# Fuzzy SEIR Epidemic Model of Amoebiasis Infection in Human

Authors: Saul C. Mpeshe
Journal: Advances in Fuzzy Systems
 (2022)

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## Abstract

Amoebiasis is a disease caused by the protozoanEntamoeba histolytica. It is often asymptomatic but may cause dysentery and invasive extraintestinal infection in humans. To gain an understanding of the disease, a fuzzy SEIR model of amoebiasis infection is presented with the fuzziness introduced in the disease transmission rate, the disease-induced death rate, and the rate of recovery from infection. The basic reproduction number ℛ0v is computed and used to make a comparison with the fuzzy basic reproduction number ℛ0f at different amounts of cysts as well as to analyse the stability of equilibrium points. Stability results indicate that the disease-free equilibrium and the endemic equilibrium are locally and globally asymptotically stable. Further, numerical simulations are carried out to illustrate the analytical results of the fuzzy model system. The result shows that whenever there is an outbreak, the disease is likely to persist in the first month and thereafter it will start to slow down to disease-free equilibrium.


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## Body

## 1. Introduction

Amoebiasis is a human infection of the large intestine caused by a protozoan parasiteEntamoeba histolytica (E. histolytica) [1]. It is the most common worldwide cause of mortality from a protozoan after malaria [2–4], causing about 55,000 deaths worldwide in 2010 [5, 6]. Amoebiasis has been a major health problem in China, and some parts of Asia, Latin America, and Africa [7].A brief history ofamoebiasis leads us to Lambe (or Lambl) in 1859 who reported amoebae in the discharge of a patient with dysentery [8]. The name Entamoeba histolytica was established in 1903 by Schaudinn due to the parasite ability to cause tissue lysis, and he described it as the causal agent of amoebic dysentery [9].Transmission ofamoebiasis occurs through the faecal-oral route, either directly from person-to-person contact or indirectly by eating or drinking faecally contaminated food or water. Sexual transmission by oral-rectal contact is also possible especially among male homosexuals [10–13]. Depending on the immune system of the infected individual and other biological factors which are yet to be known, 90% of the persons infected by Entamoeba histolytica develop an asymptomatic amoebiasis while 10% of them develop symptoms that identify the amoebiasis colitis [14–16].Due to the resistant character of the cysts ofEntamoeba histolytica, mature cysts can remain alive for new infection for some weeks in soils, 12 days in a moist and cool environment, and 30 days in water. However, they can stand alive for the temperatures under 50°C and greater than 400°C [14]. The ingestion of one viable cyst may cause an infection [12]. The incubation period varies from a few days to several months or years but is generally 2 to 4 weeks [17, 18].Modelling approach has played a major role in better understanding the global behaviour of infectious diseases. Modelling of infectious diseases using ordinary or integer-order differential equations has proven to be valuable in analysing the dynamics of various diseases includingamoebiasis [19, 20]. However, these methods are crisp, deterministic, and precise. Crisp, in this sense, mean dichotomous, that is, yes-or-no-type, or true or false type rather than more-or-less type or anything in between [21]. According to Zimmenmann [21], precision assumes that the parameters of a model represent exactly either our perception of the phenomenon modelled or the features of the real system that has been modelled and there is no ambiguities.Many real situations are not crisp and deterministic, and therefore, they cannot be described precisely. The complete description of a real situation will often require more detailed information than a human being could process, and understand. Lack of information and vagueness of the situations may cause the future state of the system not to be known completely [21].Fuzziness can be found in many areas of daily life, such as in engineering, medicine, meteorology, manufacturing, and others where human judgment, evaluation, reasoning and decision making are important [21–23]. The reasons for such fuzziness is mostly due to our daily communication which uses natural languages, where the meaning of words is very often vague [21, 23]. For example, when we say there is an infection in a certain population, the statement is not necessarily to be true or false. It may be true to some degree, the degree to which an infected person belong to that population. Therefore, the statement is vague and we need to introduce the concept of vagueness to understand the truthness of this statement.The concept of vagueness is well established through the idea of fuzzy set which involve assigning to each possible individual in the population a value representing its degree of membership. For example, a fuzzy set representing our concept of infection can assign a degree of membership of 1 to a high infection, 0.5 to a medium infection, and 0 to a low infection. Fuzzy sets representing linguistic concepts such as low, medium, and high, are often employed to define states of a variable called a fuzzy variable.Modelling with fuzzy sets and logic has motivated many researchers in science and social sciences including epidemiology. This is evident from the work by Abdy et al. [24], Li et al. [25], Shi et al. [26], Stiegelmeier and Bressan [27], Shaban et al. [28], Chowdhury et al. [29], and Bhuju et al. [30], just to mention a few. Motivated by their work, a fuzzy model of amoebiasis infection is presented in this paper. The paper begins with the definition of some fuzzy concepts followed by the model formulation of a model and a description of some fuzzy parameters used in the model. The feasibility solution, basic reproduction number, fuzzy reproduction number and global stability analysis of the model are also discussed. Numerical simulations of the model are also established to study the behaviour of the disease over a certain time period.


## 2. The Fuzzy Model of Amoebiasis

### 2.1. Definitions

Definition 1.
(see [21]). A fuzzy set AinX is a set of ordered pairs:(1)A=x,μAx|x∈X,where X is the universe of discourse, μAX is called the membership function or grade of membership (also degree of compatibility or degree of truth) of x in A that maps X to the membership space M=0,1, A is nonfuzzy and μAX is the memmbership function.Definition 2.
(see [21]). A membership function μAX is a mapping from the universe X to the interval 0,1 denoted by μ:X⟶0,1Definition 3.
(see [31]). Let u be a fuzzy subset. The fuzzy integral, or fuzzy expected value of u,FEVu, is defined by(2)FEVu=sup0≤α≤1infα,μu≥α,whereu≥α=x∈ℝ:ux≥α.The value ofFEVu is a real number and it provides a good estimate of mathematics expectancy of the random variable u. It is also a good estimate for the classical expectancy Eu when the fuzzy subset u is also a random variable [31].Definition 4.
(see [32]). A linguistic variable is a quintuple x,Tx,U,G,M in which x is the name of the variable; Tx is the set of names of linguistic values of x, with each value being a fuzzy variable denoted generically by X and ranging over a universe of discourse U that is associated with the base variable u; G is a syntactic rule for generating the name, X, of values of x; and M is a semantic rule for associating with each X its meaning, MX, which is a fuzzy subset of U.


### 2.2. Model Formulation

The model consider a human population subdivided into four compartments namely: susceptibleS, exposed E, infected I, and recovered R. The total human population N is therefore given by N=S+E+I+R. It is assumed that individuals who have recovered from amoebiasis will experience a temporal immunity induced by the disease. The transmission rate β, recovery rate γ, and the disease-induced rate d are considered to be fuzzy parameters. The parameters and their description as they have been used in the model are given in Table 1. The mode of transmission of amoebiasis is presented by Figure 1 with the parameters used in this model described in Table 1.Using the parameters described in Table 1 and the transmission diagram in Figure 1, an SEIR model is formulated using ordinary differential equations as follows:(3a)dSdt=Λ−μS−βIS,(3b)dEdt=βIS−ε+μE,(3c)dIdt=εE−μ+d+γI,(3d)dRdt=γI−μR,where β,γ, and d are fuzzy parameters.Table 1
Description of parameters used in the model.
ParameterDescriptionΛRecruitment rate of humanλTransmission rate in humanεInfectious rateμNatural death rate of humandDisease-induced death rate in humanγRecovery rate in humanFigure 1
Transmission diagram for Amoebiasis.


#### 2.2.1. Membership Function forβ

The transmission rateβ depends on the amount of cysts v present in the environment. Using Barros et al. [31, 33] we define the membership function for β as(4)βv=0,ifv>vmin,v−vminvm−vmin,ifvmin≤v≤vm,1,ifvm≤v≤vmax,where vmin is the minimum amount of cysts needed for the disease transmission to occur, vm is the amount of cysts where the transmission of the disease is maximum and equal to 1.


#### 2.2.2. Membership Function forγ

The recovery rateγ depends on the amount of cysts ingested by a human. When the amount of cysts is high it will take a longer time to recover from the disease. Using Verma et al. [34] we define the membership of γ as(5)γv=γ0−1vmaxv+1if0≤v≤vmax,where 0<γ0<1 is the lowest recovery rate.


#### 2.2.3. Membership Function for Induced-Death Rated

The disease-induced death rate occurs due to infection of the disease. When the disease transmission is negligible for a lower amount of cysts ingested, there is only natural death because there is no death due to infection. When the amount of cysts is highvm<V the death will also be high. We, therefore, define the membership of d according to Verma et al. [35] as(6)dv=1−d0vmv+d0,if0≤v≤vm,1,ifvm<v,where 0<d0<1 is the lowest death rate.


#### 2.2.4. The Amount of CystsΓ Membership Function

The amount of cysts in the environment is assumed to be different in each population considered which can be seen as a linguistic variable classified as weak, medium and strong. The membership function for the amount of cysts is defined as in Barros et al. [31] as(7)Γv=0,ifv<v¯−d,v−v¯+δδ,ifv¯−d≤v≤v¯,−v−v¯−δδ,ifv¯<v≤v¯+d,1,ifv>v¯+d,where v¯ is the central value and δ is the dispersion of cyst charge in the population.


### 2.3. Feasibility of the Model Solution

The feasible region is the positive regionℝ+4. Thus, we show that dXi/dt≥0 in the region ℝ+4. From the model system (3) we see that(8a)dSdt|S=0=Λ>0,(8b)dEdt|E=0=βIS≥0,(8c)dIdt|I=0=εE≥0,(8d)dRdt|R=0=γI≥0.Thus, the solution is feasible and positive in the regionΩ=S,E,I,R≥0∈ℝ+4.Furthermore,(9)dNdt=Λ−μN−dI,≤Λ−μN.Solving this differential inequality gives(10)Nt≤N0e−μt+Λμ1−e−μt.Ast⟶∞, we have(11)Nt≤Λμ.Thus, the model solution is bounded and positively invariant inℝ+4.


### 2.4. Equilibrium Points

The disease-free equilibrium and endemic equilibrium are determined by setting the left-hand side of the model system (2.2) equal to zero. SettingE=I=R=0 we obtain the disease-free equilibrium of the model as(12)P0=Λμ,0,0,0.For the caseE≠0,I≠0,R≠=0, we have the endemic equilibrium P∗=S∗,E∗,I∗,R∗ where(13a)S∗=ε+μμ+d+γβε,(13b)E∗=βεΛ−με+μμ+d+γβεε+μ,(13c)I∗=βεΛ−με+μμ+d+γβε+μμ+d+γ,(13d)R∗=γμI∗


### 2.5. The Basic Reproduction Number

ForP∗ to exist in the feasible region, the condition 0<S∗<Λ/μ or equivalently Λ/μ1/S∗≥1 is sufficiently necessary [36]. Now define the basic reproduction number ℛ0 by(14)ℛ0=Λμ1S∗.Then,(15)ℛ0=Λμβεε+μμ+d+γ.Notice thatℛ0 consists of β,γ and d which are both functions of the amount of cysts v, hence ℛ0 is also a function of v and can be denoted as ℛ0v. However, ℛ0v is not a fuzzy set for its value can be greater than one (1). As more individuals recover from the infection, ℛ0v is reduced, and hence 0≤γ0ℛ0v≤1 for γ0∈0,1. Therefore, γ0ℛ0v is fuzzy set and the fuzzy expected value FEVγ0ℛ0v is well defined. The fuzzy basic reproduction number ℛ0f is therefore defined as(16)ℛ0f=1γ0FEVγ0ℛ0v.Define(17)FEVγ0ℛ0v=sup0≤α≤1infα,kα,where kα=μv:γ0ℛ0v≥α=μX is a fuzzy measure. Then μX=supv∈XΓv and X∈v∗,vm where v∗ is the solution to the equation(18)γ0ℛ0v=α.Thus,(19)kα=μv∗,vm=supv∗≤v≤vmΓv,where k0=1 and k1=Γvm.


### 2.6. The Impact of Virus Load onℛ0f

We analyse the impact ofv on ℛ0f based on linguistic interpretation of the amount of cysts on whether there is weak, medium or strong amount of cysts to cause infection.Case 1.
Weak Amount of Cysts. v¯−δ≤vmin If v¯−δ≤vmin, then βv=0 and ℛ0=0. So FEVγ0ℛ0v=0<γ0 and hence ℛ0f<1. Therefore, the disease will die out.Case 2.
Medium Amount of Cysts. v¯−δ≥vmin and v¯+δ≤vm In this case, then we have βv=v−vmin/vm−vmin and ℛ0v=βvεΛ/με+μμ+dv+γv. The fuzzy measure kα is then given by(20)kα=1,if0<α≤γ0ℛ0v¯,Γv∗,ifγ0ℛ0v¯<α≤γ0ℛ0v¯+δ,0,ifγ0ℛ0v¯+δ<α≤1.
Thus, ifδ>0, then kα is a continuous decreasing function with k0=1 and k1=9. Hence, FEVγ0ℛ0v is a fixed point of k and(21)γ0ℛ0v¯≤FEVγ0ℛ0v≤γ0ℛ0v¯+δ.
It follows therefore(22)ℛ0v¯≤1γ0FEVγ0ℛ0v≤ℛ0v¯+δ,and that(23)ℛ0v¯≤ℛ0f≤ℛ0v¯+δ.
Sinceℛ0v¯ is also a continuous increasing function, there exist v with v¯<v<v¯+δ such that ℛ0f=ℛ0v>ℛ0v¯. That is ℛ0f and ℛ0v coincide and ℛ0f>ℛ0v¯. Therefore, the fuzzy average number of secondary infection ℛ0f is higher than the number of infection ℛ0v¯ due to the medium amount of cysts.Case 3.
Strong Amount of Cysts. v¯−δ≤vm and v¯+δ≤vmax In this case βv=1 and ℛ0v=εΛ/με+μμ+dv+γv. The fuzzy measure kα is then given by(24)kα=1,if0<α≤γ0ℛ0v¯,Γv∗,ifγ0ℛ0v¯<α≤γ0ℛ0v¯+δ,0,ifγ0ℛ0v¯+δ<α≤1.
It follows that(25)γ0ℛ0v¯≤FEVγ0ℛ0v≤γ0ℛ0v¯+δ,or(26)ℛ0v¯≤ℛ0f≤ℛ0v¯+δ.
Thus,ℛ0f>1 and the disease will persist.


## 2.1. Definitions

Definition 1.
(see [21]). A fuzzy set AinX is a set of ordered pairs:(1)A=x,μAx|x∈X,where X is the universe of discourse, μAX is called the membership function or grade of membership (also degree of compatibility or degree of truth) of x in A that maps X to the membership space M=0,1, A is nonfuzzy and μAX is the memmbership function.Definition 2.
(see [21]). A membership function μAX is a mapping from the universe X to the interval 0,1 denoted by μ:X⟶0,1Definition 3.
(see [31]). Let u be a fuzzy subset. The fuzzy integral, or fuzzy expected value of u,FEVu, is defined by(2)FEVu=sup0≤α≤1infα,μu≥α,whereu≥α=x∈ℝ:ux≥α.The value ofFEVu is a real number and it provides a good estimate of mathematics expectancy of the random variable u. It is also a good estimate for the classical expectancy Eu when the fuzzy subset u is also a random variable [31].Definition 4.
(see [32]). A linguistic variable is a quintuple x,Tx,U,G,M in which x is the name of the variable; Tx is the set of names of linguistic values of x, with each value being a fuzzy variable denoted generically by X and ranging over a universe of discourse U that is associated with the base variable u; G is a syntactic rule for generating the name, X, of values of x; and M is a semantic rule for associating with each X its meaning, MX, which is a fuzzy subset of U.


## 2.2. Model Formulation

The model consider a human population subdivided into four compartments namely: susceptibleS, exposed E, infected I, and recovered R. The total human population N is therefore given by N=S+E+I+R. It is assumed that individuals who have recovered from amoebiasis will experience a temporal immunity induced by the disease. The transmission rate β, recovery rate γ, and the disease-induced rate d are considered to be fuzzy parameters. The parameters and their description as they have been used in the model are given in Table 1. The mode of transmission of amoebiasis is presented by Figure 1 with the parameters used in this model described in Table 1.Using the parameters described in Table 1 and the transmission diagram in Figure 1, an SEIR model is formulated using ordinary differential equations as follows:(3a)dSdt=Λ−μS−βIS,(3b)dEdt=βIS−ε+μE,(3c)dIdt=εE−μ+d+γI,(3d)dRdt=γI−μR,where β,γ, and d are fuzzy parameters.Table 1
Description of parameters used in the model.
ParameterDescriptionΛRecruitment rate of humanλTransmission rate in humanεInfectious rateμNatural death rate of humandDisease-induced death rate in humanγRecovery rate in humanFigure 1
Transmission diagram for Amoebiasis.


### 2.2.1. Membership Function forβ

The transmission rateβ depends on the amount of cysts v present in the environment. Using Barros et al. [31, 33] we define the membership function for β as(4)βv=0,ifv>vmin,v−vminvm−vmin,ifvmin≤v≤vm,1,ifvm≤v≤vmax,where vmin is the minimum amount of cysts needed for the disease transmission to occur, vm is the amount of cysts where the transmission of the disease is maximum and equal to 1.


### 2.2.2. Membership Function forγ

The recovery rateγ depends on the amount of cysts ingested by a human. When the amount of cysts is high it will take a longer time to recover from the disease. Using Verma et al. [34] we define the membership of γ as(5)γv=γ0−1vmaxv+1if0≤v≤vmax,where 0<γ0<1 is the lowest recovery rate.


### 2.2.3. Membership Function for Induced-Death Rated

The disease-induced death rate occurs due to infection of the disease. When the disease transmission is negligible for a lower amount of cysts ingested, there is only natural death because there is no death due to infection. When the amount of cysts is highvm<V the death will also be high. We, therefore, define the membership of d according to Verma et al. [35] as(6)dv=1−d0vmv+d0,if0≤v≤vm,1,ifvm<v,where 0<d0<1 is the lowest death rate.


### 2.2.4. The Amount of CystsΓ Membership Function

The amount of cysts in the environment is assumed to be different in each population considered which can be seen as a linguistic variable classified as weak, medium and strong. The membership function for the amount of cysts is defined as in Barros et al. [31] as(7)Γv=0,ifv<v¯−d,v−v¯+δδ,ifv¯−d≤v≤v¯,−v−v¯−δδ,ifv¯<v≤v¯+d,1,ifv>v¯+d,where v¯ is the central value and δ is the dispersion of cyst charge in the population.


## 2.2.1. Membership Function forβ

The transmission rateβ depends on the amount of cysts v present in the environment. Using Barros et al. [31, 33] we define the membership function for β as(4)βv=0,ifv>vmin,v−vminvm−vmin,ifvmin≤v≤vm,1,ifvm≤v≤vmax,where vmin is the minimum amount of cysts needed for the disease transmission to occur, vm is the amount of cysts where the transmission of the disease is maximum and equal to 1.


## 2.2.2. Membership Function forγ

The recovery rateγ depends on the amount of cysts ingested by a human. When the amount of cysts is high it will take a longer time to recover from the disease. Using Verma et al. [34] we define the membership of γ as(5)γv=γ0−1vmaxv+1if0≤v≤vmax,where 0<γ0<1 is the lowest recovery rate.


## 2.2.3. Membership Function for Induced-Death Rated

The disease-induced death rate occurs due to infection of the disease. When the disease transmission is negligible for a lower amount of cysts ingested, there is only natural death because there is no death due to infection. When the amount of cysts is highvm<V the death will also be high. We, therefore, define the membership of d according to Verma et al. [35] as(6)dv=1−d0vmv+d0,if0≤v≤vm,1,ifvm<v,where 0<d0<1 is the lowest death rate.


## 2.2.4. The Amount of CystsΓ Membership Function

The amount of cysts in the environment is assumed to be different in each population considered which can be seen as a linguistic variable classified as weak, medium and strong. The membership function for the amount of cysts is defined as in Barros et al. [31] as(7)Γv=0,ifv<v¯−d,v−v¯+δδ,ifv¯−d≤v≤v¯,−v−v¯−δδ,ifv¯<v≤v¯+d,1,ifv>v¯+d,where v¯ is the central value and δ is the dispersion of cyst charge in the population.


## 2.3. Feasibility of the Model Solution

The feasible region is the positive regionℝ+4. Thus, we show that dXi/dt≥0 in the region ℝ+4. From the model system (3) we see that(8a)dSdt|S=0=Λ>0,(8b)dEdt|E=0=βIS≥0,(8c)dIdt|I=0=εE≥0,(8d)dRdt|R=0=γI≥0.Thus, the solution is feasible and positive in the regionΩ=S,E,I,R≥0∈ℝ+4.Furthermore,(9)dNdt=Λ−μN−dI,≤Λ−μN.Solving this differential inequality gives(10)Nt≤N0e−μt+Λμ1−e−μt.Ast⟶∞, we have(11)Nt≤Λμ.Thus, the model solution is bounded and positively invariant inℝ+4.


## 2.4. Equilibrium Points

The disease-free equilibrium and endemic equilibrium are determined by setting the left-hand side of the model system (2.2) equal to zero. SettingE=I=R=0 we obtain the disease-free equilibrium of the model as(12)P0=Λμ,0,0,0.For the caseE≠0,I≠0,R≠=0, we have the endemic equilibrium P∗=S∗,E∗,I∗,R∗ where(13a)S∗=ε+μμ+d+γβε,(13b)E∗=βεΛ−με+μμ+d+γβεε+μ,(13c)I∗=βεΛ−με+μμ+d+γβε+μμ+d+γ,(13d)R∗=γμI∗


## 2.5. The Basic Reproduction Number

ForP∗ to exist in the feasible region, the condition 0<S∗<Λ/μ or equivalently Λ/μ1/S∗≥1 is sufficiently necessary [36]. Now define the basic reproduction number ℛ0 by(14)ℛ0=Λμ1S∗.Then,(15)ℛ0=Λμβεε+μμ+d+γ.Notice thatℛ0 consists of β,γ and d which are both functions of the amount of cysts v, hence ℛ0 is also a function of v and can be denoted as ℛ0v. However, ℛ0v is not a fuzzy set for its value can be greater than one (1). As more individuals recover from the infection, ℛ0v is reduced, and hence 0≤γ0ℛ0v≤1 for γ0∈0,1. Therefore, γ0ℛ0v is fuzzy set and the fuzzy expected value FEVγ0ℛ0v is well defined. The fuzzy basic reproduction number ℛ0f is therefore defined as(16)ℛ0f=1γ0FEVγ0ℛ0v.Define(17)FEVγ0ℛ0v=sup0≤α≤1infα,kα,where kα=μv:γ0ℛ0v≥α=μX is a fuzzy measure. Then μX=supv∈XΓv and X∈v∗,vm where v∗ is the solution to the equation(18)γ0ℛ0v=α.Thus,(19)kα=μv∗,vm=supv∗≤v≤vmΓv,where k0=1 and k1=Γvm.


## 2.6. The Impact of Virus Load onℛ0f

We analyse the impact ofv on ℛ0f based on linguistic interpretation of the amount of cysts on whether there is weak, medium or strong amount of cysts to cause infection.Case 1.
Weak Amount of Cysts. v¯−δ≤vmin If v¯−δ≤vmin, then βv=0 and ℛ0=0. So FEVγ0ℛ0v=0<γ0 and hence ℛ0f<1. Therefore, the disease will die out.Case 2.
Medium Amount of Cysts. v¯−δ≥vmin and v¯+δ≤vm In this case, then we have βv=v−vmin/vm−vmin and ℛ0v=βvεΛ/με+μμ+dv+γv. The fuzzy measure kα is then given by(20)kα=1,if0<α≤γ0ℛ0v¯,Γv∗,ifγ0ℛ0v¯<α≤γ0ℛ0v¯+δ,0,ifγ0ℛ0v¯+δ<α≤1.
Thus, ifδ>0, then kα is a continuous decreasing function with k0=1 and k1=9. Hence, FEVγ0ℛ0v is a fixed point of k and(21)γ0ℛ0v¯≤FEVγ0ℛ0v≤γ0ℛ0v¯+δ.
It follows therefore(22)ℛ0v¯≤1γ0FEVγ0ℛ0v≤ℛ0v¯+δ,and that(23)ℛ0v¯≤ℛ0f≤ℛ0v¯+δ.
Sinceℛ0v¯ is also a continuous increasing function, there exist v with v¯<v<v¯+δ such that ℛ0f=ℛ0v>ℛ0v¯. That is ℛ0f and ℛ0v coincide and ℛ0f>ℛ0v¯. Therefore, the fuzzy average number of secondary infection ℛ0f is higher than the number of infection ℛ0v¯ due to the medium amount of cysts.Case 3.
Strong Amount of Cysts. v¯−δ≤vm and v¯+δ≤vmax In this case βv=1 and ℛ0v=εΛ/με+μμ+dv+γv. The fuzzy measure kα is then given by(24)kα=1,if0<α≤γ0ℛ0v¯,Γv∗,ifγ0ℛ0v¯<α≤γ0ℛ0v¯+δ,0,ifγ0ℛ0v¯+δ<α≤1.
It follows that(25)γ0ℛ0v¯≤FEVγ0ℛ0v≤γ0ℛ0v¯+δ,or(26)ℛ0v¯≤ℛ0f≤ℛ0v¯+δ.
Thus,ℛ0f>1 and the disease will persist.


## 3. Stability Analysis of Equilibrium Points

### 3.1. Local Stability of the Disease-Free Equilibrium

Theorem 5.
The disease-free equilibrium of the fuzzy amoebiasis model (3) is locally asymptotically stable ifℛ0<1and unstable ifℛ0>1.Proof.
We need to show that the Jacobian matrixJP0 of the fuzzy model (3) evaluated at the disease-free equilibrium has negative eigenvalues. Further computation shows that JP0 of the model (3) at P0 is(27)JP0=−μ0−βS∗00−ε+μβS∗00ε−μ+d+γ000γ−μ.
From the matrixJP0, the diagonal entry −μ is one of the eigenvalues of JP0. Eliminating the column containing −μ and its corresponding row, we remain with a 2×2 matrix(28)J∗P0=−ε+μβS∗ε−μ+d+γ.
The characteristic equation ofJ∗P0 is(29)λ2+Aλ+B=0,where A=ε+μ+μ+d+γ,B=ε+μμ+d+γ1−ℛ0. Since at disease-free equilibrium ℛ0<1, then B is positive. Using Routh-Hurwitz criteria we conclude that the roots of the quadratic equation are negative real numbers or complex numbers with real parts. Hence, JP0 has negative eigenvalues, and thus, the disease-free equilibrium is locally asymptotically stable.


### 3.2. Global Stability of the Disease-Free Equilibrium

Definition 6.
(Lyapunov function [37]). Let x=0 be an equilibrium point of x˙t=fxt. A continuously differentiable function Vx is a Lyapunov function if the following conditions hold:A1:V0=0A2:Vx>0,x∈N\0A3:V˙x<0,x∈N\0
HereN\0∈ℝn is the neighbourhood of 0 excluding the origin itself.Theorem 7.
The disease-free equilibrium of the fuzzyamoebiasis model (3) is globally asymptotically stable if ℛ0<1 and unstable if ℛ0>1.Proof.
Here we investigate the global stability of the disease-free equilibrium by mean of Lyapunov direct method. In this method, we define a positive definite Lyapunov functionV=ω1E+ω2I. Clearly,(30)dVdt=ω1dEdt+ω2dIdt,=ω1βIS−ε+μE+ω2εE−μ+d+γI≤ω1βIS∗−ε+μE+ω2εE−μ+d+γI.
If we chooseω1=ε and ω2=ε+μ, we have(31)dVdt≤εβIS∗−ε+μμ+d+γI≤εβS∗ε+μμ+d+γ−1ε+μμ+d+γI≤ℛ0−1ε+μμ+d+γI.
Since at disease-free equilibriumℛ0<1, then dV/dt≤0 and hence, the disease-free equilibrium is globally asymptotically stable.


### 3.3. Bifurcation and Fuzzyℛ0

The stability of disease-free equilibrium changes from stable to unstable asℛ0 increases through 1. When ℛ0=1, the system obtain a bifurcation at the disease-free equilibrium. Since ℛ0 depends on the viral load, v, we need to find the value v∗ such that ℛ0v∗=1 and that value is the bifurcation point.From (15), we have(32)ℛ0v∗=βv∗εΛμε+μμ+dv∗+γv∗.Also, from (4)–(6), we have(33a)βv∗=v∗−vminvm−vmin,ifvmin≤v∗≤vm,(33b)γv∗=γ0−1vmaxv∗+1,if0≤v∗≤vmax,(33c)dv∗=1−d0vmv∗+d0,if0≤v∗≤vm.Using (32) and (29), we can solve for v∗ at ℛ0v∗=1. Further computation shows that v∗ exist and is given by(34)v∗=vmvmaxεΛvmin+με+μvm−vminμ+d0+1εΛvmvmax+με+μvm−vmind0−1vmax+1−γ0vm.Since there is only one endemic equilibrium, the model exhibit forward bifurcation as shown in Figure2.Figure 2
Variation ofℛ0v with βv and γv.


### 3.4. Global Stability of the Endemic Equilibrium

The local stability of the disease-free equilibrium implies local stability of the endemic equilibrium for the reverse condition [38]. Therefore, we only establish the global stability of the endemic equilibrium.Theorem 8.
The endemic equilibrium of the fuzzyamoebiasis model (3) is globally asymptotically stable if ℛ0>1.Proof.
We prove the global stability of the endemic equilibrium via construction of a suitable Lyapunov function of the form(35)Vxi=∑ωixi−xi∗−xi∗lnxixi∗,where ωi>0 is constant to be chosen, xi is the population of the ith compartment, and xi∗ is the equilibrium point [39].
Note that the equation containingR in the model system (3) is independent of the other three equations. Therefore, it can be removed from the analysis since the value of R can be obtained when S,E, and I are known. For that case, we now, consider the Lyapunov function(36)V=ω1S−S∗−S∗lnSS∗+ω2E−E∗−E∗lnEE∗+ω3I−I∗−I∗lnII∗.
It is easily seen thatVS,E,I≥0 and VS∗,E∗,I∗=0. The time derivative of V is then given by(37)dVdt=ω11−S∗SdSdt+ω21−E∗EdEdt+ω31−I∗IdIdt.
From the model system (3) we have(38)dVdt=ω11−S∗SΛ−μS−βSI+ω21−E∗EβSI−ε+μE+ω31−I∗IεE−μ+d+γI.
At endemic equilibrium,Λ=μS∗+βS∗I∗,ε+μ=βS∗I∗/E∗, and μ+d+γ=εE∗/I∗. Therefore,(39)dVdt=ω11−S∗SμS∗+βS∗I∗−μS−βSI+ω21−E∗EβSI−βS∗I∗E∗E+ω31−I∗IεE−εE∗I∗I.
Further simplification gives,(40)dVdt=−ω1μS1−S∗S2+ω11−S∗S1−SIS∗I∗βS∗I∗+ω21−E∗ESIS∗I∗−EE∗βS∗I∗+ω31−I∗IEE∗−II∗εE∗,
On expanding, we have(41)dVdt=−ω1μS1−S∗S2+ω11−S∗S+II∗−SIS∗I∗βS∗I∗+ω21−E∗E+SIS∗I∗−SIE∗S∗I∗EβS∗I∗+ω31−I∗I+EE∗−I∗EIE∗εE∗,
If we chooseω2=ω1,ω3=ω1βS∗I∗/εE∗, we obtain(42)dVdt=−ω1μS1−S∗S2+ω11−S∗S+II∗−SIS∗I∗βS∗I∗+ω11−E∗E+SIS∗I∗−SIE∗S∗I∗EβS∗I∗+ω11−I∗I+EE∗−I∗EIE∗βS∗I∗.
Further simplification gives(43)dVdt=−ω1μS1−S∗S2+ω13−S∗S−I∗EIE∗−SIE∗S∗I∗EβS∗I∗.
From the property of arithmetic mean, we have3−S∗/S−I∗E/IE∗−SIE∗/S∗I∗E≤0, and hence(44)dVdt=−ω1μS1−S∗S2+ω13−S∗S−I∗EIE∗−SIE∗S∗I∗EβS∗I∗≤0.
Following the LaSalle’s invariant principle, it is concluded that the endemic equilibrium is globally asymptotically stable.


## 3.1. Local Stability of the Disease-Free Equilibrium

Theorem 5.
The disease-free equilibrium of the fuzzy amoebiasis model (3) is locally asymptotically stable ifℛ0<1and unstable ifℛ0>1.Proof.
We need to show that the Jacobian matrixJP0 of the fuzzy model (3) evaluated at the disease-free equilibrium has negative eigenvalues. Further computation shows that JP0 of the model (3) at P0 is(27)JP0=−μ0−βS∗00−ε+μβS∗00ε−μ+d+γ000γ−μ.
From the matrixJP0, the diagonal entry −μ is one of the eigenvalues of JP0. Eliminating the column containing −μ and its corresponding row, we remain with a 2×2 matrix(28)J∗P0=−ε+μβS∗ε−μ+d+γ.
The characteristic equation ofJ∗P0 is(29)λ2+Aλ+B=0,where A=ε+μ+μ+d+γ,B=ε+μμ+d+γ1−ℛ0. Since at disease-free equilibrium ℛ0<1, then B is positive. Using Routh-Hurwitz criteria we conclude that the roots of the quadratic equation are negative real numbers or complex numbers with real parts. Hence, JP0 has negative eigenvalues, and thus, the disease-free equilibrium is locally asymptotically stable.


## 3.2. Global Stability of the Disease-Free Equilibrium

Definition 6.
(Lyapunov function [37]). Let x=0 be an equilibrium point of x˙t=fxt. A continuously differentiable function Vx is a Lyapunov function if the following conditions hold:A1:V0=0A2:Vx>0,x∈N\0A3:V˙x<0,x∈N\0
HereN\0∈ℝn is the neighbourhood of 0 excluding the origin itself.Theorem 7.
The disease-free equilibrium of the fuzzyamoebiasis model (3) is globally asymptotically stable if ℛ0<1 and unstable if ℛ0>1.Proof.
Here we investigate the global stability of the disease-free equilibrium by mean of Lyapunov direct method. In this method, we define a positive definite Lyapunov functionV=ω1E+ω2I. Clearly,(30)dVdt=ω1dEdt+ω2dIdt,=ω1βIS−ε+μE+ω2εE−μ+d+γI≤ω1βIS∗−ε+μE+ω2εE−μ+d+γI.
If we chooseω1=ε and ω2=ε+μ, we have(31)dVdt≤εβIS∗−ε+μμ+d+γI≤εβS∗ε+μμ+d+γ−1ε+μμ+d+γI≤ℛ0−1ε+μμ+d+γI.
Since at disease-free equilibriumℛ0<1, then dV/dt≤0 and hence, the disease-free equilibrium is globally asymptotically stable.


## 3.3. Bifurcation and Fuzzyℛ0

The stability of disease-free equilibrium changes from stable to unstable asℛ0 increases through 1. When ℛ0=1, the system obtain a bifurcation at the disease-free equilibrium. Since ℛ0 depends on the viral load, v, we need to find the value v∗ such that ℛ0v∗=1 and that value is the bifurcation point.From (15), we have(32)ℛ0v∗=βv∗εΛμε+μμ+dv∗+γv∗.Also, from (4)–(6), we have(33a)βv∗=v∗−vminvm−vmin,ifvmin≤v∗≤vm,(33b)γv∗=γ0−1vmaxv∗+1,if0≤v∗≤vmax,(33c)dv∗=1−d0vmv∗+d0,if0≤v∗≤vm.Using (32) and (29), we can solve for v∗ at ℛ0v∗=1. Further computation shows that v∗ exist and is given by(34)v∗=vmvmaxεΛvmin+με+μvm−vminμ+d0+1εΛvmvmax+με+μvm−vmind0−1vmax+1−γ0vm.Since there is only one endemic equilibrium, the model exhibit forward bifurcation as shown in Figure2.Figure 2
Variation ofℛ0v with βv and γv.


## 3.4. Global Stability of the Endemic Equilibrium

The local stability of the disease-free equilibrium implies local stability of the endemic equilibrium for the reverse condition [38]. Therefore, we only establish the global stability of the endemic equilibrium.Theorem 8.
The endemic equilibrium of the fuzzyamoebiasis model (3) is globally asymptotically stable if ℛ0>1.Proof.
We prove the global stability of the endemic equilibrium via construction of a suitable Lyapunov function of the form(35)Vxi=∑ωixi−xi∗−xi∗lnxixi∗,where ωi>0 is constant to be chosen, xi is the population of the ith compartment, and xi∗ is the equilibrium point [39].
Note that the equation containingR in the model system (3) is independent of the other three equations. Therefore, it can be removed from the analysis since the value of R can be obtained when S,E, and I are known. For that case, we now, consider the Lyapunov function(36)V=ω1S−S∗−S∗lnSS∗+ω2E−E∗−E∗lnEE∗+ω3I−I∗−I∗lnII∗.
It is easily seen thatVS,E,I≥0 and VS∗,E∗,I∗=0. The time derivative of V is then given by(37)dVdt=ω11−S∗SdSdt+ω21−E∗EdEdt+ω31−I∗IdIdt.
From the model system (3) we have(38)dVdt=ω11−S∗SΛ−μS−βSI+ω21−E∗EβSI−ε+μE+ω31−I∗IεE−μ+d+γI.
At endemic equilibrium,Λ=μS∗+βS∗I∗,ε+μ=βS∗I∗/E∗, and μ+d+γ=εE∗/I∗. Therefore,(39)dVdt=ω11−S∗SμS∗+βS∗I∗−μS−βSI+ω21−E∗EβSI−βS∗I∗E∗E+ω31−I∗IεE−εE∗I∗I.
Further simplification gives,(40)dVdt=−ω1μS1−S∗S2+ω11−S∗S1−SIS∗I∗βS∗I∗+ω21−E∗ESIS∗I∗−EE∗βS∗I∗+ω31−I∗IEE∗−II∗εE∗,
On expanding, we have(41)dVdt=−ω1μS1−S∗S2+ω11−S∗S+II∗−SIS∗I∗βS∗I∗+ω21−E∗E+SIS∗I∗−SIE∗S∗I∗EβS∗I∗+ω31−I∗I+EE∗−I∗EIE∗εE∗,
If we chooseω2=ω1,ω3=ω1βS∗I∗/εE∗, we obtain(42)dVdt=−ω1μS1−S∗S2+ω11−S∗S+II∗−SIS∗I∗βS∗I∗+ω11−E∗E+SIS∗I∗−SIE∗S∗I∗EβS∗I∗+ω11−I∗I+EE∗−I∗EIE∗βS∗I∗.
Further simplification gives(43)dVdt=−ω1μS1−S∗S2+ω13−S∗S−I∗EIE∗−SIE∗S∗I∗EβS∗I∗.
From the property of arithmetic mean, we have3−S∗/S−I∗E/IE∗−SIE∗/S∗I∗E≤0, and hence(44)dVdt=−ω1μS1−S∗S2+ω13−S∗S−I∗EIE∗−SIE∗S∗I∗EβS∗I∗≤0.
Following the LaSalle’s invariant principle, it is concluded that the endemic equilibrium is globally asymptotically stable.


## 4. Numerical Results and Discussion

In this section, we perform numerical simulation for the fuzzy model system 2.2 to study the persistence of the disease when an outbreak occur. The results of numerical simulations are presented in the form of figures followed by discussion. Where necessary, numerical values are provided.In numerical simulations, the initial values used to perform simulations areS=1000,E=10,I=1,R=1. Since ingestion of one viable cyst can cause infection then V is taken from 1 to 10. The incubation period is general 2 to 4 weeks [17, 18], but for simulation, we use 2 weeks and hence, the infection rate ε=1/14. The recovery period for amebiasis depends on the severity of the disease. However, the recovery period after medical treatment varies from about 1 to 2 weeks or more [40]. For simulation, we use 2 weeks, and therefore, γ=1/14. The parameter values and their source are as given in Table 2.Table 2
Parameters and their description.
ParameterDescriptionValueSourceΛRecruitment rate of human0.0361[41]λTransmission rate in human0.00045EstimatedεInfectious rate1/14[17, 18].μNatural death rate of human1/65/365[41]dDisease-induced death rate in human0.001EstimatedγRecovery rate in human1/14[40]Figure3 shows the variations of human subpopulation with time. In Figure 3(a), the parameters β,γ and d are crisp, they do not depend on amount of cysts v present in the environment. In Figure 3(b), the parameters β,γ and d are fuzzy, they depend on amount of cysts v present in the environment. The graphs in Figure 3(a) are the S-curves of the classical SEIR model while the graphs in Figure 3(b) are log-normal curves. Both graphs confirm the theoretical result presented from the stability analysis of the model. That is, amoebiasis infection is persistence if ℛ0>1 and it dies away when ℛ0<1. Thus, applying control measures that reduces ℛ0 may to manage the transmission of amoebiasis.Figure 3
Variations of human subpopulation with time. (a) Crispβ, γ and d. (b) Fuzzy β, γ and d.
(a)(b)Figure4 shows the variations of ℛ0v with βv and γv. When v increases it increase βv while reducing γv which in turn reduces recovery rate and increase ℛ0v. Since βv and γv depends on v, that is, the amount of cyst v ingested by a human, personal protection and hygiene to reduce the amount of cyst ingested is crucial.Figure 4
Variation ofℛ0v with βv and γv.Figure2 shows the variations of ℛ0v with v. When v increases from 0 to 3, ℛ0v remains constant, that is, ℛ0v=1. For v>3,ℛ0v is greater 1 showing that there is a point of bifurcation vv∗=3 where the equilibrium shift from disease-free equilibrium to endemic equilibrium.


## 5. Conclusion

Amoebiasis will remain a potentially transmissible protozoan infection worldwide nature of transmission. A fuzzy modelling approach has been used to investigate the dynamics of amoebiasis infection in humans. The basic reproduction number ℛ0 was computed and used to make a comparison with the fuzzy basic reproduction number ℛ0f, and study the stability of the disease-free equilibrium and endemic equilibrium. The stability analysis of equilibrium points indicated that both the disease-free equilibrium and endemic equilibrium of the model system are locally and globally asymptotically stable. The stability of the disease-free equilibrium implies that the amoebiasis infection can be controlled provided that ℛ0<1. Further analysis indicates that the fuzzy parameters β,γ and d depends on the amount of cyst v ingested by a human. The more cysts ingested the higher the transmission rate and the disease-induced rate, thus, making the recovery to be slow. Effective control mechanisms such as personal prevention and hygiene may help to reduce the number of cysts ingested and hence reduce disease transmission. These results call for a study on the optimal control mechanism to look for the best control strategy for the disease.



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*Source: 5292830-2022-04-11.xml*