A mathematical model for Shigellosis including disease carriers with multiple control strategies is developed. We compute the effective reproductive number , 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 . 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.

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 [512]. 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 studied Shigellosis in different ways including developing mathematical models (e.g., see [1619]). 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 [1619]). 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. 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.

2. Model Formulation

The model considered here follows the basic Susceptible-Exposed-Infected-Recovered () 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 () and the number of susceptible individuals () 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 time is subdivided into five mutually exclusive subpopulations: susceptible (), exposed (), infectious (), carrier () (i.e., infected individuals who are contagious but do not show any disease symptoms), and recovered (). To incorporate a real biological phenomenon, an additional compartment, , 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, . Susceptible individuals may acquire Shigella infection following effective direct contact with infectious individuals or carriers at the time-dependent rate or after ingesting environmental pathogens from contaminated aquatic reservoirs at the time-dependent rate . Here the term represents direct transmission between individuals, and it is modelled by standard mass action principle, whereas represents indirect transmission which is modelled by Holling type-II functional response (Michaelis–Menten function). These forces of infection are given bywhere and are the transmission rates for infectious and carrier individuals, respectively. We define as the number of contacts infectious and carrier individuals make with susceptibles per unit time, respectively, whereas are the probabilities that the contacts will cause infection. Likewise, is the half-saturation constant of the bacteria population in water that yields a 50% chance of catching the disease and is the ingestion rate of Shigella bacteria by individuals. The term 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 , and thus the total force of infection will be given by , where measures the efficacy of the education campaign. If , then it implies that health education has been ignored as an intervention strategy, whereas when , it means that education is 100% efficient in limiting the spread of Shigellosis. Exposed individuals may either join infectious or carrier classes. A fraction of the exposed individuals may progress to the infectious stage at the rate while the complement of the exposed may become carriers at the same rate . Carrier individuals do not show any symptoms of Shigellosis even though they remain infectious; this complicates efforts to eliminate Shigellosis. A fraction of carriers are screened at the rate and join the infectious individuals where they are finally treated at the rate . The remaining fraction of the carriers recover naturally at the rate . Together with treatment measures, infectious individuals may experience natural recovery at the rate . Shigellosis-induced mortality rate for infectious individuals is denoted by , while the natural death rate of humans is represented by . Shigellosis induces temporal immunity that wanes at the rate . Therefore, recovered individuals may join the susceptible class when they lose their immunity. Infected individuals from both states and excrete bacteria into the environment at the reduced rates and , respectively. It is assumed that the rate of excretion by the infectious individuals, , is significantly higher than that by the carrier group, . 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 while bacteria deplete naturally at a rate or by sanitation measures at the rate . It is assumed that the growth rate of bacteria () cannot exceed its death rate () (that is, ). 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.

From Figure 1, assumptions and model description the following system of differential equations are

The initial conditions for the model system (2) are . Model (2) is biologically meaningful in the invariant region

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:where

Solving 4th equation of system (4), we get

Substitute equation (6) into 3rd equation of system (4) to get

Substitute equations (6) and (7) into fifth equation and solve for to getwhere

From second equation of system (4), we have

Substitute from equation (8) and 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 , expression (11) can only be performed when , 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

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 from (15) and from (11) into second equation of system (4) to getresulting into a polynomial of degree three of the form ofwhere

One of the solutions of equation (17) is , which confirms the existence of disease-free equilibrium (DFE) while the existence of the endemic equilibrium (EE) is guaranteed by the nonzero solution () of the quadratic equation:

To determine the DFE, substitute into equations (6)–(13) to get the DFE as

If , then the EE denoted by is given bywhere is the positive solution of equation (19).

3.2. Reproduction Number

We compute the reproduction number, , using the next-generation operator approach [23]. The reproduction number is obtained by taking the largest (dominant) eigenvalue (spectral radius) of the matrixwhere is the rate of appearance of new infection in compartment , is the transfer of infections from one compartment to another, and is the disease-free equilibrium. From system (2), we rewrite the equations with infectious classes, , and . This leads to the system where

From system (23), we obtain

Partial derivative of and with respect to , and evaluated at gives

The model reproduction number in the presence of control measures (treatment, education campaign, and sanitation) is now given bywhere

Additionally, 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 . Therefore, the basic reproduction number is given bywhere

Each term characterizes the contribution from infectious individuals, carriers, and environment, respectively, whereas and 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 and unstable if .

Proof. The partial differentiation of system (4) with respect to at the DFE gives the Jacobian matrix aswhere , and have been defined in equation (24).
Matrix (32) has two trivial negative eigenvalues and .
If we set , then the remaining submatrix is given asThe remaining eigenvalues are the roots of the polynomial , which is given bywhere the constants are such thatEquivalently, can be split into partswhereTo ensure that all roots of equation (34) have negative real parts, the Routh–Hurwitz stability criterion requires thatIt is obvious that . In addition, if , it implies that , and hence .
Also, can be shown to be positive as follows:where , and hence is positive.
The only remaining condition to show isTo prove inequality (41), it is sufficient to establish the following two inequalities:To show (42), we write into the sum of the following parts:Similarly, to show (43), we write into the sum of parts as follows:It can be noted that if , then each and therefore and . Thus, equations (42) and (43) hold and so does condition (41). In the same fashion, the proof for condition can be established from the fact that . Fortunately, we have already proved that ; therefore, it is clear that . Hence, all conditions of Routh–Hurwitz for this case (equations (38) and (39)) are satisfied; then, the disease-free equilibrium is locally asymptotically stable whenever .

3.3.2. Global Stability of the Disease-Free Equilibrium

We have the following results on the global stability of the DFE.

Theorem 2. If , the DFE is globally asymptotically stable and unstable if .

Proof. By the comparison theorem, the rate of change of the variables representing the infected components of the model system (2) can be rewritten asimplying thatwhere and are Jacobian matrices as in (26) and (27). Since the eigenvalues of the matrix have negative real parts (this comes from the stability results in Theorem 1), system (2) is stable whenever . So, and as . By the comparison theorem (see [24]), as . Therefore, is globally asymptotically stable whenever .

3.3.3. Global Stability of the Endemic Equilibrium

Theorem 3. The endemic equilibrium for the model (2) is globally asymptotically stable on if .

Proof. Here we construct an explicit Lyapunov function of the formwhere is a properly selected positive constant, is the population of the ith compartment, and is the equilibrium level. We define the Lyapunov function candidate for model system (2) asThe time derivative of the Lyapunov function is given byIt can be noted that at endemic equilibrium (see equation (4)), we haveand substituting equation (51) into equation (50) and simplifying can result into the following equation:where is the balance of the right-hand terms of equation (52). Following the approach by [2527], is a nonpositive function for . Thus, for and is zero if and . Therefore, if , model (2) has a unique endemic equilibrium point 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 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. 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, , and , the reproduction number-induced by treatment, education, and sanitationwhere , , and have been defined in equation (29).

4.1.2. Effects of Treatment and Education Campaign

In the absence of sanitation (), the treatment and education-induced reproduction number iswherewhere and have been defined in equation (29) and 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, , and have been defined in equation (24). Since , then , and hence , which shows that even though treatment and education campaigns can reduce the spread of infections, multiple control strategy that accounts for all three controls () will yield a better result.

4.1.3. Effects of Sanitation and Treatment

In the absence of education campaign (), the sanitation and treatment-induced reproduction number iswherewhere , and represent the contributions from infectious individuals, carriers, and environment-to-host transmission, respectively. It must be noted that , and have been defined in equation (24). It is possible to express . Since , then it is clear to see that and ; therefore, . 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 () 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 (), the sanitation and education-induced reproduction number iswherewhere and represent the contributions from infectious individuals and contribution from environment-to-host transmission, respectively, whereas has been defined in equation (29). Since , then and ; therefore, . 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 , 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 () and sanitation (), the treatment-induced reproduction number iswherewhere represents the contribution from the surroundings to Shigellosis transmissions in absence of sanitation effort, whereas, have been defined in equation (57). From equation (59), it has been shown that . Since , it implies that ; hence, . Since , we can summarize the inequality as . 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 () and water purification (), the education-induced reproduction number iswherewhere represents the contribution from the surroundings to Shigellosis transmissions and is defined in equation (59), whereas is defined in equation (57).

4.2.3. Effects of Sanitation Strategy

In the absence of treatment () and education campaigns (), the sanitation-induced reproduction number iswherewhere represents the contribution from the surroundings to Shigellosis transmissions, whereas is defined in (30) and is defined in equation (57). It can be seen that since and . Besides that, one can note that , which implies . We use numerical simulations (Figure 2) to compare all reproduction numbers.

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: .

From Figure 3, if the screening rate , 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 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 4 shows that an increase in or has an effect of increasing and hence the number of infections. One can note that the change in causes a sharp rise while the corresponding change in causes a minimal change 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 () is the best-case scenario followed by a strategy that takes care of sole education campaign () and the worst case being sanitation strategy (). 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 () 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 () 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, Figure 2(b) shows that a combination of treatment and education campaign () yields the best results among the dual strategies followed by the combination of sanitation and treatment () and the last combination being sanitation and education ().

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 and ) 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 (), treatment (), screening (), and sanitation (),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.

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 given by equation (28). Since 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:

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 with respect to 21 parameters.

From Table 3, we can obtain ; this means that an increase in will cause a decrease in . Similarly, a decrease in will cause an increase in , as they are inversely proportional. We can also note that , , , and are all negative; hence, these parameters are inversely proportional to . This means that an increase (decrease) in any of these parameters will cause a decrease (increase) in .

On the other hand, one can see that ; this means that an increase in will cause an increase of exactly the same proportion in . Correspondingly, a decrease in will cause a decrease in , as they are directly proportional. We can also note that , or ; hence, these parameters are directly proportional to . This means that an increase (decrease) in any of these parameters will cause an increase (decrease) in . We can arrange these parameters in the order of their magnitude from the largest to the smallest as follows: and the least sensitive parameter is . The effective reproduction number decreases with an increase in the parameters catering for control measures (). 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 minimize Shigellosis transmission in a population, this study recommends that the general public should avoid visiting endemic areas (), reducing unnecessary contact with infectious and carriers to reduce the transmission rates (), increase Shigellosis specific education () on its transmission and control, and avoid contact with contaminated food or water; this will reduce the transmission rate () as such contacts increase the chance of contracting this epidemic. Likewise, screening of the carriers () would help to contain the spread of the disease as screened individuals will receive medical attention immediately after being identified, and increased treatment rates () 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 (), and people need to keep their bodies healthier by opting to a healthy lifestyle (eating a balanced diet and regular exercises). Increased environmental sanitation () has benefits of killing bacteria; efforts such as water purification and safe disposal of wastes would deteriorate bacterial in the environment. Growth rate () 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., , 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.

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 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 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 has the highest influence on the reproduction number , followed in decreasing order by the parameters , and .