Abstract
A nine-compartment deterministic cholera model was formulated, and the model describes interactions between human, Vibrio cholerae bacteria, and the enviroment that warrant the interaction. Realities and socioeconomic burden influence spread and control mechanism of the disease. The model investigated some effective ways of hindering cholera outbreak and spread. The existence and uniqueness of solution of the system of equations that the model comprises were ascertained. The basic reproduction number of the model was obtained using “next-generation matrix” method, and the most sensitive parameters were identified using “normalised forward sensitivity index” method. Three controls, hygiene consciousness denoted by X1, cholera vaccine X2, and cholera awareness programme X3, were chosen. Optimal control theory is applied to ascertain the level of effect of the controls in reducing susceptible, exposed, infected individuals and causative pathogen population. Pontryagin’s maximum principle is used to prove the optimal solution of the model, and the optimal system was derived and numerically solved. Simulations were made with graphs that show the effects of the controls on susceptible, exposed, infected, and Vibrio cholerae population. The findings are that simultaneous application of the three controls can be one of the fast and effective ways of controlling cholera. If two controls are to be selected, hygiene consciousness and vaccine are the best combination.
1. Introduction
Cholera is a profuse diarrhoeal disease that can lead to death within short period of onset, if prompt treatment measures are not taken. It was estimated in [1], to be annually causing 21,000 to 143,000 deaths from 1,300,000 to 4,000,000 cholera cases worldwide, representing to of the reported cases. Natural disasters, climate change, social conflicts (terrorism), and economic meltdown usually force people to live in slums and refugee camps. The camps usually have very poor or insufficient infrastructures, especially good drinking water, and this serves as igniting factors of cholera outbreaks [2].
Severe cholera symptoms include vomiting, profuse rice-water stool, cramps, sunken eyes, high dehydration, and shock. Individuals that ingest incomplete cholera-causing dose do not usually manifest any cholera symptom and are usually referred as Vibrio cholerae carriers [3]. Severe cholera usually leads to death within short period of time that ranges between hours and three days. Chance of exposed susceptible individuals catching cholera will be half, if the concentration of Vibrio cholerae is cells per millilitre (i.e., ) and the least daily consumption of untreated water peg is a minimum of 1 litre per day [4].
Pontryagin’s maximum principle is one of the few methods of obtaining analytical solutions to optimal control problems. [5] derived optimal control model , with two controls, and [6] derived an optimal intervention strategies model. [7] presented and analysed optimal control model for cholera disease with three controls, and [8] derived cholera model and introduced an exposed individual compartment. They all used Pontryagin’s maximum principle to analyse their models.
Big risk in cholera disease and its outbreaks is that about 75% of Vibrio cholerae carriers do not have symptoms of the disease, but can spread the bacteria in the community through their faeces for one to two weeks after infection [9]. Cholera infection can be asymptomatic, mild or moderate, and severe; among the infected individuals that manifest symptoms, only 20% develop severe watery diarrhoea, while 80% have mild or moderate symptoms [10].
This calls for the formulation of cholera model that will analyse dynamics of Vibrio cholerae carriers during and after cholera outbreak, to optimise application of hygiene, vaccination, and cholera awareness programme controls. The paper is organised as follows: Section 2 describes the model formulation, which comprises the basic assumptions, description of state variables and parameters, system of differential equations, existence and uniqueness of solution, positivity of solution, and disease-free and endemic equilibria. Section 3 discusses the calculation of basic reproduction number and sensitivity of its parameters. Section 4 defines optimal control. Section 5 illustrates numerical simulation, and Section 6 explains summary, conclusion, and recommendations.
2. Model Formulation
We derive a mathematical model of the dynamics of cholera disease, with homogeneous total number of humans denoted by representing combined human population at time . This human population of the model is divided into eight mutually exclusive compartments and one nonhuman compartment.
2.1. Basic Assumptions of the Model
(i)Exposed individuals imply a very short period at which individuals have ingested Vibrio cholerae bacteria but yet to develop symptoms and cannot spread cholera pathogens(ii)Infected individuals cannot be vaccinated(iii)Infection can be as follows:(a)Moderate cholera infection, individuals that have the pathogen in their system but did not develop into cholera disease(b)Severe cholera infection, infected individuals that have manifested symptoms(iv)Moderate infected individuals can recover without treatment(v)Severe cholera infection must be treated before recovery(vi)Recovered individuals have no permanent cholera immunity
2.2. State Variables and Parameters of the Model
Description of the model’s state variables and parameters is shown in Tables 1 and 2.
2.3. The Model’s Equations
Schematic diagram of the flow between different classes of the state variables S, V, E, I, C, U, T, R, and B of the model is shown in Figure 1.
A system of differential equations (1)–(9) was obtained from the model’s assumptions, descriptions of variables and parameters, and the compartmental flow diagram.
2.4. Properties of the Model
Properties that ensure the correctness of the system of equations of the model, such as existence, uniqueness, boundedness, positivity, and equilibria, are ascertained.
2.4.1. Existence and Uniqueness of Solution
The existence and uniqueness of solution in formulated mathematical models need to undergo surety test to ascertain whether the solution exists, and if it exists, it is unique. Using the Lipschitz criteria, it is established as follows.
Let
Considering the region , bounded solution of system of equations (1)–(9) will be looked for in the same region whose partial derivatives satisfy , where and are positive constants.
Theorem 1. Let the region be denoted by , and then, the system of equations (1)–(9) has a unique solution provided that it is established thatis continuous and bounded in .
Proof. Partial derivatives with respect to each state variables of the system of equations (1)–(9) are obtained as follows: for ,Similarly, for , , , , , , and , the system of equations (1)–(9) exists and has a unique solution in since all the partial derivatives are and therefore bounded and defined.
2.5. Positivity of Solutions
Let equations (1)–(9) be subject to initial conditions:
For the cholera model to be meaningfully epidemiological, it is mandatory to prove that all the state variables are nonnegative at time , since the population of human is non negative but positive. This implies that the solutions of the system of equations (1)–(9) have nonnegative initial values and will remain nonnegative for all time .
Letbe solutions to the system of the equations (1)–(9).
Theorem 2. (a) Solution (11) is positive for all if , , , , , , , , and . (b) Solution (11) is nonnegative for all with respect to initial conditions (10).
Proof. (a) Suppose that (11) is defined for all , where , to prove that (11) is positive for all .
Considering (1), it is obvious thatIntegrating both sides of (15) with respect to , with the range to , and applying initial condition, it givesSince , .
Therefore,must be greater than zero.
Also, from (2)Integrating both sides within the interval and applying initial conditions, it givesand since , must be greater than zero.
Similarly, from equations (3)–(9), their respective results are as follows:(b) With the aid of proof of (a) and since the solution of the system of equations (1)–(9) continuously depends on the respective initial conditions in neighbourhood of zero, obviously it implies that (14) is nonnegative for all .
The model is dealing with population of living things, their number cannot be negative, and hence they are positive.
Theorem 3. Solutions of cholera model are bounded in with initial conditions (10), and hence, dynamics of the model is a dynamical system in the biologically feasible compact set.
Proof. Let be the solutions of the model with nonnegative initial conditions. Total human population is as follows:Without cholera infection, all the cholera-related deaths will be zero; therefore, .Using differential inequality theorem from [17] and the Birkhoff and Rota version as used in [11, 18],Multiplying both sides of (23) by the integrating factor and integrating, it givesSubstituting initial conditions are as follows: and .
This further gives
As , , which gives , but ; therefore, for all ,Hence, is bounded.
The solution set of human population of the system of equations (1)–(9) belongs to the feasible region:Also, letOn the other hand, the equation for the dynamics of the pathogen is (9) as follows:Since can be less than , then from (27),Integrating (29) givesAs , then .
The population of Vibrio cholerae bacteria is as follows:provided . Hence, the feasible solution set of Vibrio cholerae population of (9) enters the regionThis shows that the cholera model governed by the system of equations (1)–(9) is epidemiological well posed in a feasible region , defined by and .
It then implies that the entire human and nonhuman population of the model enter the feasible region . Also, since and , for all , then and cannot blow up to infinity in the finite time. Consequently, the model system is dissipative; that is, its solutions are bounded and exist for all in the invariant and compact set .The region is positively invariant and attracts all solutions in .
2.5.1. Disease-Free Equilibrium
Analysis of cholera model governed by a system of equations (1)–(9) is qualitatively carried out to obtain features of the dynamics and impact of control strategies, spread, and transmission dynamics of the cholera causative bacteria.
Disease-free equilibrium of the model governed by system of equations (1)–(9) is algebraically calculated by setting
In the absence of cholera disease, it is assumed that , and .
The DFE point is as follows:
2.5.2. Existence of Endemic Equilibrium State (EES)
This is a state where the disease cannot be completely eliminated, but stay in the concern region. The state variables of the model with exception of are not zero.
Let be the endemic equilibrium point, and let
Substituting (37) into the system of equations (1)–(9), the following is obtained:
From (25),
Substituting (42) and (43) into (24) gives
Let
Substituting related terms of (45) into (44) givesand from (43),
From (20),
Putting (31), (35), and (36) in (26), we obtain
Also let
Substituting (29), (37), (38), and related terms of (50) in (39) gives
Substituting (51) into (48) and simplifying, (48) becomes
Substituting (29), (40), and (41) into (21) gives the following:setting
This giveswhich means eitheror
For , it implies a disease-free equilibrium; when , it implies and ; hence, product of the roots , which means either or , and the positive root will then be
Therefore, there exists an endemic equilibrium state defined as follows:where
Note that must be negative for endemic.
3. Basic Reproduction Number
Basic reproduction number denoted by is the average number of secondary infections produced by a single infective introduced into the whole susceptible population.
is a threshold quantity that can be used to predict infection; when , it implies that only less than one person can be infected, which means the disease dies out. When implies that more than one person can be infected, this means the disease will persist.
There are many methods of obtaining for mathematical models, and “the next-generation matrix method,” a technique introduced by [19], which was also reported by [20], was used in this work, simply because the model is divided into compartments.
Taking to be the rate of appearance of new infection in compartment , is the transfer of individuals out of compartment by other means and substituting values of state variables at disease-free equilibrium point . will then be the spectral radius of
Note that is nonnegative, is a nonsingular M matrix, is nonnegative, and is also nonnegative as in [19]. The matrix was referred as the next-generation matrix as in [20] that gave the method its name. Finally, , which is the spectral radius of , considered as maximum real part of the matrix . Rearranging the equations of the model as , , and , the is calculated and obtained as follows:with .
Substituting terms given by (37), it gives
The Jacobian matrices of and about the disease-free equilibrium point are obtained by partially differentiating and with respect to , and to get matrices and .
Let and .
Let and ; hence,
3.1. Sensitivity Analysis
Sensitivity analysis is determining the level of effectiveness of parameters. The sensitivity analysis was carried out on model’s . The most sensitive parameters in this model are and ; increasing reduces , thereby reducing infection. On the other hand, increasing increases , thereby increasing infection. The normalised forward sensitivity indexes were performed using the following formula:where .
Using (68), the level of sensitivity of the parameters was obtained as given in Table 3. Level of sensitivity of each parameter in the obtained is shown in Figure 2.
4. Optimal Control Problem of the Model
Minimising susceptible, exposed, and infected individuals through hygiene consciousness, vaccine coverage, and cholera awareness strategies is the goal set to achieve using optimisation. The objectives of optimizing the model are: to reduce as much as possible, cholera susceptible individuals, exposed individuals, and cholera infection through three controls, hygiene consciousness of human, vaccinating susceptible individuals and awareness of measures to be taken to avoid cholera infection. The model is reformulated to identify permissible time-dependent controls , , and , which are now functions of time , and are, respectively, replacing parameters , , and of the system of equations (1)–(9) that is reformulated as follows:
Considering biological reasons, initial conditions of the state variables are assumed to be as follows: let similarly for the remaining state variables
Let the terminal time be .
The objective functional or performance index of the model is as follows:
The set of controls is Lebesgue measurable where
, , and , respectively, denote the weight constant of susceptible, exposed, and infected, while , , and , respectively, denote the constant relative cost weight of hygiene consciousness, rate of vaccine, and public awareness programme on cholera. The controls take their values in a closed interval .
implies inhibition of cholera outbreak and spread, while means no cholera outbreak and spread inhibition. implies vaccination coverage and effective cholera immune population. From general experience in vaccinations, as in [21], about of susceptible is usually the maximum to be covered in vaccination programme. means the susceptible have only natural cholera immunity (which might be weak) and are not complimented with vaccine, and this in turn increases chance of catching cholera. implies full cholera awareness in the population, which is assumed to alert the community to observe preventive measures to contain and curtail cholera outbreak and spread. will render the population unaware of the best ways to avoid and prevent the disease outbreak and spread.
The main aim is to minimise susceptible, exposed, and infected individuals to a level that will reduce cholera disease escalation through these three controls, such that
4.1. Existence of an Optimal Control Set
From (33), all feasible solutions of system (69) enter into the region . The system (69) can be written in the formwherewhere
satisfieswith the positive constant , independent to the state variables , , , , , , , , and .
It follows that optimality system and the optimal controls exist and the solution of system is bounded, since where
This follows that the function is uniformly Lipchitz continuous. Hence, the control variables and nonnegative initial conditions reveal that a solution of the system (69) exists (see [22, 23]).
Theorem 4. Given the objective functional (48) with set defined in (49) subject to the system (46) with initial conditions (47), then there exists an optimal control set such that (50) is true if the following conditions hold ():(i)The set of controls and corresponding state variables is nonempty(ii)The control set is closed and convex(iii)Each right-hand side of (46) is continuous bounded above by a sum of the bounded control and state and can be written as a linear function with coefficients depending on time and state(iv)The integrand of the objective functional is convex on (v)There exist nonnegative constants and and satisfying the following expression
Proof. Using established approach as in [6, 24, 25], the existence of solution of (46) is satisfied, the bounded coefficients and all solutions are bounded on a finite time interval, and first condition is satisfied.
It suffices to state that , where and are closed and convex sets defined as follows: Hence, by definition of convex, the control set is convex and closed, satisfying second condition.
By definition, each right-hand side of the system (69) is continuous and can be written as a linear function of with coefficients depending on time and state. All variables , and are bounded on . Using the boundedness of the solution and its positivity, as (69) is bilinear in , and , its right-hand side satisfies condition 3. The integrand in the objective functional (71) is convex on , and the fourth condition is satisfied.
Finally, it can be easily seen that there exist a constant and positive numbers and satisfyingThe Hamiltonian equation is formed accordingly by allowing each adjoint variable to correspond to each state variable, all combined with the objective functional.
Applying Pontryagin’s maximum principle, system (69) together with equations (48) and (49) is converted into problem of minimising the Hamiltonian defined below.
Since necessary conditions that an optimal control and state must satisfy can be generated from Hamiltonian , see [26].
Hence, the Hamiltonian function of the control problem of the model will bewhere , and are the adjoint functions, which can also be the co-state functions with respect to andLet , and be the optimal state solutions with associated optimal control variables for the optimal control of system (69).
To achieve optimal control, the adjoint or co-state functions , and must satisfy
4.2. Optimality Equations of the Model’s Controls
Partial derivative of the Hamiltonian with respect to each admissible control is all equal to zero, ease solving the optimal controls; thus,
Now, , , and .
The characterisation is as follows:where
5. Numerical Simulations of the Model
The numerical algorithm that simulates the optimal control of system (69) is presented below using a semi-implicit finite-difference method. This method is known as improved Gauss–Seidel-like implicit finite-difference method, and it was introduced by [27] and denoted it as GSS1. It was successfully applied by [5, 28].
Time interval where is discretely calibrated into points , where is the step. State and adjoint variables together with controls are defined in terms of nodal points.
State variables and have nodal points and .
Adjoint variables and have nodal points , and controls have nodal points with as the final time. The approximation of first-order time derivative can be numerically given as follows:
Using the GSS1 method, the forward-difference approximation of this optimal control problem is as follows:
The backward-difference is obtained using similar technique as follows:
The initial values of the state variables of system (46) used in the simulation are as follows:
Values of the parameters used in the simulations are in Table 2, with exception of the values of controls , and . Time used in the model is fixed at 35 days in the simulation of optimal control graphs except Figure 3(a), whose maximum value is 25 days. Values of weighting constants are as follows: The graphs from simulations of the model of system of equation (69) here are used to display the impact of the control measures on four state variables: susceptible, exposed, infected, and population of Vibrio cholerae.
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See Figure 4 shows the template of the three controls. Figure 5 displays impact of the three controls on susceptible, and it shows how the controls reduce population of susceptible to below 20 throughout the period of 35 days, but without the controls, population of the susceptible rise up to 900 in less than 10 days of the onset of the disease.
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Figures 6–8 show how the three controls reduced population of the exposed, infected, and Vibrio cholerae population. On the other hand without controls, the population of the exposed, infected, and the Vibrio cholerae rises 150000, 60000, and 100000, respectively.
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6. Summary and Conclusion
The study has succeeded in obtaining the model's threshold R0 and the most sensitive parameters in it; The interpretation of the figures and conclusion shows the final result.
6.1. Summary
The model’s basic reproduction number is as follows:
The most sensitive parameters in this obtained are and , increasing , which denotes hygiene consciousness of individuals will definitely reduce the spread of Vibrio cholerae, which will in turn reduce rate of contracting cholera disease. Since reducing implies making it less than unity, this is interpreted as “the disease reduces and dies out.” Increasing is increasing Vibrio cholerae, thereby spreading it to many individuals. On the other hand, denotes the rate of getting Vibrio cholerae through human-to-human interaction and its sensitivity index value is a positive, that is, 0.9978. Increasing it by 10% will increase by 9.978%. Literally increasing rate of ingesting Vibrio cholerae will definitely increase , the increment that will make it greater than unity, which implies escalation of the disease and persistence.
Simulations of SVEICUTRB model displayed in graphs show the impact of the control measures on four state variables: susceptible, exposed, infected, and population of Vibrio cholerae. Though [29] suggested the need to eradicate epidemics and its spread using appropriate measures in approximately 40 days, Figure 4 shows a template of the three controls the model use: hygiene consciousness, cholera vaccine, and cholera awareness programme. Impact of the three controls on four of the nine state variables of the model was presented as follows. Figure 5(a) shows how the controls reduce population of susceptible to zero throughout the period of 35 days, but without the controls, population of the susceptible rises up to 900 in less than 10 days of the onset of the disease. Figures 5(b)–5(d) show how the three controls reduced population of the exposed, infected, and Vibrio cholerae population. On the other hand, without controls, the population of the exposed, infected, and the Vibrio cholerae bacteria rises infinitely.
Figures 6(a)–6(d) show results of applying only one control “hygiene consciousness,” compared with when no control is used. The outcome shows that using the single control alone will increase the population of susceptible from 1000 when there is no control applied to 4000. Applying hygiene consciousness alone can suppress the population of the exposed, infected, and cholera pathogens (Vibrio cholerae bacteria).
Using single control “cholera vaccine” alone, results in Figures 7(a)–7(d) were obtained. Administering cholera vaccine as the only control for cholera can reduce the number of susceptible, exposed, rate of infection, and population of Vibrio cholerae bacteria to zero. It is hence an effective measure of eradicating cholera. Cholera awareness programme administered alone yields the results in Figures 8(a)–8(d). Application of single control 'cholera awareness' alone and not applying any controlat all seem to give the same result. The graphs show that population of the susceptible, exposed, infected, and Vibrio cholerae bacteria will increase the same way, when no controls were adopted. The population of susceptible will even increase the more, because cholera awareness programme is a preventive measure, not curative.
Application of two controls, hygiene measures and cholera awareness programme, gives result in Figures 9(a)–9(d). Figure 9(a) shows how susceptible individuals increase up to 5000 individuals, which is five times more than when there is no administered control. Application of two controls generally reduced exposed and infected individuals and population of Vibrio cholerae bacteria. In particular, two controls, hygiene measures and cholera vaccine, as shown in Figures 10(a)–10(d) show how the two controls were able to reduce number of susceptible, exposed, and infected individuals and population of Vibrio cholerae bacteria to a very small rate that is less than when no control is administered. Figures 3(a)–3(d) also display two controls: cholera vaccine and awareness programme; they show how the two controls reduced number of susceptible, exposed, and infected individuals and reduced the number of Vibrio cholerae bacteria population.
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6.2. Conclusion
Effects of the controls on susceptible, exposed, and infected individuals together with population of cholera pathogens were obtained and plotted. The effects show that the fastest way to control cholera quickly and effectively is application of all the three controls, at least above average rates.
6.3. Recommendations
After treatment measures, application of cholera vaccines, hygiene, and cholera awareness programmes are the best measures of preventing the spread of cholera and entire full control of the disease. Keeping hygiene consciousness rate always high and hindering ingestion of Vibrio cholerae bacteria by any means to keep its rate zero will surely prevent cholera disease. This has been justified by the model’s .
Data Availability
The values of variables and parameters data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported in part of S. F. Abubakar’s Ph.D. thesis. The authors acknowledge the contribution of Kebbi State University of Science and Technology, Aliero, for sponsoring the Ph.D. course.