Journal of Applied Mathematics

Volume 2018, Article ID 2528513, 14 pages

https://doi.org/10.1155/2018/2528513

## Analysis and Optimal Control Intervention Strategies of a Waterborne Disease Model: A Realistic Case Study

^{1}Institute of Systems Science, Durban University of Technology, Durban 4000, South Africa^{2}School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa

Correspondence should be addressed to Obiora Cornelius Collins; moc.liamg@snilloc.c.aroibo

Received 14 May 2018; Accepted 20 September 2018; Published 21 November 2018

Academic Editor: Wan-Tong Li

Copyright © 2018 Obiora Cornelius Collins and Kevin Jan Duffy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A mathematical model is formulated that captures the essential dynamics of waterborne disease transmission under the assumption of a homogeneously mixed population. The important mathematical features of the model are determined and analysed. The model is extended by introducing control intervention strategies such as vaccination, treatment, and water purification. Mathematical analyses of the control model are used to determine the possible benefits of these control intervention strategies. Optimal control theory is utilized to determine how to reduce the spread of a disease with minimum cost. The model is validated using a cholera outbreak in Haiti.

#### 1. Introduction

Waterborne diseases which include Cholera, Hepatitis A and Hepatitis E, Giardia, Cryptosporidium, and Rotavirus are among the serious health problems of people globally. This is especially so in developing countries where there is limited access to clean water. Unsafe water supply, poor sanitation and poor hygiene are major causes of waterborne diseases [1]. According to WHO [2], approximately 1.1 billion people globally do not have access to sources of reliable water. About 700,000 children die every year from diarrhoea caused by unsafe water and poor sanitation [3]. The prevalence of waterborne diseases could be controlled especially in developing countries through access to safe water, provision of adequate sanitation facilities, and better hygiene practices [1]. Control measures such as water purification, vaccination, and treatment of infected individuals are among the most effective ways of reducing the spread of these diseases [4–6]. In this study, we investigate the impact of these types of control measures in reducing the spread of waterborne diseases.

Even with the availability of control measures, affordability is often the greatest obstacle for many communities where diseases are endemic. The spread of waterborne diseases are often associated with poverty, limited resources, and low socioeconomic status [2, 7]. Optimal control theory can point to efficient approaches to reduce the spread of a disease with minimum costs [4, 8]. In this study, we consider optimal control theory to investigate how to reduce the spread of waterborne diseases with minimum costs.

Some of the essential factors that influence the dynamics of waterborne diseases include sanitation [9], different transmission pathways [10, 11], water treatment efforts [12, 13], pathogen ecology outside of human hosts [14], climatological factors or seasonal fluctuations [15–18], and heterogeneity in disease transmission [19, 20]. Understanding how these factors interact to influence the dynamics of waterborne diseases are challenging, making the dynamics of waterborne diseases complex. Several theoretical studies have taken some of these factors into account to improve the understanding of waterborne disease dynamics and subsequently investigate the possible means of reducing the diseases [7, 11, 19, 21–26]. Even though these studies have contributed immensely in improving the understanding of waterborne disease dynamics, theoretical studies for waterborne disease dynamics and control are not complete. In this study, we consider mathematical models to investigate the dynamics and control of waterborne diseases. The findings complement existing results in the literature on the dynamics and control of waterborne diseases.

The remaining parts of this paper are organized as follows. In Section 2, a waterborne disease model, which underpins the essential dynamics, is presented and analysed. To determine the possible benefits of control measures a multiple control model (with all controls imposed simultaneously) is presented and analysed in Section 3. In Section 4, optimal control analyses are used to investigate how to reduce the spread of a disease with minimum costs. The model is validated by using it to study a cholera outbreak in Haiti in Section 5. We conclude the paper by discussing our results in Section 6.

#### 2. Waterborne Disease Model and Analysis

In this section we present a waterborne disease model that underlies the dynamics for a homogeneous population without any control intervention measures. Analyses of this model are necessary as a comparison to understand the effects of the control intervention strategies included in subsequent sections.

##### 2.1. Formulation of the Control-Free Model

We consider an extension of the standard SIR model under the assumption of constant human population size by adding a compartment that measures pathogen concentration in a water reservoir [11, 27]. As usually done, the total human population is partitioned into susceptible , infected , and recovered individuals such that . Individuals enter the susceptible class through birth at a rate . Susceptible individuals become infected with waterborne disease through contact with contaminated water at a rate . Direct person-to-person transmissions are not considered because water-to-person transmissions have been shown to be the major route of waterborne disease transmissions [6, 23, 28]. Infected individuals shed pathogens into water at a rate and recover naturally at a rate . Pathogens are generated naturally in the water at a rate and decay at a rate . Natural human deaths occur at a rate . With these assumptions we obtain the model where is the natural decay rate of pathogens in the water reservoir. Note that our model (1) is in the form considered by Tien and Earn [11] to study the multiple transmission pathways for waterborne diseases. A difference is that they considered infections to be generated through both direct person-to-person and indirect water-to-person contacts. Our approach that considers infections through indirect water-to-person contacts only is particularly relevant with waterborne diseases such as cholera which are primarily transmitted through contaminated water.

For the qualitative analyses of model (1), we consider a dimensionless version given by where , , , , and

All parameters are realistically assumed positive and the initial conditions are as follows:All the solutions of model (2) considered are in the feasible region is positively invariant and the existence and uniqueness of solutions of model (2) hold in this region. Thus, model (2) is well posed mathematically and epidemiologically in .

##### 2.2. Basic Reproduction Number

The control-free model (2) has a unique disease-free equilibrium (DFE) given by The basic reproduction number of the control-free model (2) is determined using the next generation matrix method [29]:

##### 2.3. Stability Analysis of the Disease-Free Equilibrium

For a dynamical infectious disease model, stability about its disease-free equilibrium (DFE) describes the short-term dynamics of the disease [30]. Therefore to determine the short-term dynamics of the waterborne disease considered here, it is necessary to investigate the stability of the control-free model (2) about its DFE. From Theorem 2 in van den Driessche and Watmough [29], the following result holds.

Theorem 1. *The DFE of the control-free model (2) is locally asymptotically stable if and unstable if .*

Theorem 1 implies that waterborne disease can be eliminated from the entire population (when ) if the initial size of the infected population is in the basin of attraction of the DFE (5). On the other hand, the disease will establish in the population if .

Theorem 2. *The DFE of the control-free model (2) is globally asymptotically stable provided that .*

This theorem can be established using a global stability result by Castillo-Chavez et al. [31]. This global stability ensures that disease elimination is independent of the initial size of the population of infected individuals if . The epidemiological implication is that in this case waterborne disease can be eradicated from the entire community irrespective of the initial number of infected people in the community.

##### 2.4. Outbreak Growth Rate

If , then the DFE (5) becomes unstable and a disease outbreak occurs in the population. The positive (dominant) eigenvalue of the Jacobian at the DFE is typically referred to as the initial outbreak growth rate [11]. The eigenvalues of the Jacobian matrix of model (2) evaluated at the DFE (5) are Clearly, . Thus, the positive (dominant) eigenvalue is given by From the above results, when the outbreak growth rate vanishes. Also, if all three eigenvalues become negative confirming Theorem 1. So the outbreak growth rate () exists only when . Epidemiologically, this result demonstrates how for an outbreak will occur in the community and the growth rate of that outbreak is determined by . In particular, this might occur when there are no control measures. Next it is necessary to obtain the likely magnitude of an outbreak, often called the expected final size of the outbreak [32].

##### 2.5. Final Outbreak Size

Our analyses have shown that when a waterborne disease outbreak occurs and grows at the rate . The final outbreak size of SIR epidemiological models and other similar models are given by the relation where denotes the proportion of the population who become infected at some point during the outbreak. This relation applies to our control-free model (2) [11] and so if there is no control intervention and an outbreak occurs, with then the final outbreak size of the epidemic can be determined by (9).

##### 2.6. Stability Analysis of the Endemic Equilibrium

The long-term dynamics of a dynamical system is characterized by the stability about its endemic equilibrium [30]. To determine the long-term dynamics of the control-free model (2) we investigate its stability about this endemic equilibrium (EE). Algebraically, it can be demonstrated that when , a unique EE occurs in model (2) given by Obviously, will vanish if . This confirms that the disease cannot be endemic when . The stability analyses of the EE (10) are summarized as follows using [11, 33–36].

Theorem 3. *The unique endemic equilibrium (10) is locally and globally asymptotically stable whenever .*

The proof of Theorem 3 can be established using the approach in [11] and implies that whenever any outbreak in the population will persist in the population (remain endemic). So, to minimize the chances of a disease outbreak the intervention of control measures can be used such that the basic reproduction number is kept below unity (i.e., ).

#### 3. Multiple Control Model and Analyses

In this section we present a control model to investigate the impact of introducing control measures in the spread of diseases. Three different types of control measure are considered: vaccination, water purification, and treatment. The impact of these control measures is investigated by extending the original control-free model (2) to include these control measures.

##### 3.1. Formulation of the Multiple Control Model

A multiple control model is formulated as follows.

Vaccination is one control strategy for reducing the spread of waterborne diseases such as cholera. For example, a cholera vaccine can offer about 60-90% protection against the disease. Thus in the control model we assume that susceptible individuals are vaccinated at rate with a vaccine whose efficacy is . Effectively the model now has new class of individual that are vaccinated.

Effective treatment of waterborne disease is also very important in reducing the spread of the disease. Some waterborne diseases like cholera can kill within hours of contracting the disease if there is no proper treatment. If people infected with cholera are treated quickly and properly, the mortality rate is less than 1% but if they are left untreated, the mortality rate rises to 50-60% [37, 38]. We introduce treatment in the control-free model (2) by assuming that infected individuals are treated at rate and treated individuals recover due to treatment at rate .

According to the World Health Organization [1], unsafe water, poor sanitation, and poor hygiene are the major causes of waterborne diseases. A significant number of cases of a disease could be reduced through access to clean water supplies, provision of adequate sanitation facilities, and better hygiene practices. Here we extend the model (2) by assuming that provision of clean water reduces pathogen concentrations at a rate .

Based on these assumptions, and by introducing these control intervention strategies simultaneously, we obtain the multiple control model where are vaccinated individuals, are treated individuals, are the proportion of vaccinated individuals, and are the proportion of treated individuals.

All the solutions of model (11) enter the feasible region The region is positively invariant; thus it is sufficient to consider the solutions of model (11) within it.

##### 3.2. Basic Reproduction Number for the Multiple Control Model

The multiple control intervention strategy model (11) has a DFE given by and a basic reproduction number given by where The basic reproduction number can be defined as the expected number of secondary infections that result from introducing a single infected individual into an otherwise susceptible population [29]. Thus, the threshold quantity is a measure of the number of secondary infections of the population in the presence of vaccination, treatment, and water purification. From (15) and (14), we have that This implies that the multiple control measures in the model do have an impact in reducing the number of secondary infections.

##### 3.3. Stability Analysis of the DFE for Multiple Control Measures

Again, to determine the short-term dynamics in the presence of the multiple control measures we investigate the stability of the multiple control model at the DFE. The results are summarized in the theorem below.

Theorem 4. *If , the DFE (13) of model (11) is globally asymptotically stable and unstable if .*

The epidemiological implication of this result is that waterborne diseases can be eradicated from the entire community using multiple control measures irrespective of the initial size of the infected people provided that .

##### 3.4. Outbreak Growth Rate for Multiple Control Measures

Suppose that the multiple control measures are not effective, then and the DFE (13) becomes unstable and a disease outbreak occurs. The outbreak growth rate of the multiple control model is given by Comparing the outbreak growth rate of the multiple control model with the no control model is summarized in the theorem below.

Theorem 5. *If , , , and , then . Furthermore, if and only if .*

The proof of this theorem can be established by simple algebraic manipulation. This show that introducing multiple controls can reduce the outbreak growth rate.

##### 3.5. Single and Double Control Measures

Waterborne disease outbreaks are often associated with poverty and limited resources to control the disease [1]. Often such communities cannot afford to introduce more than one control measure such as the three considered here. Thus, it is important to investigate the impact of introducing a single control or double controls. By comparing the impact of a single control, double control, and multiple controls, we determine the control (or combination of controls) that can yield the best results. These comparisons will be done at both the epidemic stage as well as the endemic stage of the outbreak. Theoretically, for the epidemic stage of the outbreak the basic reproduction number is used, while at the endemic stage the outbreak growth rate is used. These results can help in advising communities with limited resources.

###### 3.5.1. Basic Reproduction Number for a Single Control Measure

Suppose the community can afford vaccinations only, then the basic reproduction number is This threshold quantity can be understood as a measure of the number of secondary infections in the presence of vaccination [11, 39]. By elementary algebraic manipulations, the equations hold. From (19) the number of secondary infections is less when vaccination is introduced provided , and this implies that introducing vaccination decreases the spread of the infection.

For communities that can afford only treatment the basic reproduction number is Clearly, the equations hold. Epidemiologically, the treatment of infected individuals reduces the number of secondary infections in the population provided that .

Finally, we investigate the impact of introducing water purification as the only control measure. The water purification induced basic reproduction number is where Again the following equations hold:and introducing water purification reduces the number of secondary infections in the community provided that .

Having shown the impact of each of the single controls using the basic reproduction number, it is important to compare each of these single controls with the multiple control. Since, is the product of , , and and each of these is less than 1, then using the calculations of each reproduction number (19), (22), (26), and (16) can be written in compact form as These results show that even though each of the single controls has some influence in reducing the number of secondary infections, the multiple control always has at least the greatest influence. The above results agree with intuitive expectation and justify why multiple controls are encouraged whenever an outbreak occurs in any community.

###### 3.5.2. Outbreak Growth Rate for a Single Control Measure

For communities that consider only treatment as a control measure, our analyses show that, if infected individuals are not properly treated such that , then an outbreak occurs in the community. The treatment-induced outbreak growth rate is given by To determine the strength of this outbreak, we compare it with the outbreak growth rate in the absence of control intervention. The result of the comparison is summarized in the theorem below.

Theorem 6. *If , then . Furthermore, if and only if .*

Theorem 6 can be established by algebraic manipulations. Thus, the outbreak growth rate in the presence of treatment is always lower than that with no control.

Similarly, for communities that can afford only vaccination or water purification the outbreak growth rates are given by and respectively.

Since and then andThis shows that vaccination or introducing water purification reduces the outbreak growth rate more than when no control is introduced.

Epidemiologically, these results demonstrate that even when a control does not prevent a disease from invading the population, the outbreak will be less compared to when no control is considered.

We have shown that each of the single control intervention strategies and the multiple control intervention strategy reduce the outbreak growth rate. Next, the multiple control intervention strategy is shown to reduce the outbreak growth rate more than each of the single control intervention strategy. The details are given in Theorem 7,

Theorem 7. *Suppose that , , , and , then Furthermore, *

which is easily demonstrated.

##### 3.6. Two Control Measures

Suppose a community can only afford two control intervention strategies which, for example, could be (i) vaccination + treatment, (ii) vaccination + treatment, and (iii) treatment + water purification. Qualitative analyses of these cases can be important for communities that can afford up to two control measures. Similar to the single control, the basic reproduction number and outbreak growth rate are used to investigate the impact of introducing double control measures. Using the same approach, the basic reproduction numbers induced by vaccination + treatment , vaccination + water purification , and treatment + water purification are, respectively, given by By a similar reasoning, the outbreak growth rates associated with vaccination + treatment , vaccination + water purification , and treatment + water purification are, respectively, given by Again, we compare these results with the cases for no control, a single control, or multiple control measures. In each comparison, the case with less controls must be a subset of the case with more controls. Comparing these basic reproduction numbers and outbreak growth rates show that considering two control measures is always better than no control or a single control in reducing the disease. On the other hand, the multiple control is better than the two control measures. Thus, if an outbreak occurs in any community, multiple controls are highly recommended. However, if a community has limited resources to control an outbreak, then the double or single control methods can be recommended depending on availability of resources.

To compare other combinations of controls (one case not necessarily a subset of the other) we consider numerical simulations. The parameter values for the numerical simulations are given in Table 1. The results are presented in Figures 1(a)–1(d). Again, as expected, in each comparison the multiple control has the greatest impact in reducing infections while no control has the least impact. The multiple control can prevent close to 25% more people from the disease as compared with no control.