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

A mathematical model for Shigellosis 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 contributingmore infections than indirect transmission.%e 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.


Introduction
Shigellosis is an enteric infectious disease which is caused by Shigella bacteria. ese 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][6][7][8][9][10][11][12]. e 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.
e 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. erefore, it is necessary to explore the role played by the carrier in the transmission dynamics of Shigellosis infections.
Several scholars have studied Shigellosis in different ways including developing mathematical models (e.g., see [16][17][18][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 humanto-human infection routes. ey 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 personto-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][17][18][19]). erefore, 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. e rest of the paper is organized as follows. Section 2 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.

Model Formulation
e model considered here follows the basic Susceptible-Exposed-Infected-Recovered (SEIR) model. is 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. is 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. e total human population at time t is subdivided into five mutually exclusive subpopulations: susceptible (S(t)), exposed (E(t)), infectious (I(t)), carrier (C(t)) (i.e., infected individuals who are contagious but do not show any disease symptoms), and recovered (R(t)). To incorporate a real biological phenomenon, an additional compartment, B(t), 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 λ h (t) or after ingesting environmental pathogens from contaminated aquatic reservoirs at the timedependent rate λ p (t). Here the term λ h (t) represents direct transmission between individuals, and it is modelled by standard mass action principle, whereas λ p (t) represents indirect transmission which is modelled by Holling type-II functional response (Michaelis-Menten function). ese forces of infection are given by where β 1 � c 1 q 1 and β 2 � c 2 q 2 are the transmission rates for infectious and carrier individuals, respectively. We define c 1 , c 2 as the number of contacts infectious and carrier individuals make with susceptibles per unit time, respectively, whereas q 1 , q 2 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. e 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. erefore, 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] 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 c.
e remaining fraction l of the carriers recover naturally at the rate η 2 . Together with treatment measures, infectious individuals may experience natural recovery at the rate η 1 . Shigellosis-induced mortality rate for infectious individuals is denoted by d 1 , while the natural death rate of humans is represented by μ h . Shigellosis induces temporal immunity that wanes at the rate ω. erefore, recovered individuals may join the susceptible class when they lose their immunity. Infected individuals from both states I(t) and C(t) excrete bacteria into the environment at the reduced rates (1 − ρ)ϵ 1 and (1 − ρ)ϵ 2 , respectively. It is assumed that the rate of excretion by the infectious individuals, ϵ 1 , is significantly higher than that by the carrier group, ϵ 2 . 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. e per capita growth rate of Shigella bacteria is denoted by r while bacteria deplete naturally at a rate μ b or by sanitation measures at the rate σ. It is assumed that the growth rate of bacteria (r) cannot exceed its death rate (μ b ) (that is, r < μ b ).

2
International Journal of Mathematics and Mathematical Sciences A full description of the variables and parameters to be used in the model is shown in Tables 1 and 2, respectively. e flow diagram for the dynamics is given in Figure 1. From Figure 1, assumptions and model description the following system of differential equations are dS dt e initial conditions for the model system (2) are

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: where Solving 4 th equation of system (4), we get Substitute equation (6) into 3 rd equation of system (4) to get Substitute equations (6) and (7) into fifth equation and solve for R * to get where From second equation of system (4), we have Substitute R * from equation (8) and λ * S * from equation (10) into the first equation of system (4) to get Since the solution of system (4) is feasible only in the invariant region T, expression (11) can only be performed when with equality at the disease-free equilibrium (DFE). From the 6 th equation of system (4), ignore the logistic growth of bacteria for simplicity to get Solving equation (12), we get We know that the force of infection is given by Substitute equations (6), (7), and (13) into equation (14) to get Substitute λ * from (15) and S * from (11) into second equation of system (4) to get resulting into a polynomial of degree three of the form of   International Journal of Mathematics and Mathematical Sciences where 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: To determine the DFE, substitute E * � 0 into equations (6)- (13) to get the DFE as If E * > 0, then the EE denoted by E 1 is given by where E * is the positive solution of equation (19).

Reproduction Number.
We compute the reproduction number, R e , using the next-generation operator approach [23]. e reproduction number is obtained by taking the largest (dominant) eigenvalue (spectral radius) of the matrix where F i is the rate of appearance of new infection in compartment i,V i is the transfer of infections from one compartment i to another, and E 0 is the disease-free equilibrium. From system (2), we rewrite the equations with infectious classes, E, I, C, and B. is leads to the system where From system (23), we obtain International Journal of Mathematics and Mathematical Sciences Partial derivative of F i and V i with respect to E, I, C, and B evaluated at E 0 gives e model reproduction number in the presence of control measures (treatment, education campaign, and sanitation) is now given by where Additionally, R 0i (i � 1, 2, 3) are partial basic reproduction number induced by susceptible-to-infectious transmission, susceptible-to-carrier transmission, and environment-to-susceptible transmission, respectively.

e 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, and c � 0. erefore, the basic reproduction number is given by where Each term R i ′ (i � 1, 2, 3) characterizes the contribution from infectious individuals, carriers, and environment, respectively, whereas a 0 and b 2 have been defined in equation (24).

Local Stability of the Disease-Free Equilibrium.
Here we establish the stability of the DFE that is obtained in equation (20). is is stated in eorem 1 as follows.

Theorem 1.
e DFE of model (4) is locally asymptotically stable if R e < 1 and unstable if R e > 1.

Proof.
e partial differentiation of system (4) with respect to (S, E, I, C, R, B) at the DFE gives the Jacobian matrix J as 6 International Journal of Mathematics and Mathematical Sciences where a 0 , b 1 , and b 2 have been defined in equation (24).
e remaining eigenvalues are the roots of the polynomial |J 1 (E 0 ) − λ| � 0, which is given by where the constants are such that Equivalently, R e can be split into parts To ensure that all roots of equation (34) have negative real parts, the Routh-Hurwitz stability criterion requires that

International Journal of Mathematics and Mathematical Sciences
It is obvious that D 1 � c 3 > 0. In addition, if R e < 1, it implies that R a , R b , R c , R d , R f , R g < 1, and hence c 0 , c 1 , c 2 > 0.
Also, D 2 can be shown to be positive as follows: To prove inequality (41), it is sufficient to establish the following two inequalities: To show (42), we write c 2 c 3 − 2c 1 into the sum of the following parts: Similarly, to show (43), we write c 1 c 2 − 2c 0 c 3 into the sum of parts as follows: It can be noted that if R e < 1, then each R a , R b , R c , R d , R f , R g < 1 and therefore c 1 c 2 − 2c 0 c 3 > 0 and c 2 c 3 − 2c 1 > 0. us, equations (42) and (43) hold and so does condition (41). In the same fashion, the proof for condition D 4 can be established from the fact that D 4 � c 0 D 3 . Fortunately, we have already proved that D 3 > 0; therefore, it is clear that D 4 � c 0 D 3 > 0. Hence, all conditions of Routh-Hurwitz for this case (equations (38) and (39)) are satisfied; then, the disease-free equilibrium E 0 is locally asymptotically stable whenever R e < 1.

Global Stability of the Disease-Free Equilibrium.
We have the following results on the global stability of the DFE.

Theorem 2.
If R e < 1, the DFE is globally asymptotically stable and unstable if R e > 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 implying that dE dt dI dt dC dt dB dt where F and V are Jacobian matrices as in (26) and (27).

Theorem 3. e endemic equilibrium for the model (2) is globally asymptotically stable on
Proof. Here we construct an explicit Lyapunov function of the form where w i is a properly selected positive constant, x i is the population of the i th compartment, and x * i is the equilibrium level. We define the Lyapunov function candidate L for model system (2) as e time derivative of the Lyapunov function L is given by It can be noted that at endemic equilibrium (see equation (4)), we have and substituting equation (51) into equation (50) and simplifying can result into the following equation: International Journal of Mathematics and Mathematical Sciences where F is the balance of the right-hand terms of equation (52). Following the approach by [25][26][27], F is a nonpositive function for S, E, I, C, R, B > 0. us, 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 * . erefore, if R e > 1, model (2) has a unique endemic equilibrium point E 1 which is globally asymptotically stable.

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 combating Shigellosis 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. e primary purpose of the comparison was to determine which of them has a significant influence in diminishing Shigellosis.

Effects of Multiple Control Intervention Strategies.
We focus on the effects of a combination of two or three strategies simultaneously.

Effects of Treatment, Education Campaign, and
Sanitation. In the presence of three interventions, that is, ρ ≠ 0, σ ≠ 0, and c ≠ 0, the reproduction number-induced by treatment, education, and sanitation where R 01 , R 02 , and R 03 have been defined in equation (29).

Effects of Treatment and Education Campaign.
In the absence of sanitation (σ � 0), the treatment and educationinduced reproduction number is where where R 01 and R 02 have been defined in equation (29) and R 03 ″ 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, a 0 , b 1 , and b 2 have been defined in equation (24).
, and hence R e < R Ted , which shows that even though treatment and education campaigns can reduce the spread of infections, multiple control strategy that accounts for all three controls (R e ) will yield a better result.

Effects of Sanitation and Treatment.
In the absence of education campaign (ρ � 0), the sanitation and treatmentinduced reproduction number is where where R 1 , R 2 , and R 3 represent the contributions from infectious individuals, carriers, and environment-to-host transmission, respectively. It must be noted that a 0 , b 1 , and b 2 have been defined in equation (24). It is possible to express Since ρ ∈ [0, 1], then it is clear to see that R 01 ≤ R 1 , R 02 ≤ R 2 and R 03 ≤ R 3 ; therefore, R e < R ST . is 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. e strategy that takes care of all three controls (R e ) will yield a far better result. is result agrees with intuitive expectations.

Effects of Sanitation and Education.
In the absence of treatment (c � 0), the sanitation and education-induced reproduction number is where where R 01 ′ and R 03 ′ represent the contributions from infectious individuals and contribution from environment-to-host transmission, respectively, whereas R 02 has been defined in equation (29). Since μ h + d 1 + η 1 < b 1 , then R 01 ′ > R 01 and R 03 ′ > R 03 ; therefore, R e < R Sed . e 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 (R e ), the result is more appealing.

Effects of Single Control Intervention Strategy.
We focus on the effects of each of the controls individually.

Effects of Treatment.
In the absence of education campaign (ρ � 0) and sanitation (σ � 0), the treatment-induced reproduction number is where where R 3 ″ represents the contribution from the surroundings to Shigellosis transmissions in absence of sanitation effort, whereas, R 1 , R 2 have been defined in equation (57). From equation (59), it has been shown that R 1 > R 01 , R 2 > R 02 .
Since R e < R ST , we can summarize the inequality as R e < R ST < R T . is inequality suggests that three control strategies are far better than two strategies, and two strategies are far better than single strategy.

Effects of Education Campaign.
In the absence of treatment (c � 0) and water purification (σ � 0), the education-induced reproduction number is where International Journal of Mathematics and Mathematical Sciences where R ″ ′ 03 represents the contribution from the surroundings to Shigellosis transmissions and R 01 ′ is defined in equation (59), whereas R 2 is defined in equation (57).

Effects of Sanitation Strategy.
In the absence of treatment (c � 0) and education campaigns (ρ � 0), the sanitation-induced reproduction number is where where R ″ ′ 3 represents the contribution from the surroundings to Shigellosis transmissions, whereas R 1 ′ is defined in (30) and R 2 is defined in equation (57). It can be seen that R e < R S since R 1 ′ > R 01 , R 2 > R 02 and R ″ ′ 3 > R 03 . Besides that, one can note that R ″ ′ 3 < R 3 ′ , which implies R S < R 0 . We use numerical simulations (Figure 2) to compare all reproduction numbers.

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 fourthorder methods. e aim was to validate the analytical results obtained in the previous sections. e 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.
e parameters used for simulation are shown in Table 2, while initial values for subpopulations are given as follows: From Figure 3, if the screening rate α � 30%, then the number of infectious individuals (I) decreases, but the number of asymptomatic carriers is still high. is 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. is shows that high testing coverage of Shigellosis could contribute to the detection of infectious individuals, thereby mitigating the transmission dynamics of the disease. Figure 4 shows that an increase in ϕ or β 1 has an effect of increasing R e and hence the number of infections. One can note that the change in β 1 causes a sharp rise R e while the corresponding change in ϕ causes a minimal change R e which means that direct transmission plays a significant role in the transmission dynamics of Shigellosis as compared with the indirect transmission.
From Figure 2, one can see the comparison between the strategies, in particular, Figure 2(a) shows the comparison among single strategies from which treatment strategy (R T ) is the best-case scenario followed by a strategy that takes care of sole education campaign (R ed ) and the worst case being sanitation strategy (R S ). is implies that Shigellosis can be better controlled with treatment efforts than with education or sanitation. ough 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 (R ed ) 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. e last scenario is sanitation (R S ) which principally entails water sanitation. Treating water with chlorine plays a vital role in combating Shigellosis. is 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. erefore, 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. at 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, Figure 2(b) shows that a combination of treatment and education campaign (R Ted ) yields the best results among the dual strategies followed by the combination of sanitation and treatment (R ST ) and the last combination being sanitation and education (R Sed ).
Also, it can be seen from Figure 2(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 R T and R Sed ) in Figure 2(c). Reproduction number  , and three control strategies (c). Basic reproduction number (R 0 ) is the largest followed by single induced reproduction numbers and then the dual controls and the least being reproduction number capturing all three controls (R e ). Note that reproduction number with education and treatment efforts (R Ted ) has coincided with R e . 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 (c), 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. is 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.

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

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 R e given by equation (28). Since R e 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) e 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 R e with respect to 21 parameters.
From Table 3, we can obtain Γ R e μ h � − 1.002; this means that an increase in μ h will cause a decrease in R e . Similarly, a decrease in μ h will cause an increase in R e , as they are inversely proportional. We can also note that q, l, α, d 1 , η 1 , η 2 , K, μ b , c, ρ, and σ are all negative; hence, these parameters are inversely proportional to R e . is means that an increase (decrease) in any of these parameters will cause a decrease (increase) in R e .
On the other hand, one can see that Γ 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. e mean values of parameters are listed in Table 3, and the range values of these parameters are given in Table 4. It 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 R e 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. e 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 R e and each parameter of model (2). e 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 R e , followed in decreasing order by the parameters μ h , q, K, η 1 , β 2 , σ, β 1 , μ b , ϕ, η 2 , ρ, ϵ 2 , and c. e other parameters are α, r, d 2 , ϵ 1 , l, d 1 , and δ. ese parameters have almost no effect on R e . It can be noted that parameters such as β 2 , β 1 , σ, ρ, and c allow us to considerably  Table 4. Parameters with PRCC > 0 and PRCC < 0 increase and decrease values of R e , respectively. 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 R e from the local sensitivity analysis does not match those from the global sensitivity analysis; this shows that the local results are not robust enough.

Conclusion
In the current work, we have developed a deterministic mathematical model for transmission dynamics of Shigellosis that captures both direct and indirect transmission. e 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 R e < 1, whereas the EE was proved to be globally asymptotically stable when R e > 1.
e 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 rate (μ h ), recruitment rate (Λ), a fraction of exposed individuals who progress to infectious class (q), the transmission rate for carriers (β 2 ), the transmission rate for infectious individuals (β 1 ), education efficacy (ρ), and screening and testing (α).
ese parameters need great attention if one needs to eradicate Shigellosis from the community. Other parameters are natural recovery for infectious individuals (η 1 ), 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 (c), Shigella death rate (μ b ), natural recovery for infectious carriers (η 2 ), sanitation rate (σ), contribution by infectious individuals (ϵ 1 ), Shigella growth rate (r), contribution by carriers (ϵ 2 ), a fraction of carriers who recover naturally (l), infectioninduced death rate (d 1 ), and rate of incubation (δ).
One of our impressive results is that direct transmission between human-to-human (β 1 , β 2 ) 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 β 1 and β 2 ). 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.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.