COST-EFFECTIVENESS ANALYSIS OF A CHOLERA MODEL WITH AWARENESS PROGRAMS AND LIMITED RESOURCES

In this paper, we developed a mathematical model to describe the dynamics of Cholera with awareness programs under limited resources. The model was fully analysed and the threshold for the eradication of the disease was given. It was proven that the model undergoes a backward bifurcation under certain conditions. The model is then extended to include optimal controls, namely: vaccination of susceptible individuals, killing of vectors, and water sanitation. Numerical simulations for the application of a single control, combinations of two controls and all three controls were given. Cost-effectiveness analysis was carried out in order to determine the most cost-effective control strategies, which was found to be the combination of vaccination and water sanitation.


INTRODUCTION
Cholera is a notable public health problem in many major parts of the world which can kill tens of thousands of individuals [25,12,36,37]. Symptoms include severe acute watery diarrhea. However, it is possible to be a carrier of cholera without exhibiting symptoms since most infected people do not show symptoms for 1-10 days after infection, even though Vibro.
cholerae is present in their faeces and they can shed back into the environment and infect other people. On the other hand, among people who develop symptoms, the major have mild or moderate symptoms, while rare cases develop acute watery diarrhea with severe dehydration.
Cholera is acquired through direct or indirect transmission either through physical contact with an infected host or contact with a contaminated environment [9,4,38,6]. Also, Cholera can be obtained by vector transmission when a living organism carries an infectious agent on its body (mechanical) or as an infection host itself (biological), to a new host [30,28,2]. Despite recent progress in the provision of clean water and good sewage systems especially in developed countries, cholera is still endemic in many developing countries and remains a considerable public health problem with approximately 1.3 to 4.0 million cases, and 21 000 to 143 000 cholerarelated deaths annually worldwide [24].
Several mathematical models have been published in order to describe the dynamics of Cholera [9,4,38,6,25,12,30,28,2]. The first mathematical model was published in 1927 by Kermack and McKendrick [35]. The model includes three compartmental classes for susceptible state S, infected people I and recovered individuals R. Capasso [33] proposed a model which was introduced in 1973 to describe the Cholera epidemics in Italy. Then Codeco [4] in 2001 extend Capasso's model and explicitly accounted for the Vibro Cholerae concentration in the water supply which represents the environmental component into a basic SIR model and he modeled the incidence by a logistic function to represent the saturation effect. Hartley et al. in 2006 extended Codeco's model by adding a hyperinfectious state of the pathogen, representing the "explosive" infectivity of freshly shed Vibro Cholerae, based on the laboratory observations [7]. Codeco's model also modified by Joh et al. in 2009 [14] by assuming that there is an explicit incorporation of a minimum infectious dose of pathogen to cause infection. Mukandavire et al. [38] proposed a model to study the 2008-2009 Cholera outbreak in Zimbabwe in which they explicitly considered both human-to-human and environment-to-human transmission pathways. Tien et.al. in 2010 [17] published a SIWR model which also included the water state pathogen concentration in water reservoir. Their model included the dual transmission pathways, with bilinear incidence rates employed for both the environment-to-human and human-to-human infection transmission. Moreover, Neilan et. al. [29] modified the Cholera model proposed by Hartley [7] and added three control measures into the model. Misra et. al. in 2012 citemisra2012modeling proposed a delay mathematical model for the control of cholera epidemic by assuming that the disease spreads through carriers, which makes the human food contaminated by transporting bacteria from the environment. Recently, Al-Shanfari et.al. [30] proposed a mathematical model that describes the dynamics of Cholera by investigating the role of houseflies in the transmission of Cholera which carry Vibro Cholerae on its body and transfer it from unsafe environment to safe environment. These models differ from each other in some aspects, however, the recovery rate is assumed to be a constant in most of these models.
Farai et.al. in 2019 studied the transmission dynamics of cholera in the presence of limited resources by including a nonlinear recovery rate [10]. In real application, recovery rate depends on many factors. The first factor is the number of infectious individuals seeking treatment. Recovery rate also depends on the availability of health resources to the public which are considered very sufficient for the infectious disease [1,10,16,5]. The resources of the health system includes the number of health care workers (physicians, nurses, pharmacists, etc.), number of hospital beds and medicines and the effectiveness and efficiency of the treatment. In many developing countries, resources of treatment are very limited. So assuming constant recovery rate cannot reflect the real cure rate and the nonlinear recovery rate is normally developed to account for the role of the availability of medical resources.
The awareness programs play an important role in the transmission of the infectious disease.
Obviously, awareness can reduce the opportunity and probability of contact transmission among the alerted susceptible populations, which in turn helps to control and prevent the disease from further spreading. In recent time, information about the outbreaks is spreading quickly because of the significant use in social media and the increase the travel around the world. Awareness can have a reflective effect on the actual epidemic dynamics. It can have a positive influence on the disease spread [27,14,22,26,32]. For instance, in Mexico (2009) the media coverage of transmissibility and mortality rate associated with the virus of swine influenza strongly influenced by behavioral changes and hence it helped to control the epidemic [14]. Also, awareness can lead to negative consequences such as in Romania (2007Romania ( -2012 although the implementation of two HPV vaccination programs was started, the uptake of vaccination remained low and the programs were failed because the information about the side effects and insufficient testing of the vaccine was negatively spread [26].
In this study, in order to understand the impact of awareness programs on the spread and control of cholera, a mathematical model will be proposed to capture the dynamics of the effect of awareness programs. The awareness about the disease will alert the aware individuals so that they isolate themselves, reducing the contact with the infected and avoid unsafe environments.
Hence, the probability of contracting infection for individuals in aware class is less than those who are in unaware class.
The rest of this paper is organised as follows; in section 2 we present the model formulation and analysis, in section 3 we move on to present the numerical simulations and bifurcation analysis of the model, in section 4 we study the optimal control analysis and cost-effectiveness, and finally, in section 5 we draw the conclusion of our studies.

MODEL FORMULATION
We use the model in [30] and divide human population into five compartments: aware susceptible S hu (t), uaware susceptible S ha (t), aware infectious I hu (t), unaware infectious I ha (t), and recovered R h (t). The vector population is divided into two compartments: susceptible S v (t), infectious I v (t). Furthermore, we consider two compartments B(t) and E(t) that reflects the bacterial concentration at time t in safe and unsafe environments respectively. It is assumed that the disease spreads due to the direct contact between susceptible and infective and by indirect transmission either by vectors or by ingesting environmental vibrios from unsafe environment.
Also, it is considered that due to the awareness, aware susceptible individuals avoid being in contact with the unsafe environment.
Also, we incorporate aspects of limited resources in the model by assuming that the recovery rate (γ) depends on both the number of infectious individuals (I h ) and the hospital bedpopulation ratio (b > 0) such that, it is a function of both b and I h . The recovery rate γ(b, I h ) is thus given as follows: where γ 1 is the maximum per capita recovery rate due to the sufficient health care resource and few infectious individuals as well as the inherent property of a specific disease, γ 0 is the minimum per capita recovery rate due to the function of basic clinical resources [5]. These assumptions are translated into the following mathematical model: The new system becomes: The feasible region for model ( The set Γ is compact. To show that Γ is positively invariant set, i.e., all the variables of model (3.1) stay non-negative if it starts with non-negative initial data, we rewrite model (3.1) as ..9, then it is obvious that F i (x) ≥ 0 and hence Γ is positively invariant set.
We assume that all parameters are positive and the initial conditions of system (2.2) are given The C 1 smoothness of the right side of system (3.1) implies the local existence and uniqueness of the solution with the initial values in ℜ 9+ .

3.2.
Equilibria and the basic reproduction number. Obviously, model (3.1) admits a unique disease-free equilibrium DFE ξ 0 . A positive equilibrium of (3.1) in the interior of Γ, if one exists, is called an endemic equilibrium EE, and denoted by ξ * = The disease-free equilibrium (DFE) of model (3.1)is: The basic reproduction number, R 0 , is a threshold parameter that allows us to predict whether the disease will die out or persist. Here, we used the constant term of characteristic equation method to find R 0 since the next generation matrix failed to find its value.
The Jacobian matrix at the DFE is given by: The characteristic equation of model (3.1) around the DFE is given by where, c 0 is given by: and R 0 is given by: The disease-free equilibrium Obviously c 0 > 0 iff R 0 < 1.
• more then one endemic equilibria otherwise.

Bifurcation.
We establish conditions for the existence of backward bifurcation following Theorem 4.1 proven in [3]. We shall make the following change of variables: We now use the vector notation X = where Φ is the bifurcation parameter, we assume R 0 = 1 and then find the The Jacobian matrix, after linearizing model system (3.1) around the disease-free equi- It can be shown that J(ξ 0 ), has a right eigenvector is given by w = We compute a and b in order to apply Theorem 4.1 in [3]. For system (3.8), the associated non-zero partial derivatives of F at the disease-free equilibrium are as follows: It thus follows that: Here the coefficient b is obviously positive and the coefficient a will decide the backward bifurcation of the model (3.8). In particular the backward bifurcation in the model would occur if the coefficient a is positive. Therefore, since there is a possibility for the model to exhibits backward bifurcation, then, reducing R 0 below unity is not sufficient to control the cholera epidemic.

NUMERICAL SIMULATION
We perform some numerical simulations of system (3.1) to see the effects of awareness programs parameters (δ , ε 1 , ε 2 ).
We list the values for the parameters in the system in Table ( µ e (30d) −1 day −1 [38] ε, ε 1 , ε 2 0.001 − 0.01 day −1 Assumed α 0 1 − 150 Cells mL −1 day −1 per vector [30] α 2 1 − 150 Cells mL −1 day −1 per person [38,10]  The limited resource parameter b, is varied in Figure (4). It is shown that as b is increased, the human infection population decrease. This means that increasing the number of hospital-beds during an outbreak, will give a high opportunity for the disease to persist. Hence, the number of hospital beds plays an important role in controlling the spread of infectious diseases.

EXTENDED MODEL WITH OPTIMAL CONTROL
Optimal control theory is a powerful tool to make decisions involving complex dynamical systems. Optimal control theory is a tool to find an optimal path or value that gives either maximum or minimum points of functions. Optimal control is a set of ordinary differential equations describing the paths of the control variables that minimize the cost functions. Studies have indicated that epidemiological models may give some fundamental rules to public health practitioners to compare the effectiveness of different potential management strategies. We introduce three time dependent control variables u 1 (t), u 2 (t) and u 3 (t) in the model (3.1). u 1 (t) represents vaccination of susceptible individuals, u 2 (t) represents the killing of vectors, and u 3 (t) represents water sanitation. The control functions u 1 (t), u 2 (t) and u 3 (t) are bounded and Lebesgue integrable functions. Vaccination is introduced to the susceptible populations at a rate of u 1 (t), so that u 1 (t)s hu (t) and u 1 (t)s ha (t) individuals per time are removed from the susceptible classes and added to the recovered class. The elimination of the vector populations is applied to the infected people at a rate of u 2 (t). Water sanitation leads to the death of vibrios at a rate of u 3 (t). As a result, we obtain the following dynamical system: A control scheme is assumed to be optimal if it minimizes the objective functional: Here the parameters c i and a i (i = 1, 2, 3), with appropriate units, define the balancing cost factors for control strategies. We introduced the quadratic terms to show nonlinear costs which are arising at high intervention levels [?, 31, 29].Hence, we want to minimize the objective function (5.2) subject to the system (5.1). For this purpose, we have to find the values of u 1 (t), u 2 (t) and u 3 (t) which minimize the objective functional (5.2), i.e. to find u * 1 , u * 2 and u * 3 such that where u 1 , u 2 , u 3 are Lebesgue measurable and the control set Ω is defined as: Note that, , the conditions for the existence of optimal control are satisfied since the control set Ω is closed and convex, and the integrand of the objective functional (5.2) is convex [31].
To find the optimal control solution, we will use the Pontryagin's Maximum/Minimum principle [34]. We define the adjoint functions λ i where i = s hu , s ha , i hu , i ha , s v , i v , B, E and associated with the state equations for s hu , s ha , i hu , i ha , s v , i v , B and E, respectively. We then form the Hamiltonian, H, by multiplying each adjoint function with the right-hand side of its corresponding state equation, and adding each of these products to the integrand of the objective functional. As a result, we obtain Theorem 5.1. Given an optimal control quintuple (u * 1 , u * 2 , u * 3 ) and solutions over Ω. Then there exists adjoint variables λ s hu , λ s ha , λ i hu , λ i ha , λ i v , λ B , and λ E satisfying and with transversality conditions λ i (t f ) = 0 where i = a hu , s ha , i hu , i ha , i v , B and E with the control quadruple (u * 1 , u * 2 , u * 3 ) is given by: Proof. The adjoint equations and the transversality conditions are obtained from the Pontryagin's Maximum Principle, such that The characterizations of the optimal controls u 1 ,u 2 and u 3 ,are based on the conditions Solving equation (5.9) for (u * 1 , u * 2 , u * 3 ) gives the characterization (5.7).

Numerical illustrations.
The optimal control problem consists of the state system • Single optimal control scheme.
• Two optimal control schemes.
• All the control schemes.   . two optimal schemes in the absence of the third one. It can be seen form figures (8,9,10) that, all the method the number of aware infected individuals effectively while the effects on unaware infected vary. The infection level of unaware people has been significantly reduced due to vaccination and vector elimination (figure (8)). Using vector elimination and water sanitation schemes reduce number of unaware infected at the beginning of the infection period (figure (9)) but then the number if unaware infected will increase. Also, it can be seen from figure (10) using vaccination and water sanitation schemes have effects only at the onset of the infections.  Right: The optimal controls.

COST-EFFECTIVENESS ANALYSIS
Cost-Effectiveness Analysis calculates the costs and effects gains of alternative interventions by providing a method for organizing interventions for a given set of resource constraints. It identifies which control profile that have the potential which gives the optimal improvement in health with less cost.1 [21,20]. A variety of methods have been used in applied costeffectiveness studies to estimate the costs and effects of different interventions.
6.1. Efficacy function. The efficacy function is given by: where i h (0) is the values of infected human obtained at the endemic equilibrium of the system (5.1) before the introduction of interventions ( i.e. u 1 = 0; u 2 = 0; u 3 = 0), and i * h (0) is the values of infected human obtained at the endemic equilibrium of the system (5.1) after the introduction of the corresponding intervention.
Using the above formula (6.1), the efficacy of each intervention is given by the following computations:   Using the above formula (6.2), the IAR of each intervention is given by table (6.2).
Intervention IAR i hu (t) IAR i ha (t)  However it is also clear from the table that the differences between the IAR of theses two interventions is very small.

Incremental Cost-Effectiveness Ratio.
Incremental cost-effectiveness ratio is used to compare the differences between the costs and health outcomes of two intervention strategies that compete for the same resources. It is generally described as the additional cost per additional health outcome [23]. The ICER numerator includes the differences in intervention costs (total cost TC), while, the denominator is the difference in health outcomes ( infections averted (A i ) ). We start the analysis by ranking the total infections averted of the control interventions in increasing order. Next, we define and calculate the ICERs as follows: The remaining calculations of ICER for both aware and unaware infected population are given in    Repeating the entire process, we can determine the next most cost and less effective strategy for unaware infected population. Thus, we found that the combination of vaccination and vector elimination (u 1 and u 2 ) is the next more costly and less effective strategy; then it is followed by using all controls strategy (tables (6.3) and (6.3)). From table (6.3), it is concluded that using vaccination (u 1 only) has the least ICER and therefore is more cost-effective control strategy for unaware infected population.

CONCLUSION
The main contribution of this work is that we have developed a cholera model with two distinct groups (aware and unaware) in the susceptible and infected classes. The disease dynamics is formulated by presuming a general representation for both the direct transmission and the pathogen shedding, and the interaction between environmental vibrios and human vibrios. In addition, we study a nonlinear recovery rate which gives us more insight into the dynamics of the disease transmission in resource limited health settings. We consider a parameter b representing the number of available hospital bed over the human population. The model exhibits disease-free equilibrium which is locally asymptotically stable. Then we study the existence of endemic equilibrium. The numerical results show that dynamics of cholera infection decreases as the number of hospital beds and awareness programs increase. Therefore, in order to eliminate the disease, effort must be targeted to increasing hospital resources as well as media coverage.
Then, we applied optimal control theory to cholera model. We introduced three time-dependent control variables into the model and investigated the associated benefits of different control strategies using cost-effectiveness analysis. Using efficacy analysis, we found that the control strategy utilizing all three control variables is the most efficient strategy to eliminate cholera from both aware and unaware infected populations. On the other hand, IAR analysis of costeffectiveness showed that the most effective way to eliminate the disease is by using vaccination and water sanitation for both aware and unaware infected populations. ICER analysis showed that vaccination only is the most effective strategy to control cholera for aware infected population, while, using water sanitation only for is the best way to control the disease for unaware infected population. Therefore, we can conclude that, the most efficient and cost-effective control strategy for the entire population is the strategy involving vaccination and water sanitation.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.