# Shigellosis Dynamics: Modelling the Effects of Treatment, Sanitation, and Education in the Presence of Carriers

Authors: Stephen Edward; Eunice Mureithi; Nyimvua Shaban
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

A mathematical model forShigellosis including disease carriers with multiple control strategies is developed. We compute the effective reproductive number Re, which is used to analyze the local stability of the equilibria, while the comparison theorem is used to prove global stability. By constructing a suitable Lyapunov function, the model endemic equilibrium is globally asymptotically stable when Re>1. Sensitivity analysis is performed to investigate the parameters that have a high impact on the transmission dynamics of the disease with direct transmission contributing more infections than indirect transmission. The effects of control measures are then investigated both analytically and numerically. Numerical results show that there is a reduction in the number of infections when at least a single control measure is applied efficiently. However, as the number of control interventions increases, Shigellosis elimination is more possible. Results also show that carriers play a potential role in the prevalence of Shigellosis and ignoring these individuals could potentially undermine the efforts of containing this epidemic.


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## Body

## 1. Introduction

Shigellosis is an enteric infectious disease which is caused by Shigella bacteria. These bacteria encompass four subgroups, namely, S. flexneri, S. sonnei, S. dysenteriae 1, and S. boydii [1]. It is responsible for approximately 1.1 million deaths per year worldwide. Approximately two-thirds of those who die from the disease are children under five years of age. It is one of the most common diarrhoea-related causes of morbidity and mortality in children in developing countries [2, 3]. Severe epidemics of dysentery can be caused by S. dysenteriae 1 which produces Shiga toxins, whereas the endemic form of the disease is caused essentially by S. flexneri and S. sonnei [4]. Shigellosis epidemics usually occur in areas with crowding and poor sanitary conditions, where direct transmission or contamination of food or water by the organism is common [5–12]. The disease is marked by fever, violent abdominal cramps, and rectal urgencies. Resistance to multiple antibiotics has recently been observed, including fluoroquinolones [13], which increases the threat of the occurrence of severe disease forms due to lack of efficient treatments. Unfortunately, no vaccine for the disease is available despite multiple and diverse vaccine design strategies [14, 15]. Asymptomatic carriers pose a potential problem when it comes to controlling infectious diseases such as Shigellosis. The problem usually arises because carriers do not show clinical symptoms, as a result they continue infecting others unknowingly. Since they do not show symptoms, efforts to control the disease such as treatment and quarantine/isolation will ignore these individuals. On the contrary, initiatives such as vaccination will wrongly include these individuals because it is difficult to distinguish them from susceptible individuals. Therefore, it is necessary to explore the role played by the carrier in the transmission dynamics of Shigellosis infections.Several scholars have studiedShigellosis in different ways including developing mathematical models (e.g., see [16–19]). Tien and Earn [16] developed a waterborne pathogen model termed as Susceptible-Infectious-Recovered-Water (SIRW); the model incorporated a dual transmission pathway with bilinear incidence rates employed for both the environment-to-human and human-to-human infection routes. They used the model to investigate the distinction between the different transmission routes in the dynamics of waterborne diseases. Chaturvedi et al. [17] studied Shigellosis by a SIRS model, with the assumption that transmission occurs solely via the person-to-person pathway. Nonetheless, Shigellosis can also be contracted indirectly through person-to-environment or vice versa mainly, through food and water. Chen et al. [18] developed a Susceptible-Exposed-Asymptomatic-Infectious- Recovered-Water (SEARW) model that included a water compartment. Berhe et al. [19] developed an SIRB model that included a water compartment, but the model did not capture the role played by carriers and exposed individuals in Shigellosis transmission. Moreover, the model did not capture detection as an intervention in the study.Most previous works have ignored the role played by carriers as such they could not capture interventions like screening (e.g., see [16–19]). Therefore, this study intends to explore the effects of control measures such as sanitation, treatment, and health education campaign on the dynamics of Shigellosis in the presence of carriers.The rest of the paper is organized as follows. Section2 focuses on model formulation, whereas Section 3 is based on the analysis of the model. In Section 4, the effects of control strategies are discussed. Numerical simulation is presented in Section 5, while Section 6 is devoted to sensitivity analysis, and lastly, Section 7 winds up by giving concluding remarks.


## 2. Model Formulation

The model considered here follows the basic Susceptible-Exposed-Infected-Recovered (SEIR) model. This model is an extension of the work done by Tien and Earn [16]; in our case, we add extra classes for carriers and exposed. Indirect transmission is captured by nonlinear incidence function, contrary to previous models whose incidence were captured by linear functions. This is more realistic because nonlinear incidence function shows the presence of a gradual increase in disease incidence between the number of bacteria (B) and the number of susceptible individuals (S) than the counterpart which exhibits a sharp rise of incidence. On top of that, this ensures that the contact rate is bounded.The total human population at timet is subdivided into five mutually exclusive subpopulations: susceptible (St), exposed (Et), infectious (It), carrier (Ct) (i.e., infected individuals who are contagious but do not show any disease symptoms), and recovered (Rt). To incorporate a real biological phenomenon, an additional compartment, Bt, which represents the reservoir of Shigella bacteria in the environment is considered.It is assumed that susceptible individuals are recruited into the population at a constant rate,Λ. Susceptible individuals may acquire Shigella infection following effective direct contact with infectious individuals or carriers at the time-dependent rate λht or after ingesting environmental pathogens from contaminated aquatic reservoirs at the time-dependent rate λpt. Here the term λht represents direct transmission between individuals, and it is modelled by standard mass action principle, whereas λpt represents indirect transmission which is modelled by Holling type-II functional response (Michaelis–Menten function). These forces of infection are given by(1)λht=β1I+β2C,λpt=ϕBK+B,where β1=c1q1 and β2=c2q2 are the transmission rates for infectious and carrier individuals, respectively. We define c1,c2 as the number of contacts infectious and carrier individuals make with susceptibles per unit time, respectively, whereas q1,q2 are the probabilities that the contacts will cause infection. Likewise, K is the half-saturation constant of the bacteria population in water that yields a 50% chance of catching the disease and ϕ>0 is the ingestion rate of Shigella bacteria by individuals. The term B/K+B represents the probability of a susceptible individual to develop Shigellosis per contact. It is assumed that the education campaign has an effect of reducing the number of Shigellosis infections. Therefore, both direct and indirect transmission will be reduced by the rate 1−ρ, and thus the total force of infection will be given by 1−ρλht+λpt, where ρ∈0,1 measures the efficacy of the education campaign. If ρ=0, then it implies that health education has been ignored as an intervention strategy, whereas when ρ=1, it means that education is 100% efficient in limiting the spread of Shigellosis. Exposed individuals may either join infectious or carrier classes. A fraction q of the exposed individuals may progress to the infectious stage at the rate δ while the complement 1−q of the exposed may become carriers at the same rate δ. Carrier individuals do not show any symptoms of Shigellosis even though they remain infectious; this complicates efforts to eliminate Shigellosis. A fraction 1−l of carriers are screened at the rate α and join the infectious individuals where they are finally treated at the rate γ. The remaining fraction l of the carriers recover naturally at the rate η2. Together with treatment measures, infectious individuals may experience natural recovery at the rate η1. Shigellosis-induced mortality rate for infectious individuals is denoted by d1, while the natural death rate of humans is represented by μh. Shigellosis induces temporal immunity that wanes at the rate ω. Therefore, recovered individuals may join the susceptible class when they lose their immunity. Infected individuals from both states It and Ct excrete bacteria into the environment at the reduced rates 1−ρϵ1 and 1−ρϵ2, respectively. It is assumed that the rate of excretion by the infectious individuals, ϵ1, is significantly higher than that by the carrier group, ϵ2. Note that despite low excretion of bacteria by the carrier group, because of its extremely long duration without showing any disease symptoms, the carrier group plays an important role in infection dynamics of Shigellosis. The per capita growth rate of Shigella bacteria is denoted by r while bacteria deplete naturally at a rate μb or by sanitation measures at the rate σ. It is assumed that the growth rate of bacteria (r) cannot exceed its death rate (μb) (that is, r<μb). A full description of the variables and parameters to be used in the model is shown in Tables 1 and 2, respectively. The flow diagram for the dynamics is given in Figure 1.Table 1
State variables and their description.
VariableDescriptionStThe number of susceptible humans at a timet.EtThe number of exposed humans at a timet.ItThe number of infectious humans at a timet.RtThe number of recovered (immune) humans at a timet.CtThe number of carriers at a timet.BtThe number ofShigella bacteria at a time t.Table 2
Parameters and their description.
ParameterDescriptionValueSourceΛRate of recruitment of individuals into the susceptible class462 humans/year[19]β1Transmission rate for infectious individuals0.4/year[19]β2Transmission rates for carriers0.6/year[19]ϕEffective transmission rate ofShigellosis due to environment to human interaction0.4465/year[17]δIncubation rate (rate at which exposed individuals,Et, progress to either class It or Ct)0.35/day[18]μhNatural human mortality rate0.4465/yearAssumedqA fraction of exposed individuals,Et, who progress to class It0.9[18]lA fraction of carriers,Ct, who recover naturally and progress to class Rt0.4AssumedαRate of screening carriers0.56Assumedd1Disease-induced death rate byIt0.02/yearAssumedη1Recovery rate of infectious humans0.14/day[16]η2Recovery rate of carrier humans0.0286/day[18]KpThe environmental carrying capacity forShigella bacteria in the food or water supply104 cells/mLAssumedKHalf-saturation rate ofShigella bacteria that can cause a 50% chance of infection60 cells/mLAssumedϵ1Bacteria shed rate into the water supply by infectious human80 cells/mL/day[20]ϵ2Bacteria shed rate into the water supply by carrier humans human70 cells/mL/day[20]rMaximum per capita growth rate ofShigella bacteria0.73/day[21]μbMortality rate ofShigella bacteria, including phage degradation0.83/dayAssumedωWaning rate of disease-induced immunity0.25/year[16]σRate at which sanitation leads to death ofShigella bacteria2μb/year[22]γTreatment rate for infectious individuals (I)0.4/yearAssumedρEducation efficacy parameter0.6/year[21]Figure 1
A flow diagram forShigellosis transmission dynamics.From Figure1, assumptions and model description the following system of differential equations are(2)dSdt=Λ+ωR−1−ρλht+λpt+μhS,dEdt=1−ρλht+λptS−μh+δE,dIdt=qδE+1−lαC−μh+d1+η1+γI,dCdt=1−qδE−μh+lη2+1−lαC,dRdt=η1+γI+lη2C−μh+ωR,dBdt=rB1−BKp+1−ρϵ1I+1−ρϵ2C−μb+σB.The initial conditions for the model system (2) are S0>0,E0>0,I0>0,C0>0,R0>0,B0>0. Model (2) is biologically meaningful in the invariant region(3)T=S,E,I,C,R,Bt≥0:S+E+I+C+R≤Λμh,Bt≤1−ρϵ1+ϵ2Λμhμb+σ−r.


## 3. Analysis of the Model

### 3.1. Existence of the Equilibrium Solutions

We establish the existence of the equilibrium points. To determine the equilibrium points, we set the right-hand side of system (2) to zero and solve the resulting system:(4)0=Λ+ωR∗−1−ρλ∗+μhS∗,0=1−ρλ∗S∗−μh+δE∗,0=qδE∗+1−lαC−μh+d1+η1+γI∗,0=1−qδE∗−μh+lη2+1−lαC∗,0=η1+γI∗+lη2C∗−μh+ωR∗,0=rB∗1−B∗Kp+1−ρϵ1I∗+1−ρϵ2C∗−μb+σB∗,where(5)λ∗=β1I∗+β2C∗+ϕB∗K+B∗.Solving 4th equation of system (4), we get(6)C∗=1−qδE∗μh+d2+lη2+1−lα.Substitute equation (6) into 3rd equation of system (4) to get(7)I∗=qb2+α1−l1−qδE∗μh+d1+η1+γμh+d2+lη2+1−lα.Substitute equations (6) and (7) into fifth equation and solve for R∗ to get(8)R∗=η1+γQ2ϵ1+lη2Q1ϵ2E∗μh+ω,where(9)Q1=1−qδμh+d2+lη2+1−lα,Q2=qb2+α1−l1−qδμh+d1+η1+γμh+d2+lη2+1−lα.From second equation of system (4), we have(10)1−ρλ∗S∗=μh+δE∗.SubstituteR∗ from equation (8) and λ∗S∗ from equation (10) into the first equation of system (4) to get(11)S∗=Λμh+η1+γQ2ϵ1+lη2Q1ϵ2ωμhμh+ω−μh+δμhE∗.Since the solution of system (4) is feasible only in the invariant region T, expression (11) can only be performed when η1+γQ2ϵ1+lη2Q1ϵ2ω/μhμh+ω−μh+δ/μh≥0, with equality at the disease-free equilibrium (DFE). From the 6th equation of system (4), ignore the logistic growth of bacteria for simplicity to get(12)r−μb+σB∗+ϵ1I∗+1−ρϵ2C∗=0.Solving equation (12), we get(13)B∗=gE∗,whereg=1−ρϵ1Q2+1−ρϵ2Q1μb+σ−r.We know that the force of infection is given by(14)λ∗=β1I∗+β2C∗+ϕB∗K+B∗.Substitute equations (6), (7), and (13) into equation (14) to get(15)λ∗=β1Q2+β2Q1+ϕgK+gE∗E∗.Substituteλ∗ from (15) and S∗ from (11) into second equation of system (4) to get(16)1−ρβ1Q2+β2Q1+ϕgK+gE∗E∗Λμh+hE∗μh=μh+δE∗,resulting into a polynomial of degree three of the form of(17)A2E∗3+A1E∗2+A0E∗=0,where(18)A0=μh+δKμh−Λ1−ρβ1KQ2+β2KQ1+ϕgμh,=μh+δK1−Re,Re=1−ρΛβ1KQ2+β2KQ1+ϕgμhμh+δK,A1=μh+δgμh−1−ρβ1Q2+β2Q1gΛ+hK+hϕgμh,=μh+δg1−1−ρβ1Q2+β2Q1gΛ+hK+hϕgμhμh+δg,A2=gh1−ρβ1Q2+β2Q1μh,h=η1+γQ2ϵ1+lη2Q1ϵ2ωμh+ω−μh+δ.One of the solutions of equation (17) is E∗=0, which confirms the existence of disease-free equilibrium (DFE) while the existence of the endemic equilibrium (EE) is guaranteed by the nonzero solution (E∗>0) of the quadratic equation:(19)A2E∗2+A1E∗+A0=0.To determine the DFE, substituteE∗=0 into equations (6)–(13) to get the DFE as(20)E0=S0,E0,I0,C0,R0,B0=Λμh,0,0,0,0,0.IfE∗>0, then the EE denoted by E1 is given by(21)E1=S∗,E∗,I∗,C∗,R∗,B∗=Λμh+hE∗μh,E∗,Q2E∗,Q1E∗,η1+γQ2ϵ1+lη2Q1ϵ2E∗μh+ω,gE∗,where E∗ is the positive solution of equation (19).


### 3.2. Reproduction Number

We compute the reproduction number,Re, using the next-generation operator approach [23]. The reproduction number is obtained by taking the largest (dominant) eigenvalue (spectral radius) of the matrix(22)FV−1=∂FiE0∂xj∂ViE0∂xj−1,where Fi is the rate of appearance of new infection in compartment i,Vi is the transfer of infections from one compartment i to another, and E0 is the disease-free equilibrium. From system (2), we rewrite the equations with infectious classes, E,I,C, and B. This leads to the system (23)dEdt=1−ρλ1+λ2S−a0E,dIdt=qδE+1−lαC−b1I,dCdt=1−qδE−b2C,dBdt=r1−BKbB+1−ρϵ1I+1−ρϵ2C−μb+σB,where(24)a0=μh+δ,b1=μh+d1+η1+γ,b2=μh+lη2+1−lα.From system (23), we obtain(25)ℱi=1−ρϕBSK+B+β1IS+β2CS000,Fi=a0Eb1I−qδE−1−lαCb2C−1−qδEμb+σB−1−ρϵ1I−1−ρϵ2C−r1−BKbB.Partial derivative ofFi and Vi with respect to E,I,C, and B evaluated at E0 gives(26)F=01−ρβ1S01−ρβ2S01−ρϕS0K000000000000,(27)V=a0000−qδb1−1−lα0−1−qδ0b200−1−ρϵ1−1−ρϵ2−r+μb+σ.The model reproduction number in the presence of control measures (treatment, education campaign, and sanitation) is now given by(28)Re=ρFV−1=R01+R02+R03,where(29)R01=δ1−ρS0b2q+α1−l1−qβ1a0b1b2,R02=δ1−ρS01−qβ2a0b2,R03=δ1−ρS01−ρϵ1b2q+α1−l1−q+b11−qϵ2ϕa0b1b2Kμb+σ−r.Additionally,R0ii=1,2,3 are partial basic reproduction number induced by susceptible-to-infectious transmission, susceptible-to-carrier transmission, and environment-to-susceptible transmission, respectively.


#### 3.2.1. The Basic Reproduction Number

In the absence of all the three control interventions, namely, treatment, education campaign, and sanitation, the basic reproduction is deduced from the reproduction number in equation (28) by setting ρ=0,σ=0,andγ=0. Therefore, the basic reproduction number is given by(30)R0=R1′+R2′+R3′,where(31)R1′=δS0b2q+α1−l1−qβ1a0μh+d1+η1b2,R2′=δS01−qβ2a0b2,R3′=δS0ϵ1b2q+α1−l1−q+μh+d1+η11−qϵ2ϕa0μh+d1+η1b2Kμb−r.Each termRi′i=1,2,3 characterizes the contribution from infectious individuals, carriers, and environment, respectively, whereas a0 and b2 have been defined in equation (24).


### 3.3. Stability Analysis of the Model Equilibria

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

Here we establish the stability of the DFE that is obtained in equation (20). This is stated in Theorem 1 as follows.Theorem 1.
The DFE of model (4) is locally asymptotically stable if Re<1 and unstable if Re>1.Proof.
The partial differentiation of system (4) with respect to S,E,I,C,R,B at the DFE gives the Jacobian matrix J as(32)JE0=−μh0−β11−ρS0−β21−ρS0−1−ρϕS0Kω0−a0β11−ρS0β21−ρS01−ρϕS0K00qδ−b11−lα0001−qδ0−b200001−ρε11−ρε2r−μb−σ000η1+γlη20−μh+σ,where a0,b1, and b2 have been defined in equation (24).
Matrix (32) has two trivial negative eigenvalues λ=−μh and λ=−μh+σ.
If we setH11=a0,H12=β11−ρS0,H13=β21−ρS0,H14=1−ρS0ϕ/K,H21=qδ,H22=b1,H23=1−lα,H31=1−qδ,H33=b2,H42=1−ρϵ1,H43=1−ρϵ2,H44=μb−r+σ, then the remaining 4×4 submatrix is given as(33)J1E0=−H11H12H13H14H21−H22H230H310−H3300H42H43−H44.
The remaining eigenvalues are the roots of the polynomialJ1E0−λ=0, which is given by(34)λ4+c3λ3+c2λ2+c1λ+c0=0,where the constants are such that(35)c3=H11+H22+H33+H44,c2=H22H33+H11H44+H22H44+H33H44+H11H221−Rg+H11H331−Rd,c1=H11H22H441−Rb+Rg+H11H33H441−Rc+Rd+H11H22H331−Rd+Rf+Rg+H22H33H44,c0=H11H22H33H441−Re.
Equivalently,Re can be split into parts(36)Re=Ra+Rb+Rc+Rd+Rf+Rg,where(37)Ra=H14H23H31H42H11H22H33H44,Rb=H14H21H42H42H11H22H44,Rc=H14H31H43H42H11H33H44,Rd=H13H31H11H33,Rf=H12H23H31H11H22H33H44,Rg=H12H21H11H22.
To ensure that all roots of equation (34) have negative real parts, the Routh–Hurwitz stability criterion requires that(38)c3>0,c2>0,c1>0,c0>0,(39)D1=c3>0,D2=c31c1c2=c3c2−c1>0,D3=c310c1c2c30c0c1=c1c2c3−c12−c0c32>0,D4=c3100c1c2c310c0c1c2000c0=c0c1c2c3−c12−c0c32>0.
It is obvious thatD1=c3>0. In addition, if Re<1, it implies that Ra,Rb,Rc,Rd,Rf,Rg<1, and hence c0,c1,c2>0.
Also,D2 can be shown to be positive as follows:(40)D2=c2c3−c1=H11H22H44Rb+2+H33Rf+2+H33H44Rc+2+H3321−Rd+p1+H112H331−Rd+H221−Rg+H44+H22+H33H22+H44H33+H44,where p1=H2221−Rg+H442, and hence D2 is positive.
The only remaining condition to show is(41)D3=c1c2c3−c1−c0c32>0.
To prove inequality (41), it is sufficient to establish the following two inequalities:(42)c1c2c3>2c12,(43)c1c2c3>2c0c32.
To show (42), we write c2c3−2c1 into the sum of the following parts:(44)c2c3−2c1=2H22H44H11Rb+2H33H44H11Rc+H33H1121−Rd+H332H111−Rd+H22H33H11Rd+H33H44H11Rd+2H22H33H11Rf+H22H1121−Rg+H222H111−Rg+H22H33H11Rg+H22H44H11Rg+H44H112+H222H11+H442H11+H22H33H11+H22H44H11+H33H44H11+H22H332+H22H442+H33H442+H222H33+H222H44+H332H44+H22H33H44.
Similarly, to show (43), we write c1c2−2c0c3 into the sum of parts as follows:(45)c1c2−2c0c3=H11H22H332H441−Rd+Rc+Rf+Rg+2Ra+2Rb+H11H222H33H441−Rg+Rb+Rd+Rf+2Ra+2Rc+H112H22H33H441−Rd+Rg+2Ra+Rb+Rc+Rf+RbRd+2RdRg+RcRg+H112H222H331−Rd+Rf+Rg+RdRg+RfRg+Rg2+H11H222H3321−Rd−Rf−Rg+H112H22H3321−Rd1−Rd−Rf−Rg+H11H222H4421−Rb−Rg+H112H22H4421−Rb−Rg+H112H222H441−Rg1−Rb−Rg+H11H332H4421−Rc−Rd+H112H332H441−Rd1−Rc−Rd+H112H33H4421−Rc−Rd+H22H332H442+H222H332H44+H11H22H33H4421+Rb+Rc+Rd+Rg+2Ra+2Rf+H222H33H442.
It can be noted that ifRe<1, then each Ra,Rb,Rc,Rd,Rf,Rg<1 and therefore c1c2−2c0c3>0 and c2c3−2c1>0. Thus, equations (42) and (43) hold and so does condition (41). In the same fashion, the proof for condition D4 can be established from the fact that D4=c0D3. Fortunately, we have already proved that D3>0; therefore, it is clear that D4=c0D3>0. Hence, all conditions of Routh–Hurwitz for this case (equations (38) and (39)) are satisfied; then, the disease-free equilibrium E0 is locally asymptotically stable whenever Re<1.


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

We have the following results on the global stability of the DFE.Theorem 2.
IfRe<1, the DFE is globally asymptotically stable and unstable if Re>1.Proof.
By the comparison theorem, the rate of change of the variables representing the infected components of the model system (2) can be rewritten as(46)dEdtdIdtdCdtdBdt=F−VEICB−01−ρβ1S01−ρβ2S01−ρϕS0K+B000000000000EICB,implying that(47)dEdtdIdtdCdtdBdt≤F−VEICB,where F and V are Jacobian matrices as in (26) and (27). Since the eigenvalues of the matrix F−V have negative real parts (this comes from the stability results in Theorem 1), system (2) is stable whenever Re<1. So, S,E,I,C,B,R⟶S0,0,0,0,0,0 and S⟶S0 as t⟶∞. By the comparison theorem (see [24]), S,E,I,C,B,R⟶E0 as t⟶∞. Therefore, E0 is globally asymptotically stable whenever Re<1.


#### 3.3.3. Global Stability of the Endemic Equilibrium

Theorem 3.
The endemic equilibrium for the model (2) is globally asymptotically stable on T if Re>1.Proof.
Here we construct an explicit Lyapunov function of the form(48)L=∑i=1nwixi−xi∗lnx,where wi is a properly selected positive constant, xi is the population of the ith compartment, and xi∗ is the equilibrium level. We define the Lyapunov function candidate L for model system (2) as(49)L=w1S−S∗lnS+w2E−E∗lnE+w3I−I∗lnI+w4C−C∗lnC+w5R−R∗lnR+w6B−B∗lnB.
The time derivative of the Lyapunov functionL is given by(50)dLdt=w11−S∗SdSdt+w21−E∗EdEdt+w31−I∗IdIdt+w41−C∗CdCdt+w51−R∗RdRdt+w61−B∗BdBdt.
It can be noted that at endemic equilibrium (see equation (4)), we have(51)Λ=−ωR∗+1−ρλ∗+μhS∗,a0=1−ρλ∗S∗E∗,b1=qδE∗+1−lαC∗I∗,b2=1−qδE∗C∗,b3=η1+γI∗+lη2C∗R∗,b4=rB∗1−B∗/Kp+1−ρϵ1I∗+1−ρϵ2C∗B∗,and substituting equation (51) into equation (50) and simplifying can result into the following equation:(52)dLdt=w1ωR1−S∗S1−R∗R−w11−ρλS1−S∗S1−λ∗S∗λS−w1μhS1−S∗S2+w21−ρλS1−E∗E1−λ∗S∗EλSE∗+w3qδE1−I∗I1−E∗IEI∗+w31−lαC1−I∗I1−C∗ICI∗+w41−qδE1−C∗C1−E∗CEC∗+w5η1+γI1−R∗R1−I∗RIR∗+w5lη2C1−R∗R1−C∗RCR∗−w6RB2Kp1−B∗B2+w61−ρϵ1I1−B∗B1−I∗BIB∗+w61−ρϵ2C1−B∗B1−C∗BCB∗=−w1μhS1−S∗S2−w6RB2Kp1−B∗B2+FS,E,I,C,R,B,where F is the balance of the right-hand terms of equation (52). Following the approach by [25–27], F is a nonpositive function for S,E,I,C,R,B>0. Thus, dL/dt<0 for S,E,I,C,R,B>0 and is zero if S=S∗,E=E∗,I=I∗,C=C∗,R=R∗ and B=B∗. Therefore, if Re>1, model (2) has a unique endemic equilibrium point E1 which is globally asymptotically stable.


## 3.1. Existence of the Equilibrium Solutions

We establish the existence of the equilibrium points. To determine the equilibrium points, we set the right-hand side of system (2) to zero and solve the resulting system:(4)0=Λ+ωR∗−1−ρλ∗+μhS∗,0=1−ρλ∗S∗−μh+δE∗,0=qδE∗+1−lαC−μh+d1+η1+γI∗,0=1−qδE∗−μh+lη2+1−lαC∗,0=η1+γI∗+lη2C∗−μh+ωR∗,0=rB∗1−B∗Kp+1−ρϵ1I∗+1−ρϵ2C∗−μb+σB∗,where(5)λ∗=β1I∗+β2C∗+ϕB∗K+B∗.Solving 4th equation of system (4), we get(6)C∗=1−qδE∗μh+d2+lη2+1−lα.Substitute equation (6) into 3rd equation of system (4) to get(7)I∗=qb2+α1−l1−qδE∗μh+d1+η1+γμh+d2+lη2+1−lα.Substitute equations (6) and (7) into fifth equation and solve for R∗ to get(8)R∗=η1+γQ2ϵ1+lη2Q1ϵ2E∗μh+ω,where(9)Q1=1−qδμh+d2+lη2+1−lα,Q2=qb2+α1−l1−qδμh+d1+η1+γμh+d2+lη2+1−lα.From second equation of system (4), we have(10)1−ρλ∗S∗=μh+δE∗.SubstituteR∗ from equation (8) and λ∗S∗ from equation (10) into the first equation of system (4) to get(11)S∗=Λμh+η1+γQ2ϵ1+lη2Q1ϵ2ωμhμh+ω−μh+δμhE∗.Since the solution of system (4) is feasible only in the invariant region T, expression (11) can only be performed when η1+γQ2ϵ1+lη2Q1ϵ2ω/μhμh+ω−μh+δ/μh≥0, with equality at the disease-free equilibrium (DFE). From the 6th equation of system (4), ignore the logistic growth of bacteria for simplicity to get(12)r−μb+σB∗+ϵ1I∗+1−ρϵ2C∗=0.Solving equation (12), we get(13)B∗=gE∗,whereg=1−ρϵ1Q2+1−ρϵ2Q1μb+σ−r.We know that the force of infection is given by(14)λ∗=β1I∗+β2C∗+ϕB∗K+B∗.Substitute equations (6), (7), and (13) into equation (14) to get(15)λ∗=β1Q2+β2Q1+ϕgK+gE∗E∗.Substituteλ∗ from (15) and S∗ from (11) into second equation of system (4) to get(16)1−ρβ1Q2+β2Q1+ϕgK+gE∗E∗Λμh+hE∗μh=μh+δE∗,resulting into a polynomial of degree three of the form of(17)A2E∗3+A1E∗2+A0E∗=0,where(18)A0=μh+δKμh−Λ1−ρβ1KQ2+β2KQ1+ϕgμh,=μh+δK1−Re,Re=1−ρΛβ1KQ2+β2KQ1+ϕgμhμh+δK,A1=μh+δgμh−1−ρβ1Q2+β2Q1gΛ+hK+hϕgμh,=μh+δg1−1−ρβ1Q2+β2Q1gΛ+hK+hϕgμhμh+δg,A2=gh1−ρβ1Q2+β2Q1μh,h=η1+γQ2ϵ1+lη2Q1ϵ2ωμh+ω−μh+δ.One of the solutions of equation (17) is E∗=0, which confirms the existence of disease-free equilibrium (DFE) while the existence of the endemic equilibrium (EE) is guaranteed by the nonzero solution (E∗>0) of the quadratic equation:(19)A2E∗2+A1E∗+A0=0.To determine the DFE, substituteE∗=0 into equations (6)–(13) to get the DFE as(20)E0=S0,E0,I0,C0,R0,B0=Λμh,0,0,0,0,0.IfE∗>0, then the EE denoted by E1 is given by(21)E1=S∗,E∗,I∗,C∗,R∗,B∗=Λμh+hE∗μh,E∗,Q2E∗,Q1E∗,η1+γQ2ϵ1+lη2Q1ϵ2E∗μh+ω,gE∗,where E∗ is the positive solution of equation (19).


## 3.2. Reproduction Number

We compute the reproduction number,Re, using the next-generation operator approach [23]. The reproduction number is obtained by taking the largest (dominant) eigenvalue (spectral radius) of the matrix(22)FV−1=∂FiE0∂xj∂ViE0∂xj−1,where Fi is the rate of appearance of new infection in compartment i,Vi is the transfer of infections from one compartment i to another, and E0 is the disease-free equilibrium. From system (2), we rewrite the equations with infectious classes, E,I,C, and B. This leads to the system (23)dEdt=1−ρλ1+λ2S−a0E,dIdt=qδE+1−lαC−b1I,dCdt=1−qδE−b2C,dBdt=r1−BKbB+1−ρϵ1I+1−ρϵ2C−μb+σB,where(24)a0=μh+δ,b1=μh+d1+η1+γ,b2=μh+lη2+1−lα.From system (23), we obtain(25)ℱi=1−ρϕBSK+B+β1IS+β2CS000,Fi=a0Eb1I−qδE−1−lαCb2C−1−qδEμb+σB−1−ρϵ1I−1−ρϵ2C−r1−BKbB.Partial derivative ofFi and Vi with respect to E,I,C, and B evaluated at E0 gives(26)F=01−ρβ1S01−ρβ2S01−ρϕS0K000000000000,(27)V=a0000−qδb1−1−lα0−1−qδ0b200−1−ρϵ1−1−ρϵ2−r+μb+σ.The model reproduction number in the presence of control measures (treatment, education campaign, and sanitation) is now given by(28)Re=ρFV−1=R01+R02+R03,where(29)R01=δ1−ρS0b2q+α1−l1−qβ1a0b1b2,R02=δ1−ρS01−qβ2a0b2,R03=δ1−ρS01−ρϵ1b2q+α1−l1−q+b11−qϵ2ϕa0b1b2Kμb+σ−r.Additionally,R0ii=1,2,3 are partial basic reproduction number induced by susceptible-to-infectious transmission, susceptible-to-carrier transmission, and environment-to-susceptible transmission, respectively.


### 3.2.1. The Basic Reproduction Number

In the absence of all the three control interventions, namely, treatment, education campaign, and sanitation, the basic reproduction is deduced from the reproduction number in equation (28) by setting ρ=0,σ=0,andγ=0. Therefore, the basic reproduction number is given by(30)R0=R1′+R2′+R3′,where(31)R1′=δS0b2q+α1−l1−qβ1a0μh+d1+η1b2,R2′=δS01−qβ2a0b2,R3′=δS0ϵ1b2q+α1−l1−q+μh+d1+η11−qϵ2ϕa0μh+d1+η1b2Kμb−r.Each termRi′i=1,2,3 characterizes the contribution from infectious individuals, carriers, and environment, respectively, whereas a0 and b2 have been defined in equation (24).


## 3.2.1. The Basic Reproduction Number

In the absence of all the three control interventions, namely, treatment, education campaign, and sanitation, the basic reproduction is deduced from the reproduction number in equation (28) by setting ρ=0,σ=0,andγ=0. Therefore, the basic reproduction number is given by(30)R0=R1′+R2′+R3′,where(31)R1′=δS0b2q+α1−l1−qβ1a0μh+d1+η1b2,R2′=δS01−qβ2a0b2,R3′=δS0ϵ1b2q+α1−l1−q+μh+d1+η11−qϵ2ϕa0μh+d1+η1b2Kμb−r.Each termRi′i=1,2,3 characterizes the contribution from infectious individuals, carriers, and environment, respectively, whereas a0 and b2 have been defined in equation (24).


## 3.3. Stability Analysis of the Model Equilibria

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

Here we establish the stability of the DFE that is obtained in equation (20). This is stated in Theorem 1 as follows.Theorem 1.
The DFE of model (4) is locally asymptotically stable if Re<1 and unstable if Re>1.Proof.
The partial differentiation of system (4) with respect to S,E,I,C,R,B at the DFE gives the Jacobian matrix J as(32)JE0=−μh0−β11−ρS0−β21−ρS0−1−ρϕS0Kω0−a0β11−ρS0β21−ρS01−ρϕS0K00qδ−b11−lα0001−qδ0−b200001−ρε11−ρε2r−μb−σ000η1+γlη20−μh+σ,where a0,b1, and b2 have been defined in equation (24).
Matrix (32) has two trivial negative eigenvalues λ=−μh and λ=−μh+σ.
If we setH11=a0,H12=β11−ρS0,H13=β21−ρS0,H14=1−ρS0ϕ/K,H21=qδ,H22=b1,H23=1−lα,H31=1−qδ,H33=b2,H42=1−ρϵ1,H43=1−ρϵ2,H44=μb−r+σ, then the remaining 4×4 submatrix is given as(33)J1E0=−H11H12H13H14H21−H22H230H310−H3300H42H43−H44.
The remaining eigenvalues are the roots of the polynomialJ1E0−λ=0, which is given by(34)λ4+c3λ3+c2λ2+c1λ+c0=0,where the constants are such that(35)c3=H11+H22+H33+H44,c2=H22H33+H11H44+H22H44+H33H44+H11H221−Rg+H11H331−Rd,c1=H11H22H441−Rb+Rg+H11H33H441−Rc+Rd+H11H22H331−Rd+Rf+Rg+H22H33H44,c0=H11H22H33H441−Re.
Equivalently,Re can be split into parts(36)Re=Ra+Rb+Rc+Rd+Rf+Rg,where(37)Ra=H14H23H31H42H11H22H33H44,Rb=H14H21H42H42H11H22H44,Rc=H14H31H43H42H11H33H44,Rd=H13H31H11H33,Rf=H12H23H31H11H22H33H44,Rg=H12H21H11H22.
To ensure that all roots of equation (34) have negative real parts, the Routh–Hurwitz stability criterion requires that(38)c3>0,c2>0,c1>0,c0>0,(39)D1=c3>0,D2=c31c1c2=c3c2−c1>0,D3=c310c1c2c30c0c1=c1c2c3−c12−c0c32>0,D4=c3100c1c2c310c0c1c2000c0=c0c1c2c3−c12−c0c32>0.
It is obvious thatD1=c3>0. In addition, if Re<1, it implies that Ra,Rb,Rc,Rd,Rf,Rg<1, and hence c0,c1,c2>0.
Also,D2 can be shown to be positive as follows:(40)D2=c2c3−c1=H11H22H44Rb+2+H33Rf+2+H33H44Rc+2+H3321−Rd+p1+H112H331−Rd+H221−Rg+H44+H22+H33H22+H44H33+H44,where p1=H2221−Rg+H442, and hence D2 is positive.
The only remaining condition to show is(41)D3=c1c2c3−c1−c0c32>0.
To prove inequality (41), it is sufficient to establish the following two inequalities:(42)c1c2c3>2c12,(43)c1c2c3>2c0c32.
To show (42), we write c2c3−2c1 into the sum of the following parts:(44)c2c3−2c1=2H22H44H11Rb+2H33H44H11Rc+H33H1121−Rd+H332H111−Rd+H22H33H11Rd+H33H44H11Rd+2H22H33H11Rf+H22H1121−Rg+H222H111−Rg+H22H33H11Rg+H22H44H11Rg+H44H112+H222H11+H442H11+H22H33H11+H22H44H11+H33H44H11+H22H332+H22H442+H33H442+H222H33+H222H44+H332H44+H22H33H44.
Similarly, to show (43), we write c1c2−2c0c3 into the sum of parts as follows:(45)c1c2−2c0c3=H11H22H332H441−Rd+Rc+Rf+Rg+2Ra+2Rb+H11H222H33H441−Rg+Rb+Rd+Rf+2Ra+2Rc+H112H22H33H441−Rd+Rg+2Ra+Rb+Rc+Rf+RbRd+2RdRg+RcRg+H112H222H331−Rd+Rf+Rg+RdRg+RfRg+Rg2+H11H222H3321−Rd−Rf−Rg+H112H22H3321−Rd1−Rd−Rf−Rg+H11H222H4421−Rb−Rg+H112H22H4421−Rb−Rg+H112H222H441−Rg1−Rb−Rg+H11H332H4421−Rc−Rd+H112H332H441−Rd1−Rc−Rd+H112H33H4421−Rc−Rd+H22H332H442+H222H332H44+H11H22H33H4421+Rb+Rc+Rd+Rg+2Ra+2Rf+H222H33H442.
It can be noted that ifRe<1, then each Ra,Rb,Rc,Rd,Rf,Rg<1 and therefore c1c2−2c0c3>0 and c2c3−2c1>0. Thus, equations (42) and (43) hold and so does condition (41). In the same fashion, the proof for condition D4 can be established from the fact that D4=c0D3. Fortunately, we have already proved that D3>0; therefore, it is clear that D4=c0D3>0. Hence, all conditions of Routh–Hurwitz for this case (equations (38) and (39)) are satisfied; then, the disease-free equilibrium E0 is locally asymptotically stable whenever Re<1.


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

We have the following results on the global stability of the DFE.Theorem 2.
IfRe<1, the DFE is globally asymptotically stable and unstable if Re>1.Proof.
By the comparison theorem, the rate of change of the variables representing the infected components of the model system (2) can be rewritten as(46)dEdtdIdtdCdtdBdt=F−VEICB−01−ρβ1S01−ρβ2S01−ρϕS0K+B000000000000EICB,implying that(47)dEdtdIdtdCdtdBdt≤F−VEICB,where F and V are Jacobian matrices as in (26) and (27). Since the eigenvalues of the matrix F−V have negative real parts (this comes from the stability results in Theorem 1), system (2) is stable whenever Re<1. So, S,E,I,C,B,R⟶S0,0,0,0,0,0 and S⟶S0 as t⟶∞. By the comparison theorem (see [24]), S,E,I,C,B,R⟶E0 as t⟶∞. Therefore, E0 is globally asymptotically stable whenever Re<1.


### 3.3.3. Global Stability of the Endemic Equilibrium

Theorem 3.
The endemic equilibrium for the model (2) is globally asymptotically stable on T if Re>1.Proof.
Here we construct an explicit Lyapunov function of the form(48)L=∑i=1nwixi−xi∗lnx,where wi is a properly selected positive constant, xi is the population of the ith compartment, and xi∗ is the equilibrium level. We define the Lyapunov function candidate L for model system (2) as(49)L=w1S−S∗lnS+w2E−E∗lnE+w3I−I∗lnI+w4C−C∗lnC+w5R−R∗lnR+w6B−B∗lnB.
The time derivative of the Lyapunov functionL is given by(50)dLdt=w11−S∗SdSdt+w21−E∗EdEdt+w31−I∗IdIdt+w41−C∗CdCdt+w51−R∗RdRdt+w61−B∗BdBdt.
It can be noted that at endemic equilibrium (see equation (4)), we have(51)Λ=−ωR∗+1−ρλ∗+μhS∗,a0=1−ρλ∗S∗E∗,b1=qδE∗+1−lαC∗I∗,b2=1−qδE∗C∗,b3=η1+γI∗+lη2C∗R∗,b4=rB∗1−B∗/Kp+1−ρϵ1I∗+1−ρϵ2C∗B∗,and substituting equation (51) into equation (50) and simplifying can result into the following equation:(52)dLdt=w1ωR1−S∗S1−R∗R−w11−ρλS1−S∗S1−λ∗S∗λS−w1μhS1−S∗S2+w21−ρλS1−E∗E1−λ∗S∗EλSE∗+w3qδE1−I∗I1−E∗IEI∗+w31−lαC1−I∗I1−C∗ICI∗+w41−qδE1−C∗C1−E∗CEC∗+w5η1+γI1−R∗R1−I∗RIR∗+w5lη2C1−R∗R1−C∗RCR∗−w6RB2Kp1−B∗B2+w61−ρϵ1I1−B∗B1−I∗BIB∗+w61−ρϵ2C1−B∗B1−C∗BCB∗=−w1μhS1−S∗S2−w6RB2Kp1−B∗B2+FS,E,I,C,R,B,where F is the balance of the right-hand terms of equation (52). Following the approach by [25–27], F is a nonpositive function for S,E,I,C,R,B>0. Thus, dL/dt<0 for S,E,I,C,R,B>0 and is zero if S=S∗,E=E∗,I=I∗,C=C∗,R=R∗ and B=B∗. Therefore, if Re>1, model (2) has a unique endemic equilibrium point E1 which is globally asymptotically stable.


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

Here we establish the stability of the DFE that is obtained in equation (20). This is stated in Theorem 1 as follows.Theorem 1.
The DFE of model (4) is locally asymptotically stable if Re<1 and unstable if Re>1.Proof.
The partial differentiation of system (4) with respect to S,E,I,C,R,B at the DFE gives the Jacobian matrix J as(32)JE0=−μh0−β11−ρS0−β21−ρS0−1−ρϕS0Kω0−a0β11−ρS0β21−ρS01−ρϕS0K00qδ−b11−lα0001−qδ0−b200001−ρε11−ρε2r−μb−σ000η1+γlη20−μh+σ,where a0,b1, and b2 have been defined in equation (24).
Matrix (32) has two trivial negative eigenvalues λ=−μh and λ=−μh+σ.
If we setH11=a0,H12=β11−ρS0,H13=β21−ρS0,H14=1−ρS0ϕ/K,H21=qδ,H22=b1,H23=1−lα,H31=1−qδ,H33=b2,H42=1−ρϵ1,H43=1−ρϵ2,H44=μb−r+σ, then the remaining 4×4 submatrix is given as(33)J1E0=−H11H12H13H14H21−H22H230H310−H3300H42H43−H44.
The remaining eigenvalues are the roots of the polynomialJ1E0−λ=0, which is given by(34)λ4+c3λ3+c2λ2+c1λ+c0=0,where the constants are such that(35)c3=H11+H22+H33+H44,c2=H22H33+H11H44+H22H44+H33H44+H11H221−Rg+H11H331−Rd,c1=H11H22H441−Rb+Rg+H11H33H441−Rc+Rd+H11H22H331−Rd+Rf+Rg+H22H33H44,c0=H11H22H33H441−Re.
Equivalently,Re can be split into parts(36)Re=Ra+Rb+Rc+Rd+Rf+Rg,where(37)Ra=H14H23H31H42H11H22H33H44,Rb=H14H21H42H42H11H22H44,Rc=H14H31H43H42H11H33H44,Rd=H13H31H11H33,Rf=H12H23H31H11H22H33H44,Rg=H12H21H11H22.
To ensure that all roots of equation (34) have negative real parts, the Routh–Hurwitz stability criterion requires that(38)c3>0,c2>0,c1>0,c0>0,(39)D1=c3>0,D2=c31c1c2=c3c2−c1>0,D3=c310c1c2c30c0c1=c1c2c3−c12−c0c32>0,D4=c3100c1c2c310c0c1c2000c0=c0c1c2c3−c12−c0c32>0.
It is obvious thatD1=c3>0. In addition, if Re<1, it implies that Ra,Rb,Rc,Rd,Rf,Rg<1, and hence c0,c1,c2>0.
Also,D2 can be shown to be positive as follows:(40)D2=c2c3−c1=H11H22H44Rb+2+H33Rf+2+H33H44Rc+2+H3321−Rd+p1+H112H331−Rd+H221−Rg+H44+H22+H33H22+H44H33+H44,where p1=H2221−Rg+H442, and hence D2 is positive.
The only remaining condition to show is(41)D3=c1c2c3−c1−c0c32>0.
To prove inequality (41), it is sufficient to establish the following two inequalities:(42)c1c2c3>2c12,(43)c1c2c3>2c0c32.
To show (42), we write c2c3−2c1 into the sum of the following parts:(44)c2c3−2c1=2H22H44H11Rb+2H33H44H11Rc+H33H1121−Rd+H332H111−Rd+H22H33H11Rd+H33H44H11Rd+2H22H33H11Rf+H22H1121−Rg+H222H111−Rg+H22H33H11Rg+H22H44H11Rg+H44H112+H222H11+H442H11+H22H33H11+H22H44H11+H33H44H11+H22H332+H22H442+H33H442+H222H33+H222H44+H332H44+H22H33H44.
Similarly, to show (43), we write c1c2−2c0c3 into the sum of parts as follows:(45)c1c2−2c0c3=H11H22H332H441−Rd+Rc+Rf+Rg+2Ra+2Rb+H11H222H33H441−Rg+Rb+Rd+Rf+2Ra+2Rc+H112H22H33H441−Rd+Rg+2Ra+Rb+Rc+Rf+RbRd+2RdRg+RcRg+H112H222H331−Rd+Rf+Rg+RdRg+RfRg+Rg2+H11H222H3321−Rd−Rf−Rg+H112H22H3321−Rd1−Rd−Rf−Rg+H11H222H4421−Rb−Rg+H112H22H4421−Rb−Rg+H112H222H441−Rg1−Rb−Rg+H11H332H4421−Rc−Rd+H112H332H441−Rd1−Rc−Rd+H112H33H4421−Rc−Rd+H22H332H442+H222H332H44+H11H22H33H4421+Rb+Rc+Rd+Rg+2Ra+2Rf+H222H33H442.
It can be noted that ifRe<1, then each Ra,Rb,Rc,Rd,Rf,Rg<1 and therefore c1c2−2c0c3>0 and c2c3−2c1>0. Thus, equations (42) and (43) hold and so does condition (41). In the same fashion, the proof for condition D4 can be established from the fact that D4=c0D3. Fortunately, we have already proved that D3>0; therefore, it is clear that D4=c0D3>0. Hence, all conditions of Routh–Hurwitz for this case (equations (38) and (39)) are satisfied; then, the disease-free equilibrium E0 is locally asymptotically stable whenever Re<1.


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

We have the following results on the global stability of the DFE.Theorem 2.
IfRe<1, the DFE is globally asymptotically stable and unstable if Re>1.Proof.
By the comparison theorem, the rate of change of the variables representing the infected components of the model system (2) can be rewritten as(46)dEdtdIdtdCdtdBdt=F−VEICB−01−ρβ1S01−ρβ2S01−ρϕS0K+B000000000000EICB,implying that(47)dEdtdIdtdCdtdBdt≤F−VEICB,where F and V are Jacobian matrices as in (26) and (27). Since the eigenvalues of the matrix F−V have negative real parts (this comes from the stability results in Theorem 1), system (2) is stable whenever Re<1. So, S,E,I,C,B,R⟶S0,0,0,0,0,0 and S⟶S0 as t⟶∞. By the comparison theorem (see [24]), S,E,I,C,B,R⟶E0 as t⟶∞. Therefore, E0 is globally asymptotically stable whenever Re<1.


## 3.3.3. Global Stability of the Endemic Equilibrium

Theorem 3.
The endemic equilibrium for the model (2) is globally asymptotically stable on T if Re>1.Proof.
Here we construct an explicit Lyapunov function of the form(48)L=∑i=1nwixi−xi∗lnx,where wi is a properly selected positive constant, xi is the population of the ith compartment, and xi∗ is the equilibrium level. We define the Lyapunov function candidate L for model system (2) as(49)L=w1S−S∗lnS+w2E−E∗lnE+w3I−I∗lnI+w4C−C∗lnC+w5R−R∗lnR+w6B−B∗lnB.
The time derivative of the Lyapunov functionL is given by(50)dLdt=w11−S∗SdSdt+w21−E∗EdEdt+w31−I∗IdIdt+w41−C∗CdCdt+w51−R∗RdRdt+w61−B∗BdBdt.
It can be noted that at endemic equilibrium (see equation (4)), we have(51)Λ=−ωR∗+1−ρλ∗+μhS∗,a0=1−ρλ∗S∗E∗,b1=qδE∗+1−lαC∗I∗,b2=1−qδE∗C∗,b3=η1+γI∗+lη2C∗R∗,b4=rB∗1−B∗/Kp+1−ρϵ1I∗+1−ρϵ2C∗B∗,and substituting equation (51) into equation (50) and simplifying can result into the following equation:(52)dLdt=w1ωR1−S∗S1−R∗R−w11−ρλS1−S∗S1−λ∗S∗λS−w1μhS1−S∗S2+w21−ρλS1−E∗E1−λ∗S∗EλSE∗+w3qδE1−I∗I1−E∗IEI∗+w31−lαC1−I∗I1−C∗ICI∗+w41−qδE1−C∗C1−E∗CEC∗+w5η1+γI1−R∗R1−I∗RIR∗+w5lη2C1−R∗R1−C∗RCR∗−w6RB2Kp1−B∗B2+w61−ρϵ1I1−B∗B1−I∗BIB∗+w61−ρϵ2C1−B∗B1−C∗BCB∗=−w1μhS1−S∗S2−w6RB2Kp1−B∗B2+FS,E,I,C,R,B,where F is the balance of the right-hand terms of equation (52). Following the approach by [25–27], F is a nonpositive function for S,E,I,C,R,B>0. Thus, dL/dt<0 for S,E,I,C,R,B>0 and is zero if S=S∗,E=E∗,I=I∗,C=C∗,R=R∗ and B=B∗. Therefore, if Re>1, model (2) has a unique endemic equilibrium point E1 which is globally asymptotically stable.


## 4. Effects of Control Intervention Strategies

We investigate the impacts of implementing control interventions, either singly or in a combination. Most governments have to invest in facilities for combatingShigellosis such as sanitation services, ensuring the availability of drugs in health care centres as well as emphasizing on education campaigns on how to avoid the disease. So, it is imperative to have a clear understanding of the benefits of implementing a different combination of intervention strategies. Different countries have different economic status as such a different ability to deal with diseases. Most low-income countries are unable to overcome various diseases such as Shigellosis because they invest fewer resources to overcome it. However, rich countries have eliminated or have minor cases of waterborne diseases like Shigellosis since they can target all possible means of transmission and are prepared in advance to tackle this epidemic once it erupts. Reproductive thresholds for all possible cases ranging from single through three control intervention were calculated and compared among themselves. The primary purpose of the comparison was to determine which of them has a significant influence in diminishing Shigellosis.


### 4.1. Effects of Multiple Control Intervention Strategies

We focus on the effects of a combination of two or three strategies simultaneously.


#### 4.1.1. Effects of Treatment, Education Campaign, and Sanitation

In the presence of three interventions, that is,ρ≠0,σ≠0, and γ≠0, the reproduction number-induced by treatment, education, and sanitation(53)Re=R01+R02+R03,where R01, R02, and R03 have been defined in equation (29).


#### 4.1.2. Effects of Treatment and Education Campaign

In the absence of sanitation (σ=0), the treatment and education-induced reproduction number is(54)RTed=R01+R02+R03″,where(55)R03″=δ1−ρS01−ρϵ1b2q+α1−l1−q+b11−qϵ2ϕa0b1b2Kμb−r,where R01and R02 have been defined in equation (29) and R03″ represents the contribution from the surroundings to Shigellosis transmissions in absence of sanitation effort but in the presence of education and medical treatment. Together with that, a0,b1, and b2 have been defined in equation (24). Since μb+σ−r>μb−r, then R03<R03′′, and hence Re<RTed, which shows that even though treatment and education campaigns can reduce the spread of infections, multiple control strategy that accounts for all three controls (Re) will yield a better result.


#### 4.1.3. Effects of Sanitation and Treatment

In the absence of education campaign (ρ=0), the sanitation and treatment-induced reproduction number is(56)RST=R1+R2+R3,where(57)R1=δS0b2q+α1−l1−qβ1a0b1b2,R2=δS01−qβ2a0b2,R3=δS0ϵ1b2q+α1−l1−q+b11−qϵ2ϕa0b1b2Kμb+σ−r,where R1,R2, and R3 represent the contributions from infectious individuals, carriers, and environment-to-host transmission, respectively. It must be noted that a0,b1, and b2 have been defined in equation (24). It is possible to express R01=1−ρR1,R02=1−ρR2,R03=1−ρ2R3. Since ρ∈0,1, then it is clear to see that R01≤R1,R02≤R2 and R03≤R3; therefore, Re<RST. This shows that sanitation and treatment alone are not sufficient to eliminate Shigellosis infections; there is a need for an additional control intervention strategy to bring the disease to an end. The strategy that takes care of all three controls (Re) will yield a far better result. This result agrees with intuitive expectations.


#### 4.1.4. Effects of Sanitation and Education

In the absence of treatment (γ=0), the sanitation and education-induced reproduction number is(58)RSed=R01′+R02+R03′,where(59)R01′=δ1−ρS0b2q+α1−l1−qβ1a0μh+d1+η1b2,R03′=δ1−ρ2S0ϵ1b2q+α1−l1−q+μh+d1+η11−qϵ2ϕa0μh+d1+η1b2Kμb+σ−r,where R01′ and R03′ represent the contributions from infectious individuals and contribution from environment-to-host transmission, respectively, whereas R02 has been defined in equation (29). Since μh+d1+η1<b1, then R01′>R01 and R03′>R03; therefore, Re<RSed. The result shows that once the strategy that combines sanitation and education campaigns is implemented thoroughly in the community, it can reduce the severity of the disease to a certain extent. However, when all three control interventions are implemented Re, the result is more appealing.


### 4.2. Effects of Single Control Intervention Strategy

We focus on the effects of each of the controls individually.


#### 4.2.1. Effects of Treatment

In the absence of education campaign (ρ=0) and sanitation (σ=0), the treatment-induced reproduction number is(60)RT=R1+R2+R3′′,where(61)R3″=δS0ϵ1b2q+α1−l1−q+b11−qϵ2ϕa0b1b2Kμb−r,where R3″ represents the contribution from the surroundings to Shigellosis transmissions in absence of sanitation effort, whereas, R1,R2 have been defined in equation (57). From equation (59), it has been shown that R1>R01,R2>R02. Since 1/μb+σ−r<1/μb−r, it implies that R3<R3″; hence, RST<RT. Since Re<RST, we can summarize the inequality as Re<RST<RT. This inequality suggests that three control strategies are far better than two strategies, and two strategies are far better than single strategy.


#### 4.2.2. Effects of Education Campaign

In the absence of treatment (γ=0) and water purification (σ=0), the education-induced reproduction number is(62)Red=R01′+R2+R03″′,where(63)R03″′=δ1−ρS01−ρϵ1b2q+α1−l1−q+μh+d1+η11−qϵ2ϕa0μh+d1+η1b2Kμb−r,where R03″′ represents the contribution from the surroundings to Shigellosis transmissions and R01′ is defined in equation (59), whereas R2 is defined in equation (57).


#### 4.2.3. Effects of Sanitation Strategy

In the absence of treatment (γ=0) and education campaigns (ρ=0), the sanitation-induced reproduction number is(64)RS=R1′+R2+R3″′,where(65)R3″′=δS0ϵ1b2q+α1−l1−q+μh+d1+η11−qϵ2ϕa0μh+d1+η1b2Kμb+σ−r,where R3″′ represents the contribution from the surroundings to Shigellosis transmissions, whereas R1′ is defined in (30) and R2 is defined in equation (57). It can be seen that Re<RS since R1′>R01,R2>R02 and R3″′>R03. Besides that, one can note that R3″′<R3′, which implies RS<R0. We use numerical simulations (Figure 2) to compare all reproduction numbers.Figure 2
The comparison of single strategies (a), double strategies (b), and three control strategies (c). Basic reproduction number (R0) is the largest followed by single induced reproduction numbers and then the dual controls and the least being reproduction number capturing all three controls (Re). Note that reproduction number with education and treatment efforts (RTed) has coincided with Re.
(a)(b)(c)


## 4.1. Effects of Multiple Control Intervention Strategies

We focus on the effects of a combination of two or three strategies simultaneously.


### 4.1.1. Effects of Treatment, Education Campaign, and Sanitation

In the presence of three interventions, that is,ρ≠0,σ≠0, and γ≠0, the reproduction number-induced by treatment, education, and sanitation(53)Re=R01+R02+R03,where R01, R02, and R03 have been defined in equation (29).


### 4.1.2. Effects of Treatment and Education Campaign

In the absence of sanitation (σ=0), the treatment and education-induced reproduction number is(54)RTed=R01+R02+R03″,where(55)R03″=δ1−ρS01−ρϵ1b2q+α1−l1−q+b11−qϵ2ϕa0b1b2Kμb−r,where R01and R02 have been defined in equation (29) and R03″ represents the contribution from the surroundings to Shigellosis transmissions in absence of sanitation effort but in the presence of education and medical treatment. Together with that, a0,b1, and b2 have been defined in equation (24). Since μb+σ−r>μb−r, then R03<R03′′, and hence Re<RTed, which shows that even though treatment and education campaigns can reduce the spread of infections, multiple control strategy that accounts for all three controls (Re) will yield a better result.


### 4.1.3. Effects of Sanitation and Treatment

In the absence of education campaign (ρ=0), the sanitation and treatment-induced reproduction number is(56)RST=R1+R2+R3,where(57)R1=δS0b2q+α1−l1−qβ1a0b1b2,R2=δS01−qβ2a0b2,R3=δS0ϵ1b2q+α1−l1−q+b11−qϵ2ϕa0b1b2Kμb+σ−r,where R1,R2, and R3 represent the contributions from infectious individuals, carriers, and environment-to-host transmission, respectively. It must be noted that a0,b1, and b2 have been defined in equation (24). It is possible to express R01=1−ρR1,R02=1−ρR2,R03=1−ρ2R3. Since ρ∈0,1, then it is clear to see that R01≤R1,R02≤R2 and R03≤R3; therefore, Re<RST. This shows that sanitation and treatment alone are not sufficient to eliminate Shigellosis infections; there is a need for an additional control intervention strategy to bring the disease to an end. The strategy that takes care of all three controls (Re) will yield a far better result. This result agrees with intuitive expectations.


### 4.1.4. Effects of Sanitation and Education

In the absence of treatment (γ=0), the sanitation and education-induced reproduction number is(58)RSed=R01′+R02+R03′,where(59)R01′=δ1−ρS0b2q+α1−l1−qβ1a0μh+d1+η1b2,R03′=δ1−ρ2S0ϵ1b2q+α1−l1−q+μh+d1+η11−qϵ2ϕa0μh+d1+η1b2Kμb+σ−r,where R01′ and R03′ represent the contributions from infectious individuals and contribution from environment-to-host transmission, respectively, whereas R02 has been defined in equation (29). Since μh+d1+η1<b1, then R01′>R01 and R03′>R03; therefore, Re<RSed. The result shows that once the strategy that combines sanitation and education campaigns is implemented thoroughly in the community, it can reduce the severity of the disease to a certain extent. However, when all three control interventions are implemented Re, the result is more appealing.


## 4.1.1. Effects of Treatment, Education Campaign, and Sanitation

In the presence of three interventions, that is,ρ≠0,σ≠0, and γ≠0, the reproduction number-induced by treatment, education, and sanitation(53)Re=R01+R02+R03,where R01, R02, and R03 have been defined in equation (29).


## 4.1.2. Effects of Treatment and Education Campaign

In the absence of sanitation (σ=0), the treatment and education-induced reproduction number is(54)RTed=R01+R02+R03″,where(55)R03″=δ1−ρS01−ρϵ1b2q+α1−l1−q+b11−qϵ2ϕa0b1b2Kμb−r,where R01and R02 have been defined in equation (29) and R03″ represents the contribution from the surroundings to Shigellosis transmissions in absence of sanitation effort but in the presence of education and medical treatment. Together with that, a0,b1, and b2 have been defined in equation (24). Since μb+σ−r>μb−r, then R03<R03′′, and hence Re<RTed, which shows that even though treatment and education campaigns can reduce the spread of infections, multiple control strategy that accounts for all three controls (Re) will yield a better result.


## 4.1.3. Effects of Sanitation and Treatment

In the absence of education campaign (ρ=0), the sanitation and treatment-induced reproduction number is(56)RST=R1+R2+R3,where(57)R1=δS0b2q+α1−l1−qβ1a0b1b2,R2=δS01−qβ2a0b2,R3=δS0ϵ1b2q+α1−l1−q+b11−qϵ2ϕa0b1b2Kμb+σ−r,where R1,R2, and R3 represent the contributions from infectious individuals, carriers, and environment-to-host transmission, respectively. It must be noted that a0,b1, and b2 have been defined in equation (24). It is possible to express R01=1−ρR1,R02=1−ρR2,R03=1−ρ2R3. Since ρ∈0,1, then it is clear to see that R01≤R1,R02≤R2 and R03≤R3; therefore, Re<RST. This shows that sanitation and treatment alone are not sufficient to eliminate Shigellosis infections; there is a need for an additional control intervention strategy to bring the disease to an end. The strategy that takes care of all three controls (Re) will yield a far better result. This result agrees with intuitive expectations.


## 4.1.4. Effects of Sanitation and Education

In the absence of treatment (γ=0), the sanitation and education-induced reproduction number is(58)RSed=R01′+R02+R03′,where(59)R01′=δ1−ρS0b2q+α1−l1−qβ1a0μh+d1+η1b2,R03′=δ1−ρ2S0ϵ1b2q+α1−l1−q+μh+d1+η11−qϵ2ϕa0μh+d1+η1b2Kμb+σ−r,where R01′ and R03′ represent the contributions from infectious individuals and contribution from environment-to-host transmission, respectively, whereas R02 has been defined in equation (29). Since μh+d1+η1<b1, then R01′>R01 and R03′>R03; therefore, Re<RSed. The result shows that once the strategy that combines sanitation and education campaigns is implemented thoroughly in the community, it can reduce the severity of the disease to a certain extent. However, when all three control interventions are implemented Re, the result is more appealing.


## 4.2. Effects of Single Control Intervention Strategy

We focus on the effects of each of the controls individually.


### 4.2.1. Effects of Treatment

In the absence of education campaign (ρ=0) and sanitation (σ=0), the treatment-induced reproduction number is(60)RT=R1+R2+R3′′,where(61)R3″=δS0ϵ1b2q+α1−l1−q+b11−qϵ2ϕa0b1b2Kμb−r,where R3″ represents the contribution from the surroundings to Shigellosis transmissions in absence of sanitation effort, whereas, R1,R2 have been defined in equation (57). From equation (59), it has been shown that R1>R01,R2>R02. Since 1/μb+σ−r<1/μb−r, it implies that R3<R3″; hence, RST<RT. Since Re<RST, we can summarize the inequality as Re<RST<RT. This inequality suggests that three control strategies are far better than two strategies, and two strategies are far better than single strategy.


### 4.2.2. Effects of Education Campaign

In the absence of treatment (γ=0) and water purification (σ=0), the education-induced reproduction number is(62)Red=R01′+R2+R03″′,where(63)R03″′=δ1−ρS01−ρϵ1b2q+α1−l1−q+μh+d1+η11−qϵ2ϕa0μh+d1+η1b2Kμb−r,where R03″′ represents the contribution from the surroundings to Shigellosis transmissions and R01′ is defined in equation (59), whereas R2 is defined in equation (57).


### 4.2.3. Effects of Sanitation Strategy

In the absence of treatment (γ=0) and education campaigns (ρ=0), the sanitation-induced reproduction number is(64)RS=R1′+R2+R3″′,where(65)R3″′=δS0ϵ1b2q+α1−l1−q+μh+d1+η11−qϵ2ϕa0μh+d1+η1b2Kμb+σ−r,where R3″′ represents the contribution from the surroundings to Shigellosis transmissions, whereas R1′ is defined in (30) and R2 is defined in equation (57). It can be seen that Re<RS since R1′>R01,R2>R02 and R3″′>R03. Besides that, one can note that R3″′<R3′, which implies RS<R0. We use numerical simulations (Figure 2) to compare all reproduction numbers.Figure 2
The comparison of single strategies (a), double strategies (b), and three control strategies (c). Basic reproduction number (R0) is the largest followed by single induced reproduction numbers and then the dual controls and the least being reproduction number capturing all three controls (Re). Note that reproduction number with education and treatment efforts (RTed) has coincided with Re.
(a)(b)(c)


## 4.2.1. Effects of Treatment

In the absence of education campaign (ρ=0) and sanitation (σ=0), the treatment-induced reproduction number is(60)RT=R1+R2+R3′′,where(61)R3″=δS0ϵ1b2q+α1−l1−q+b11−qϵ2ϕa0b1b2Kμb−r,where R3″ represents the contribution from the surroundings to Shigellosis transmissions in absence of sanitation effort, whereas, R1,R2 have been defined in equation (57). From equation (59), it has been shown that R1>R01,R2>R02. Since 1/μb+σ−r<1/μb−r, it implies that R3<R3″; hence, RST<RT. Since Re<RST, we can summarize the inequality as Re<RST<RT. This inequality suggests that three control strategies are far better than two strategies, and two strategies are far better than single strategy.


## 4.2.2. Effects of Education Campaign

In the absence of treatment (γ=0) and water purification (σ=0), the education-induced reproduction number is(62)Red=R01′+R2+R03″′,where(63)R03″′=δ1−ρS01−ρϵ1b2q+α1−l1−q+μh+d1+η11−qϵ2ϕa0μh+d1+η1b2Kμb−r,where R03″′ represents the contribution from the surroundings to Shigellosis transmissions and R01′ is defined in equation (59), whereas R2 is defined in equation (57).


## 4.2.3. Effects of Sanitation Strategy

In the absence of treatment (γ=0) and education campaigns (ρ=0), the sanitation-induced reproduction number is(64)RS=R1′+R2+R3″′,where(65)R3″′=δS0ϵ1b2q+α1−l1−q+μh+d1+η11−qϵ2ϕa0μh+d1+η1b2Kμb+σ−r,where R3″′ represents the contribution from the surroundings to Shigellosis transmissions, whereas R1′ is defined in (30) and R2 is defined in equation (57). It can be seen that Re<RS since R1′>R01,R2>R02 and R3″′>R03. Besides that, one can note that R3″′<R3′, which implies RS<R0. We use numerical simulations (Figure 2) to compare all reproduction numbers.Figure 2
The comparison of single strategies (a), double strategies (b), and three control strategies (c). Basic reproduction number (R0) is the largest followed by single induced reproduction numbers and then the dual controls and the least being reproduction number capturing all three controls (Re). Note that reproduction number with education and treatment efforts (RTed) has coincided with Re.
(a)(b)(c)


## 5. Numerical Simulations

In this section, the model system (2) was solved numerically by using the Runge–Kutta order four schemes due to the fact that they provide more stable solutions as compared with Euler’s method. Euler’s method is inadequate even for well-conditioned problems if a high degree of accuracy is required, owing to the slow first-order convergence. So, it is generally more convenient to use Runge–Kutta fourth-order methods. The aim was to validate the analytical results obtained in the previous sections. The implementation of the scheme was done using MATLAB package. Plots of the numerical solution are used to investigate the effect of some parameters on the population component of interest. The parameters used for simulation are shown in Table 2, while initial values for subpopulations are given as follows: S0=9,E0=7,I0=8,C0=6,R0=6,andB0=7.From Figure3, if the screening rate α=30%, then the number of infectious individuals (I) decreases, but the number of asymptomatic carriers is still high. This is not a desirable result because asymptomatic carriers are responsible for most of the new infections since they are unaware of their illness. When α is increased from 30% to 60% and lastly to 90%, the number of asymptomatic carriers shows a much more significant decline while the number of symptomatic infectious remains low. This shows that high testing coverage of Shigellosis could contribute to the detection of infectious individuals, thereby mitigating the transmission dynamics of the disease.Figure 3
(a, b) Simulation results showing the impact of testing and diagnosis of asymptomatic carriers and symptomatic infectious individuals.
(a)(b)Figure4 shows that an increase in ϕ or β1 has an effect of increasing Re and hence the number of infections. One can note that the change in β1 causes a sharp rise Re while the corresponding change in ϕ causes a minimal change Re which means that direct transmission plays a significant role in the transmission dynamics of Shigellosis as compared with the indirect transmission.Figure 4
(a, b) As transmission ratesϕ and β1 increase, there is an increase in disease epidemic. More people are infected with Shigellosis as a result of exposure to the environment or infectious people.
(a)(b)From Figure2, one can see the comparison between the strategies, in particular, Figure 2(a) shows the comparison among single strategies from which treatment strategy (RT) is the best-case scenario followed by a strategy that takes care of sole education campaign (Red) and the worst case being sanitation strategy (RS). This implies that Shigellosis can be better controlled with treatment efforts than with education or sanitation. Though this does not mean that the education campaign and sanitation are useless, their effectiveness is less than treatment. It must be acknowledged that Shigellosis suffers drug resistance problem; however, for this study, it is assumed that it plays a minimum role. Education in the current study (Red) includes basic knowledge on self-hygiene, the importance of using toilets, drinking boiled water, educating humans not to contaminate water, use of oral salts to help already infected individuals, avoiding direct contact with infected individuals, etc. It can be further noticed from Figure 2(a) that education campaign is more important than sanitation because individual awareness about the disease limits the spread of the epidemic better. The last scenario is sanitation (RS) which principally entails water sanitation. Treating water with chlorine plays a vital role in combating Shigellosis. This is because the addition of chlorine in water kills Shigella bacteria. Even though treatment offers the best results among the single strategies, its implementation is not practical for most communities, especially in the developing world where medical facilities are still problematic due to financial constraints. Furthermore, most Shigellosis antibiotics are resistant to treat this disease. Such constraints hinder Shigellosis eliminations in most communities. Therefore, other options of control strategies such as sanitation and education could be successfully applied in the absence of treatment and still bring forth promising results. That is why many governments today opt to offer clean water to its populace because it is not only cheaper but also healthier than treatment. In so doing they tend to limit the eruption of many waterborne diseases including, Shigellosis.In addition to that, Figure2(b) shows that a combination of treatment and education campaign (RTed) yields the best results among the dual strategies followed by the combination of sanitation and treatment (RST) and the last combination being sanitation and education (RSed).Also, it can be seen from Figure2(c) that as the number of control strategies increase from none to three control strategies, there is a significant decrease in disease eruption. Our results show that when all three interventions are implemented simultaneously, then disease control is possible. Further, it can be noted that the inclusion of treatment either singly or a combination is more beneficial as compared to when it is excluded (see RT and RSed) in Figure 2(c).Further simulations of model (2) with controls measures are depicted in Figure 5 which shows that an increase in each of the control strategies: education (ρ), treatment (γ), screening (α), and sanitation (σ),has an impact in reducing the number of Shigellosis cases from individuals or the surroundings. Figure 5(a) shows that when education coverage increases from 30% to 90%, more individuals remain susceptible since they tend to avoid getting new infections or infecting others. Also, from Figure 5(b), one can note that varying treatment rate from 30% to 90% decreases the size of infectious individuals; this suggests the significance of treatment in breaking further transmission to the safe populace. In the same manner, Figure 5(c) shows that as sanitation rate increases from 30% to 90%, there is a significant decline of Shigella bacteria as they are killed from the surroundings. This suggests the significance of undertaking sanitation if we are to bring this epidemic to an end. On the other hand, identification of carriers for Shigella bacteria is beneficial since the identified population will be treated to safeguard their life and life of others.Figure 5
Effects of controls on the susceptible (a), symptomatic infectious patients, (b) and Shigella bacteria population (c).
(a)(b)(c)


## 6. Sensitivity Analysis

Sensitivity analysis describes how the uncertainty on the model inputs influences the model output. There are two main classes of sensitivity analysis method based on local or global definitions. Both approaches will be described in this section.


### 6.1. Local Sensitivity Analysis

Local sensitivity analysis assesses the effects of individual parameters at particular points in parameter space without taking into account the combined variability resulting from considering all input parameters simultaneously.Local sensitivity for model (2) will be determined via the reproduction number Re given by equation (28). Since Re depends only on 21 parameters, we derive an analytical expression for its sensitivity to each parameter using the normalized forward sensitivity index as done in [28] as follows:(66)ΓΛRe=∂Re∂Λ×ΛRe=+1,ΓμhRe=∂Re∂μh×μhRe=−1.0020.The rest of sensitivity indices for all parameters used in equation (28) can be computed in a similar approach. Table 3 shows the sensitivity indices of Re with respect to 21 parameters.Table 3
Parameters and their sensitivity indices.
ParameterSensitivity indexΛ+1.0000β1+0.3344β2+0.4101ϕ+0.2555δ+0.0013μh−1.0021q−0.6524l−0.0158α−0.2776d1−0.0105η1−0.2625η2−0.2009K−0.2555ϵ1+0.1489ϵ2+0.1065r+0.1433μb−0.2077γ−0.2100ρ−0.3139σ−0.1911From Table3, we can obtain ΓμhRe=−1.002; this means that an increase in μh will cause a decrease in Re. Similarly, a decrease in μh will cause an increase in Re, as they are inversely proportional. We can also note that q,l,α,d1,η1,η2,K, μb,γ, ρ, and σ are all negative; hence, these parameters are inversely proportional to Re. This means that an increase (decrease) in any of these parameters will cause a decrease (increase) in Re.On the other hand, one can see thatΓΛRe=+1; this means that an increase in Λ will cause an increase of exactly the same proportion in Re. Correspondingly, a decrease in Λ will cause a decrease in Re, as they are directly proportional. We can also note that β1,β2,ϕ,δ,ϵ1,ϵ2, or r>0; hence, these parameters are directly proportional to Re. This means that an increase (decrease) in any of these parameters will cause an increase (decrease) in Re. We can arrange these parameters in the order of their magnitude from the largest to the smallest as follows: μh,Λ,q,β2,β1,ρ,α,η1,ϕ,K,γ,μb,η2,σ,ϵ1,r,ϵ2,l,d1 and the least sensitive parameter is δ. The effective reproduction number Re decreases with an increase in the parameters catering for control measures (σ,ρ,γ,α). This suggests that an optimal control measure could be the suitable combination of sanitation, education campaigns to a broad audience, the screening of asymptomatic carriers, and the treatment of symptomatic infectious individuals.More precisely, to minimizeShigellosis transmission in a population, this study recommends that the general public should avoid visiting endemic areas (Λ), reducing unnecessary contact with infectious and carriers to reduce the transmission rates (β1,β2), increase Shigellosis specific education (ρ) on its transmission and control, and avoid contact with contaminated food or water; this will reduce the transmission rate (ϕ) as such contacts increase the chance of contracting this epidemic. Likewise, screening of the carriers (α) would help to contain the spread of the disease as screened individuals will receive medical attention immediately after being identified, and increased treatment rates (γ) of already infected individuals will break the chain of transmission. Keeping our bodies healthier has an additional advantage of increasing natural immune system which would boost natural recovery (η1,η2), and people need to keep their bodies healthier by opting to a healthy lifestyle (eating a balanced diet and regular exercises). Increased environmental sanitation (σ) has benefits of killing bacteria; efforts such as water purification and safe disposal of wastes would deteriorate bacterial in the environment. Growth rate (r) tends to increase bacterial concentration in the environment, so efforts to reduce this rate should be made. Defecation in water sources done by infectious individuals or carriers, i.e., ϵ1,ϵ2, should be discouraged as this tends to amplify Shigella bacteria in the environment.


### 6.2. Uncertainty and Global Sensitivity Analysis

Global sensitivity analysis is the process of apportioning the uncertainty in outputs to the uncertainty in each input factor over their entire range of interest. A sensitivity analysis is considered to be global when all the input factors are varied simultaneously and the sensitivity is evaluated over the entire range of each input factor [29]. Here, we perform a global sensitivity analysis to examine the model responses to parameter variation within a wider range in the parameter space. The mean values of parameters are listed in Table 3, and the range values of these parameters are given in Table 4.Table 4
Parameter value ranges of model (1) used as input for the LHS method.
ParameterRangeΛ500,1500β10.2,0.6β20.3,0.8ϕ0.2465,0.6465δ0.15,0.35μh0.000257,0.000457q0.3,0.8l0.2,0.5α0.26,0.6d10.01,0.04d20.01,0.05η10.2,0.8η20.2,0.6K100,300ϵ140,80ϵ230,80r0.25,0.5μb0.252086,0.852086γ0.2,0.6ρ0.1,0.3σ0.1,0.8It is important to notice that variations of these parameters in our deterministic model lead to uncertainty to model predictions since the effective reproductive number varies with parameters. We adopted the approach in [30]; partial rank correlation coefficients (PRCCs) between the effective reproduction number Re and each parameter for model (2) are derived and are shown in Figure 6. Due to the absence of data on the distribution function, a uniform distribution is chosen for all parameters. The sets of input parameter values sampled using the Latin hypercube sampling (LHS) method were used to run 1000 simulations. We compute the partial rank correlation coefficients between Re and each parameter of model (2). The results of the PRCC are shown in Table 5. It must be noted that the parameters with large PRCC values are most influential in model (1). Table 5 shows that the parameter Λ has the highest influence on the reproduction number Re, followed in decreasing order by the parameters μh,q,K,η1,β2,σ,β1,μb,ϕ,η2,ρ,ϵ2, and γ.Figure 6
Tornado plots of PRCC of parameters that influence model (2) generated using parameter values in Table 4. Parameters with PRCC>0 and PRCC<0 increase and decrease values of Re, respectively.Table 5
Parameters and their sensitivity indices.
ParameterPRCCsΛ+0.597β1+0.184β2+0.195ϕ+0.148δ+0.00549μh−0.371q−0.267l−0.021α−0.0784d1−0.0163d2−0.065η1−0.198η2−0.144K−0.204ϵ1+0.0305ϵ2+0.104r+0.0768μb−0.174γ−0.102ρ−0.125σ−0.194The other parameters areα,r,d2,ϵ1,l,d1, and δ. These parameters have almost no effect on Re. It can be noted that parameters such as β2,β1,σ,ρ, and γ allow us to considerably reduce the reproduction number. Hence, the sensitivity analysis consistently reinforces our suggestion that the most effective manner to reduce infection is to increase education campaigns, sanitation, and treatment strategies. It can be noted that the order of the most important parameters for Re from the local sensitivity analysis does not match those from the global sensitivity analysis; this shows that the local results are not robust enough.


## 6.1. Local Sensitivity Analysis

Local sensitivity analysis assesses the effects of individual parameters at particular points in parameter space without taking into account the combined variability resulting from considering all input parameters simultaneously.Local sensitivity for model (2) will be determined via the reproduction number Re given by equation (28). Since Re depends only on 21 parameters, we derive an analytical expression for its sensitivity to each parameter using the normalized forward sensitivity index as done in [28] as follows:(66)ΓΛRe=∂Re∂Λ×ΛRe=+1,ΓμhRe=∂Re∂μh×μhRe=−1.0020.The rest of sensitivity indices for all parameters used in equation (28) can be computed in a similar approach. Table 3 shows the sensitivity indices of Re with respect to 21 parameters.Table 3
Parameters and their sensitivity indices.
ParameterSensitivity indexΛ+1.0000β1+0.3344β2+0.4101ϕ+0.2555δ+0.0013μh−1.0021q−0.6524l−0.0158α−0.2776d1−0.0105η1−0.2625η2−0.2009K−0.2555ϵ1+0.1489ϵ2+0.1065r+0.1433μb−0.2077γ−0.2100ρ−0.3139σ−0.1911From Table3, we can obtain ΓμhRe=−1.002; this means that an increase in μh will cause a decrease in Re. Similarly, a decrease in μh will cause an increase in Re, as they are inversely proportional. We can also note that q,l,α,d1,η1,η2,K, μb,γ, ρ, and σ are all negative; hence, these parameters are inversely proportional to Re. This means that an increase (decrease) in any of these parameters will cause a decrease (increase) in Re.On the other hand, one can see thatΓΛRe=+1; this means that an increase in Λ will cause an increase of exactly the same proportion in Re. Correspondingly, a decrease in Λ will cause a decrease in Re, as they are directly proportional. We can also note that β1,β2,ϕ,δ,ϵ1,ϵ2, or r>0; hence, these parameters are directly proportional to Re. This means that an increase (decrease) in any of these parameters will cause an increase (decrease) in Re. We can arrange these parameters in the order of their magnitude from the largest to the smallest as follows: μh,Λ,q,β2,β1,ρ,α,η1,ϕ,K,γ,μb,η2,σ,ϵ1,r,ϵ2,l,d1 and the least sensitive parameter is δ. The effective reproduction number Re decreases with an increase in the parameters catering for control measures (σ,ρ,γ,α). This suggests that an optimal control measure could be the suitable combination of sanitation, education campaigns to a broad audience, the screening of asymptomatic carriers, and the treatment of symptomatic infectious individuals.More precisely, to minimizeShigellosis transmission in a population, this study recommends that the general public should avoid visiting endemic areas (Λ), reducing unnecessary contact with infectious and carriers to reduce the transmission rates (β1,β2), increase Shigellosis specific education (ρ) on its transmission and control, and avoid contact with contaminated food or water; this will reduce the transmission rate (ϕ) as such contacts increase the chance of contracting this epidemic. Likewise, screening of the carriers (α) would help to contain the spread of the disease as screened individuals will receive medical attention immediately after being identified, and increased treatment rates (γ) of already infected individuals will break the chain of transmission. Keeping our bodies healthier has an additional advantage of increasing natural immune system which would boost natural recovery (η1,η2), and people need to keep their bodies healthier by opting to a healthy lifestyle (eating a balanced diet and regular exercises). Increased environmental sanitation (σ) has benefits of killing bacteria; efforts such as water purification and safe disposal of wastes would deteriorate bacterial in the environment. Growth rate (r) tends to increase bacterial concentration in the environment, so efforts to reduce this rate should be made. Defecation in water sources done by infectious individuals or carriers, i.e., ϵ1,ϵ2, should be discouraged as this tends to amplify Shigella bacteria in the environment.


## 6.2. Uncertainty and Global Sensitivity Analysis

Global sensitivity analysis is the process of apportioning the uncertainty in outputs to the uncertainty in each input factor over their entire range of interest. A sensitivity analysis is considered to be global when all the input factors are varied simultaneously and the sensitivity is evaluated over the entire range of each input factor [29]. Here, we perform a global sensitivity analysis to examine the model responses to parameter variation within a wider range in the parameter space. The mean values of parameters are listed in Table 3, and the range values of these parameters are given in Table 4.Table 4
Parameter value ranges of model (1) used as input for the LHS method.
ParameterRangeΛ500,1500β10.2,0.6β20.3,0.8ϕ0.2465,0.6465δ0.15,0.35μh0.000257,0.000457q0.3,0.8l0.2,0.5α0.26,0.6d10.01,0.04d20.01,0.05η10.2,0.8η20.2,0.6K100,300ϵ140,80ϵ230,80r0.25,0.5μb0.252086,0.852086γ0.2,0.6ρ0.1,0.3σ0.1,0.8It is important to notice that variations of these parameters in our deterministic model lead to uncertainty to model predictions since the effective reproductive number varies with parameters. We adopted the approach in [30]; partial rank correlation coefficients (PRCCs) between the effective reproduction number Re and each parameter for model (2) are derived and are shown in Figure 6. Due to the absence of data on the distribution function, a uniform distribution is chosen for all parameters. The sets of input parameter values sampled using the Latin hypercube sampling (LHS) method were used to run 1000 simulations. We compute the partial rank correlation coefficients between Re and each parameter of model (2). The results of the PRCC are shown in Table 5. It must be noted that the parameters with large PRCC values are most influential in model (1). Table 5 shows that the parameter Λ has the highest influence on the reproduction number Re, followed in decreasing order by the parameters μh,q,K,η1,β2,σ,β1,μb,ϕ,η2,ρ,ϵ2, and γ.Figure 6
Tornado plots of PRCC of parameters that influence model (2) generated using parameter values in Table 4. Parameters with PRCC>0 and PRCC<0 increase and decrease values of Re, respectively.Table 5
Parameters and their sensitivity indices.
ParameterPRCCsΛ+0.597β1+0.184β2+0.195ϕ+0.148δ+0.00549μh−0.371q−0.267l−0.021α−0.0784d1−0.0163d2−0.065η1−0.198η2−0.144K−0.204ϵ1+0.0305ϵ2+0.104r+0.0768μb−0.174γ−0.102ρ−0.125σ−0.194The other parameters areα,r,d2,ϵ1,l,d1, and δ. These parameters have almost no effect on Re. It can be noted that parameters such as β2,β1,σ,ρ, and γ allow us to considerably reduce the reproduction number. Hence, the sensitivity analysis consistently reinforces our suggestion that the most effective manner to reduce infection is to increase education campaigns, sanitation, and treatment strategies. It can be noted that the order of the most important parameters for Re from the local sensitivity analysis does not match those from the global sensitivity analysis; this shows that the local results are not robust enough.


## 7. Conclusion

In the current work, we have developed a deterministic mathematical model for transmission dynamics ofShigellosis that captures both direct and indirect transmission. The model has incorporated multiple interventions in the presence of carriers. From the model, we have derived the effective reproduction number using the next-generation matrix method, and we have used it to establish the stability of the equilibrium points. We derived both the disease-free equilibrium and the endemic equilibrium. We proved that the DFE is locally asymptotically stable (l.a.s) when Re<1, whereas the EE was proved to be globally asymptotically stable when Re>1. The model developed included intervention strategies such as sanitation, treatment, and education campaigns, and our results have shown that the integration of these control strategies into the model either singly or combined is beneficial to clear away Shigellosis epidemic.More than that, we have performed a sensitivity analysis using the effective reproduction number to determine the contribution of each parameter to this reproduction number. Most sensitive parameters are the human mortality rateμh, recruitment rate (Λ), a fraction of exposed individuals who progress to infectious class (q), the transmission rate for carriers (β2), the transmission rate for infectious individuals (β1), education efficacy ρ, and screening and testing (α). These parameters need great attention if one needs to eradicate Shigellosis from the community. Other parameters are natural recovery for infectious individuals (η1), human-to-environmental transmission rate (ϕ), half-saturation constant of Shigella bacteria that can cause a 50% chance of infection (K), treatment rate of infectious individuals (γ), Shigella death rate (μb), natural recovery for infectious carriers (η2), sanitation rate (σ), contribution by infectious individuals (ϵ1), Shigella growth rate (r), contribution by carriers (ϵ2), a fraction of carriers who recover naturally (l), infection-induced death rate (d1), and rate of incubation (δ).One of our impressive results is that direct transmission between human-to-human (β1,β2) contributes more to Shigella infection as compared with the indirect transmission (ϕ); the same result was reported in [31, 32]. Apart from that, sensitivity analysis shows that carriers are more infectious than symptomatic infectious individuals (see sensitivity indices for β1 and β2). It might be because carriers continue infecting others unknowingly as they do not know their status, and so susceptibles cannot take any precautionary measures against the carriers. By this way, transmission from carriers to the susceptibles becomes apparent. On the other hand, simulations show that treatment strategy has a more significant impact of reducing infections as compared with other strategies such as education campaigns and sanitation. However, treatment has economic and drug resistance implications.



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