε0 is asymptotically stable whenever R0<1 and unstable when R0>1 provided that
(24)π−αx+cβx1−ρθ1+ρ1−κ∑i=16κi<1.The value ofR0 may vary significantly depending on different risk dynamics studied in the population or different populations involved in similar studies [33]. Similarly, the stability of the risk-free equilibrium also changes from unstable to stable when R0 increases through 1. The system acquires a bifurcation at the risk-free equilibrium when R0=1. Suppose that the bifurcation value occurs at x∗ (see Figure 3) where x∗ is given as
(25)x∗=a4−a2−2a3+−1+u+α02x0−xmin22a1−1+u+α0a5θ1+ρκ4,provided that xmin≤x∗≤x0 and
(26)a1=c1−κκ3κ4,a2=−1+u+α0xmin+α0+κ5xmin−x0θ1a5+ρκ4+θ2δa5xmin+x02a12,a3=x0−xmin−1+u+α0−1+u+α0+α0+κ5x0+xminθ1a5+ρκ4−x0+xminθ2δa5a1,a4=u−1xmin+x0α0θ1a5+ρκ4+θ1a5+ρκ4κ5+θ2δa5x0−xmina1+−1+u+α0α0−xmin,a5=1−ρκ3+ρσ.Figure 3
Bifurcation diagram.The basic risk reproduction number presented in (21) is the function of x, that is R0=R0x, but it can not be referred to as the fuzzy number since it is open to values exceeding unity. It is clear that R0x is the function of the degree of social influence in the spread of the behavior. However, both βx and αx incline to their maximum level whenever x≥x0, and as such, we have
(27)R0u=cρκ4κ5+1−u+θ1κ5+1−u+θ2δ1−ρκ3+ρσ1−κκ3κ4κ5+1−u.We introduce a positive number∈0 such that ∈0R0x≤1, with an appropriate choices of ∈0; we have a fuzzy set ∈0R0x whose fuzzy expected value, FEV∈0R0x, can be well defined. Therefore, the fuzzy basic risk reproduction number R0f, which can be defined as the average number of secondary risk cases caused by one infected node introduced into entirely susceptible nodes [31], is given by
(28)R0f=1∈0FEV∈0R0x.We know from the definition of fuzzy expected value that(29)FEV∈0R0x=sup0≤y≤1infy,ky,where ky=Φz∈Ω:∈0R0x≥y=ΦΩ is a fuzzy measure. For the purpose of this study, the possibility measure is given by
(30)ΦΩ=supx∈X∏x,Ω⊂ℝ.FromFEV∈0R0x, for a monotonic function R0x, with set X given as an interval x∗,xmax, we let x∗∈X be the solution of the following equation:
(31)∈0cβxρκ4κ5+αx+θ1κ5+αx+θ2δ1−ρκ3+ρσ1−κκ3κ4κ5+αx=y.Thus,(32)ky=Φx∗,xmax=supx∗≤x≤xmax∏x,where k0=1 and k1=∏xmax.Now,FEV∈0R0 can be determined by considering the linguistic variable Π in three classes: “weak Π−,” “medium Π−+,” and “strong Π+.” Each of these classification is a fuzzy number based on xmin, x0, and xmax as shown in Figure 4. The classification of the degree of social influences in the community can be explained in three different cases as follows:Case 1.
In this case, we consider the weak degree of social influenceΠ− where we define that xc+d<xmin. Suppose that there exists x¯ such that R0x¯=R0f; if xc+d<x¯c, we have
(33)ky=supx¯≤x≤xmax∏x=0,∀y∈0,1.
We establish thatFEV∈0R0=0<∈0. This can be translated that the fuzzy basic risk reproduction number R0f<1, and hence, the extinction of the health risk is associated with alcoholism.Case 2.
In this case, we consider the medium degree of social influenceΠ−+ where we define that xc−d<xmin and xc+d<x0 giving.
(34)ky=1,if0≤y<∈0R0xc,∏x¯,if∈0R0xc≤y≤∈0R0xc+d,0,if∈0R0xc+d<y≤1.Figure 4
Classification of linguistic variableΠ.Ford>0, ky is the continuous function giving k0=1 and k1=0. This translates R0f as the fixed point k and R0xc<R0f<R0xc+d. Since R0x is a continuous monotonic (increasing) function, the Intermediate Value Theorem suggests that there exists x¯ with xc<x¯<xc+d in which the values of R0x¯ and R0f coincide such that R0f=R0x¯>R0xc. Further, the average number of fuzzy basic risk reproduction number R0f is higher than the number of secondary risk cases R0xc due to the medium level of social influence implying that the health risk associated with alcoholism is endemic.Case 3.
In this case, we consider the strong degree of social influenceΠ+ by defining xc−d>x0 and xc+d<xmax.
The results in (35) can be established.
(35)ky=1,if0≤y<∈0R0xc,∏x¯,if∈0R0xc≤y<∈0R0xc+d,0,if∈0R0xc+d≤y≤1.For any givend>0, ky is a monotonically decreasing and continuous function with k0=1 and k1=0.Therefore,FEV∈0R0 is established as a fixed point such that
(36)∈0R0xc<FEV∈0R0<∈0R0xc+d.By dividing by∈0 throughout, we have the following results:
(37)R0xc<R0f<R0xc+d.Now, sinceR0f>1, it translates into endemic health risks associated with alcoholism.


### 3.3. Risk Control in Fuzzy Epidemic System

In this section, we perform the control analysis of the risk estimation in the population by using the fuzzy basic risk thresholdR0f=R0x¯. The spread of health risk in the proposed fuzzy SPLMAR model (3) depends on the degree of the social influence x as an input value of the transmission factor βx. The description of existence and stability of the risk in the system is case-wise presented hereunder. Since the proposed fuzzy system (3) represents a family of systems depending on the parameter x, these family systems can be simplified by a unique system of equations with the same results. It is shown that the bifurcation occurs at x=x∗, that is, R0x∗=1.
(1)
Weak influence: in this case, we havex<xmin giving R0=0 suggesting the extinction of the health risks associated with alcoholism in the community.(2)
Medium influence: in this case, three possibilities may arise as follows:(i)
Ifx<x∗, then R0<1 suggesting the risk free community(ii)
Ifx=x∗, then R0=1 an indication of risk bifurcation(iii)
Ifx>x∗, then R0>1 implying the risk endemic in the community(3)
Strong influence: in this case, we havex∈x0,xmax giving(38)R0u=cρκ4κ5+1−u+θ1κ5+1−u+θ2δ1−ρκ3+ρσ1−κκ3κ4κ5+1−u.The spread of the health risks depends upon the parameteru. Let u∗ be an improved value of u; we can establish three possibilities in which the spread of health risks takes as follows:
(i)
If0≤u<u∗, then R0u<1 suggesting the health risks would be cleared in the community(ii)
If0≤u=u∗, then R0u=1 implying that the system passes through a bifurcation state(iii)
If0≤u∗<u, then R0u>1 suggesting that the health risk problem would spread out in the system


## 3.1. Existence of Equilibrium Points

The equilibrium points of the fuzzy model (3) can be obtained by setting the RHS to be equal to zero and solve the system simultaneously. Thus, we have
(9)0=1−ϕπN∗−κ1+λ∗xS∗+γ2P∗+ωR∗,0=ϕπN∗+γ1S∗−κ2P∗+νL∗+τM∗+ψA∗,0=λ∗xρS∗−κ3L∗,0=λ∗x1−ρS∗+σL∗−κ4M∗,0=δM∗−κ5+αxA∗,0=ξM∗+ηA∗−κ6R∗,where κ1=µ+γ1, κ2=µ+γ2, κ3=ν+µ+σ, κ4=τ+µ+δ+ξ, κ5=µ+η+ψ, and κ6=ω+μ are simplifying parameters.The risk-free equilibriumRFE point ε0 is obtained when the entire population is free from health risks associated with alcoholism, that is, L=M=A=0 implying that λx=0. Solving system (9) by maintaining risky classes at zero, we have
(10)ε0=S0∗,P0∗,L0∗,M0∗,A0∗,R0∗=1−κN0∗,κN0∗,0,0,0,0,with the positive constant κ=πϕμ+γ1/μμ+γ1+γ2.Similarly, whenL=0, M=0, A=0, and consequently λ∗x=0, the risk endemic equilibrium REE point ε1 (11) is established.
(11)ε1=S∗,P∗,L∗,M∗,A∗,R∗,where by P∗,L∗,M∗,A∗, and R∗ are defined in
(12)P∗=1γ2κ1+1−ωQ3λ∗xS∗−1−ϕπN∗,(13)L∗=Q0λ∗xS∗,(14)M∗=Q1λ∗xS∗,(15)A∗=Q2λ∗xS∗,(16)R∗=Q3λ∗xS∗,and Q0=ρ/κ3, Q1=κ3−ρν+μ/κ3κ4, Q2=κ3−ρν+μδ/κ3κ4κ5+αx, and Q3=ξκ5+αx+ηδκ3−ρν+μ/κ3κ4κ5+αxκ6 are the simplifying factors. Thus, the Theorem 1 is established.Theorem 1.
System (3) has a risk-free equilibrium ε0=1−κN0∗,κN0∗,0,0,0,0 and a unique risk endemic equilibrium ε1=S∗,P∗,L∗,M∗,A∗,R∗.


## 3.2. Bifurcation and Fuzzy Basic Risk Reproduction Number

In this section, we first compute the basic health risk reproduction number, denoted asR0 by using the Next-Generation Matrix method [39, 40]. The value of R0 serves as the determinant to indicate whether the health risk epidemic is possible or not. Based on Diekmann et al.’s study [39], the risk transmission model consists of nonnegative initial conditions together with the following system of equations:
(17)Ż=FZ−VZ=λxρSλx1−ρS0−κ3L−σL+κ4M−δM+κ5+αxA,where VZ=V−Z−V+Z, X=S,P,RT∈ℝ3, Z=L,M,AT∈ℝ3, and ⋅T denotes transpose.We then formulate the risk “transmissions” matrix F and risk “transitions” matrix V such that
(18)F=1−κcβxρcβxθ1ρcβxθ2ρcβx1−ρcβxθ11−ρcβxθ21−ρ000,V=κ300−σκ400−δκ5+αx.Thus, by direct computation, we have(19)V−1=1κ300σκ3κ41κ40σδκ3κ4κ5+αxδκ4κ5+αx1κ5+αx.Now, we have the following Next-Generation Matrix(20)FV−1=1−κcβxρκ31+θ1σκ4+θ2δσκ4κ5+αxcβxρκ4θ1+θ2δκ5+αxcβxθ2ρκ5+αxcβx1−ρκ31+θ1σκ4+θ2δσκ4κ5+αxcβx1−ρκ4θ1+θ2δκ5+αxcβxθ21−ρκ5+αx000.The basic risk reproduction number,R0, is given by the dominant eigenvalue of matrix FV−1 Therefore,
(21)R0x=cβxρκ4κ5+αx+θ1κ5+αx+θ2δ1−ρκ3+ρσ1−κκ3κ4κ5+αx.With appropriate choices ofρ,ϕ≤1, Theorem 2 is establishedTheorem 2.
The risk-free equilibriumε0 is locally asymptotic stable when R0<1and unstable otherwise.Proof.
Consider the Jacobian matrix evaluated at the risk free equilibrium point bellow(22)Jε0=J11J12J13J14J15J16J21J22J23J24J25ϕπ00J33J34J35000J43J44J450000δJ550000ξη−k6,where J11=−κ1+1−ϕπ, J12=γ2+1−ϕπ, J13=−cβx1−κ+1−ϕπ, J14=−cβxθ11−κ+1−ϕπ, J15=−cβxθ21−κ+1−ϕπ, J16=ω+1−ϕπ, J21=ϕπ+γ1, J22=ϕπ−κ2, J23=ϕπ+ν, J24=ϕπ+τ, J25=ϕπ+τ, J33=cβxρ1−κ−κ3, J34=cβxθ1ρ1−κ, J35=cβxθ1ρ1−κ, J43=1−ρcβx1−κ+σ, J44=cβxθ11−ρ1−κ−κ4, J45=cβxθ11−ρ1−κ−κ4, and J55=−κ5−αx
Using a trace determinant method, the risk-free equilibrium pointε0 is locally stable provided that TrJε0<0 and DetJε0>0 The trace and determinant of Jε0 are, respectively, given by
(23)TrJε0=π−αx+cβx1−ρθ1+ρ1−κ−∑i=16κi,DetJε0=1−R0π−μμ+γ1+γ2κ3κ4κ5+αxκ6.
Sinceμ<π, it follows that ε0 is asymptotically stable whenever R0<1 and unstable when R0>1 provided that
(24)π−αx+cβx1−ρθ1+ρ1−κ∑i=16κi<1.The value ofR0 may vary significantly depending on different risk dynamics studied in the population or different populations involved in similar studies [33]. Similarly, the stability of the risk-free equilibrium also changes from unstable to stable when R0 increases through 1. The system acquires a bifurcation at the risk-free equilibrium when R0=1. Suppose that the bifurcation value occurs at x∗ (see Figure 3) where x∗ is given as
(25)x∗=a4−a2−2a3+−1+u+α02x0−xmin22a1−1+u+α0a5θ1+ρκ4,provided that xmin≤x∗≤x0 and
(26)a1=c1−κκ3κ4,a2=−1+u+α0xmin+α0+κ5xmin−x0θ1a5+ρκ4+θ2δa5xmin+x02a12,a3=x0−xmin−1+u+α0−1+u+α0+α0+κ5x0+xminθ1a5+ρκ4−x0+xminθ2δa5a1,a4=u−1xmin+x0α0θ1a5+ρκ4+θ1a5+ρκ4κ5+θ2δa5x0−xmina1+−1+u+α0α0−xmin,a5=1−ρκ3+ρσ.Figure 3
Bifurcation diagram.The basic risk reproduction number presented in (21) is the function of x, that is R0=R0x, but it can not be referred to as the fuzzy number since it is open to values exceeding unity. It is clear that R0x is the function of the degree of social influence in the spread of the behavior. However, both βx and αx incline to their maximum level whenever x≥x0, and as such, we have
(27)R0u=cρκ4κ5+1−u+θ1κ5+1−u+θ2δ1−ρκ3+ρσ1−κκ3κ4κ5+1−u.We introduce a positive number∈0 such that ∈0R0x≤1, with an appropriate choices of ∈0; we have a fuzzy set ∈0R0x whose fuzzy expected value, FEV∈0R0x, can be well defined. Therefore, the fuzzy basic risk reproduction number R0f, which can be defined as the average number of secondary risk cases caused by one infected node introduced into entirely susceptible nodes [31], is given by
(28)R0f=1∈0FEV∈0R0x.We know from the definition of fuzzy expected value that(29)FEV∈0R0x=sup0≤y≤1infy,ky,where ky=Φz∈Ω:∈0R0x≥y=ΦΩ is a fuzzy measure. For the purpose of this study, the possibility measure is given by
(30)ΦΩ=supx∈X∏x,Ω⊂ℝ.FromFEV∈0R0x, for a monotonic function R0x, with set X given as an interval x∗,xmax, we let x∗∈X be the solution of the following equation:
(31)∈0cβxρκ4κ5+αx+θ1κ5+αx+θ2δ1−ρκ3+ρσ1−κκ3κ4κ5+αx=y.Thus,(32)ky=Φx∗,xmax=supx∗≤x≤xmax∏x,where k0=1 and k1=∏xmax.Now,FEV∈0R0 can be determined by considering the linguistic variable Π in three classes: “weak Π−,” “medium Π−+,” and “strong Π+.” Each of these classification is a fuzzy number based on xmin, x0, and xmax as shown in Figure 4. The classification of the degree of social influences in the community can be explained in three different cases as follows:Case 1.
In this case, we consider the weak degree of social influenceΠ− where we define that xc+d<xmin. Suppose that there exists x¯ such that R0x¯=R0f; if xc+d<x¯c, we have
(33)ky=supx¯≤x≤xmax∏x=0,∀y∈0,1.
We establish thatFEV∈0R0=0<∈0. This can be translated that the fuzzy basic risk reproduction number R0f<1, and hence, the extinction of the health risk is associated with alcoholism.Case 2.
In this case, we consider the medium degree of social influenceΠ−+ where we define that xc−d<xmin and xc+d<x0 giving.
(34)ky=1,if0≤y<∈0R0xc,∏x¯,if∈0R0xc≤y≤∈0R0xc+d,0,if∈0R0xc+d<y≤1.Figure 4
Classification of linguistic variableΠ.Ford>0, ky is the continuous function giving k0=1 and k1=0. This translates R0f as the fixed point k and R0xc<R0f<R0xc+d. Since R0x is a continuous monotonic (increasing) function, the Intermediate Value Theorem suggests that there exists x¯ with xc<x¯<xc+d in which the values of R0x¯ and R0f coincide such that R0f=R0x¯>R0xc. Further, the average number of fuzzy basic risk reproduction number R0f is higher than the number of secondary risk cases R0xc due to the medium level of social influence implying that the health risk associated with alcoholism is endemic.Case 3.
In this case, we consider the strong degree of social influenceΠ+ by defining xc−d>x0 and xc+d<xmax.
The results in (35) can be established.
(35)ky=1,if0≤y<∈0R0xc,∏x¯,if∈0R0xc≤y<∈0R0xc+d,0,if∈0R0xc+d≤y≤1.For any givend>0, ky is a monotonically decreasing and continuous function with k0=1 and k1=0.Therefore,FEV∈0R0 is established as a fixed point such that
(36)∈0R0xc<FEV∈0R0<∈0R0xc+d.By dividing by∈0 throughout, we have the following results:
(37)R0xc<R0f<R0xc+d.Now, sinceR0f>1, it translates into endemic health risks associated with alcoholism.


## 3.3. Risk Control in Fuzzy Epidemic System

In this section, we perform the control analysis of the risk estimation in the population by using the fuzzy basic risk thresholdR0f=R0x¯. The spread of health risk in the proposed fuzzy SPLMAR model (3) depends on the degree of the social influence x as an input value of the transmission factor βx. The description of existence and stability of the risk in the system is case-wise presented hereunder. Since the proposed fuzzy system (3) represents a family of systems depending on the parameter x, these family systems can be simplified by a unique system of equations with the same results. It is shown that the bifurcation occurs at x=x∗, that is, R0x∗=1.
(1)
Weak influence: in this case, we havex<xmin giving R0=0 suggesting the extinction of the health risks associated with alcoholism in the community.(2)
Medium influence: in this case, three possibilities may arise as follows:(i)
Ifx<x∗, then R0<1 suggesting the risk free community(ii)
Ifx=x∗, then R0=1 an indication of risk bifurcation(iii)
Ifx>x∗, then R0>1 implying the risk endemic in the community(3)
Strong influence: in this case, we havex∈x0,xmax giving(38)R0u=cρκ4κ5+1−u+θ1κ5+1−u+θ2δ1−ρκ3+ρσ1−κκ3κ4κ5+1−u.The spread of the health risks depends upon the parameteru. Let u∗ be an improved value of u; we can establish three possibilities in which the spread of health risks takes as follows:
(i)
If0≤u<u∗, then R0u<1 suggesting the health risks would be cleared in the community(ii)
If0≤u=u∗, then R0u=1 implying that the system passes through a bifurcation state(iii)
If0≤u∗<u, then R0u>1 suggesting that the health risk problem would spread out in the system


## 4. Discussion and Conclusion

In this work, we propose and analyze the fuzzySPLMAR model with the aim of using fuzzy set theory in the model system. Although all the parameters associated in the model system are important, in an uncertain environment, we consider the selected two most important parameters β and α representing the probability that a susceptible individual will drink alcohol after prolonged contact with drinking individuals and the additional death rate induced by alcoholism, respectively. In particular, the two parameters β and α are considered fuzzy numbers and functions of the degree of force of influence, whose membership functions are well defined. Later, the fuzzy basic risk reproduction numberR0fx is computed.Since the basic risk reproduction numberR0 is directly related to varied factors of drinking settings, obtained by considering the classical model system (1) with constant parameters β and α, it implied that the variation of the risky population is always positive. However, when β is considered a fuzzy set, it provide us with an additional information regarding the health risk dynamics. The health risks associated with alcoholism can be effectively controlled by controlling the fuzzy basic risk reproduction number R0f. The analysis of the fuzzy model suggests that R0f can be reduced by increasing the value of xmin. This may be done through provision of public health education which increases the resistance of susceptible individuals and prevent them from initiation of drinking behavior which would accelerate their health condition into more risky states. It is generally observed that if the amount of degree of peer of influence to an individual is low, then the alcohol-related health risks in the community may not be the most effective. The reason is natural factors in the community provide virtual immunity from deeply engaging into alcoholic behaviors. However, when the perceived most influential people in the community engage in alcoholism, it implies an increase in the degree of force of influence, and as such, the system will be endemic.In a future work, some other parameters may be considered with different functionalities under the uncertainty environment. However, the present model can be applied to those types of diseases or conditions which spread through direct contact between susceptible and infected individuals.



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