Optimal control model for the outbreak of cholera in Nigeria

In this work two mathematical models that described the dynamics of cholera in Nigeria were presented. The first model examined the bacteria population using a logistic definition for its growth in the expected habitat and their interaction with the susceptible population. The second model is an optimal control model that includes two timedependent control functions with one minimizing the contact between the susceptible and the bacteria and the other, the population of the bacteria in the water. The results from the numerical solutions of the models presented showed that increasing the susceptible pool and the infected population above some threshold values were responsible for epidemic cholera. It also showed that the difference between the growth rate (r) and the loss rate (n) of the bacteria plays a huge role in the outbreak as well as the severity of the disease.


INTRODUCTION
Cholera has become a worldwide health problem.It is an acute infection caused by the colonization and multiplication of Vibrio cholerae 01 or 0139 within the small intestine in humans (Emch et al., 2008).It is a waterborne disease that causes severe diarrhea and vomiting which leads to dehydration of the body and can prove fatal unless treated quickly.Outbreaks result from contaminated food, poor sanitation and dirty drinking water.(Codeco, 2001;Isere et al., 2009) Until the 20th century, cholera was confined to the Indian sub-continent.From this region, cholera has spread throughout the world seven times since 1817 (Capasso and Paveri-Fontana, 1979;Codeco, 2001;Lawoyin et al., 1999).The disease spread through the Asian continent during the 1960's; reached Africa in 1970 and Latin America in 1991 (Codeco, 2001;Lawoyin et al., 2004;Isere and Osemwenkhae, 2010).
In Nigeria, outbreaks of the disease have been occurring with increasing frequency since the first outbreak in modern times in 1970 (Epstein, 1993Osemwenkhae et al., 2009).Since then, cholera has continued to cause high mortality in humans, in Nigeria.The year 1999 saw the highest number of reported cases (WHO, 2009).Since then, cholera cases have been *Corresponding author.E-mail: abednis@yahoo.co.ukAuthor(s) agree that this article remain permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Mathematics Subject Classification: 53A04, 53A05.persistent in the country.Recently, in Kano, on September 22nd 2008, the United Nations office for the coordination of Humanitarian Affairs unit reported that cholera outbreak killed 97 persons in Kano.That report has it that the outbreaks were across Katsina, Zamfara, Bauchi and Kano states in northern Nigeria killing close to 100 people in just two weeks, making it the worst outbreak in the north for several years, according to an official from the National Primary Healthcare Agency (NPHA) in Abuja (UN, 2008).An earlier outbreak of cholera in Kano was reported in the work of Hutin et al. (2003).
More than 60 people have died in Zamfara State in the past two weeks, according to Zamfara's State Commissioner for Religion Affairs.The report says further; "It is quite unusual for northern Nigeria.If up to 100 people have died from cholera in just two weeks, you can imagine how many more are affected by the disease."The commissioner added, the death toll may be higher as reports of new infections are still coming in (UN, 2008).Most recently, cholera outbreak has taken toll in Adamawa State claiming a 52 year-old man, his wife, son and 36 others in the Maiha Local Government area of Adamawa state.(The Guardian Newspaper, Tuesday, August 18, 2009).It becomes significant we carry out a scientific study of this silent killing disease that has become endemic, so as to enhance its control in Nigeria using a mathematical model with a logistic growth term and an optimal control model with time dependent control functions incorporated.
Time dependent control strategies have been applied to various disease models.For instance, in Jung et al. (2002) an optimal control of treatments in a two-strain tuberculosis model was examined.Fister et al. (1998) and Kirshner et al. (1997) both studied time dependent control strategies used for HIV models.In Joshi et al. (2006), two control functions as coefficients in a system of differential equations represented treatment effects in a two-drug regime in an HIV immunology model.The goal was to maximize the concentration of T cells while minimizing the toxic effects of the drugs.However, time dependent control strategies for cholera is an innovation as few mathematical models on cholera did not include time dependent controls [Compare Capasso and Paveri-Fontana (1979) and Codeco (2001)].
In this work, we will examine two formulated models one without the control term and the other with control terms that are time-dependent.Simulations of these models are carried out to see the effect of these controls in the population dynamics of the disease.

FORMULATION OF MATHEMATICAL MODEL USING THE LOGISTIC TERM WITHOUT CONTROL
The logistic equation was first used in modeling human population by Verhulst in 1838.He followed a suggestion from his mentor Quetelet, that the resistance to growth should be quadratic and not linear (Britton, 2003).In 1940, this idea was confirmed when human population data of the USA was plotted overtime.The curve took a bell shape indicating that human population is non-linear.
The logistic equation was revived by Pearl and Reed in 1920.Pearl and others thought that fitting such a curve to a population time series would provide realistic short term forecasts as well as estimates of the ultimate steady state population K.
A careful observation however, shows that bacteria population growth is fitted excellently with a logistic growth equation (Britton, 2003).Since the bacteria population grows at a density-dependent rate and the probability of catching cholera depends on the concentration of V. cholerae in aquatic environment, then it behooves us to know the carrying capacity of the organism.The logistic growth term does that and is non-linear with respect to the bacteria population and makes the model more realistic since the population growth of bacteria is non-linear.Hence, we proposed a model using the logistic growth approach.In formulation, the dynamics of the infected is extended to include demographic factors, thus removing the assumption inherent in a closed population model.The mathematical model is proposed below: However, the parameters for the solution of the model are presented below: The symbols used: S: Number of susceptibles I: Number of infected B: Concentration of toxigenic V. cholerae in water (cells/m) (cells/ml) N: Total human population : Natural Human death rate (day -1 ) d: Disease related death rate (day -1 ) a: Rate of exposure to contaminated water (day -1 ) k: Concentration of V. cholerae in water that yields 50% chance of catching cholera (cells/ml) K: The carrying capacity of V. cholerae : Rate at which people recover from cholera (day -1 ).r: Growth rate of V. cholerae in the aquatic environment (day -1 ).n: Loss rate of V cholerae in the aquatic environment (day -1 ).e: Contribution of each infected person to the population of V. cholera (cell/ml day -1 person -1 ) α: Net mortality rate of V. cholerae in the aquatic environment : Recruitment rate into the susceptible class.(B): Probability of susceptible to catch cholera where k and B are as defined above.
Figure 1 shows the schematic representation of the flow between the different classes of the state variables (S.I.B).

The basic reproductive number and parameters for the model
The basic reproductive number is a non-dimensional quantity that measures the secondary infection caused by a typically infected person.If then there would be an epidemic since one infected is able to infect more than one person in a system.Otherwise, there would be a disease free state, that is, whenever .What happens when ; We have the threshold condition.But cholera becomes endemic whenever over a long period of time.Therefore, it becomes imperative to determine this quantity for any dynamical system.For our model it is given below: Table 1 below gives a brief description of the parameters.The result of Figure 5 is reacting to the slight increase in the net loss rate (the difference between the loss rate (n) and the growth rate (r)).The contribution from the infected is also factor.

Numerical solutions of the model without control
The factors mentioned in Figure 5 are responsible for the plot in Figure 6.Interestingly, this result is again the mirror image of the result of Figure 5.
This mild increase in the bacteria population is predicated by the infected population which could be effect of immigrants and contribution of the infected to the aquatic environment as well as the natural growth rate of the bacteria.

Optimal control model for the dynamics of cholera
Optimal control is the standard method for solving dynamic optimization problems, when those problems are expressed in continuous time (Lenhart and Workman, 2006).In this work, we use this method as part of control measures for cholera epidemics in Nigeria.
The proposed model that incorporates time dependent controls is presented below: where the objective functional to be optimized is reduce the organism growth and the contacts between infected and susceptible accordingly.
Our interest is to find a pair of control 1 u and 2 u in appropriately chosen class, and associated state variables SIB to minimize the objective functional Equation ( 10) above.
Again we minimize the contact of the infected with the susceptible and the associated cost of doing so.The population of the V. cholerae in an aquatic environment was also minimized.This was done by minimizing the objective functional above.Here, it was assumed that the cost associated with reducing contact of the infected and the cost of reducing the V. cholerae population in an aquatic environment are non-linear and take a quadratic form.This agrees with the non-linear nature of the model used.The u , such that: We assume that the controls  (Kirschner et al., 1997), we derive necessary conditions for our optimal control and corresponding state variables, including constraints on the controls.Since we have three state variables, SIB, we shall have three corresponding adjoint variables where corresponds to S, and corresponds to I and corresponds to B.

The Hamiltonian adjoint equations
The Hamiltonian equation is formed by allowing each of the adjoint variables to correspond to each of the state variables accordingly and combining the result with the objective functional as below: The adjoint equations are formed by taking the derivative of the Hamiltonian with respect to each of the state variables (SIB).Hence adjoint equations are:

The Optimality Equations
The equations are obtained by finding the derivative of the Hamiltonian equation with respect to the control variables, equating to zero and solving the resulting equation.Hence the optimal equations are: Then the optimal value for 1 u is:  The foregoing shows that the optimality conditions (taking derivatives of the Hamiltonian with respect to the controls) only hold in the interior of the control set.Next, we need to resolve the optimal control model numerically.

NUMERICAL RESULTS OF THE OPTIMAL CONTROL MODEL
Here, we are going to examine the effect of control 1 and control 2 respectively visà-vis the epidemiological classes: the susceptible, the infected and the bacteria population classes (SIB) as shown in Figure 10.It is to be noted that 1 u (control 1) is actually controlling the contact between the infected and susceptible while 2 u (control 2) is minimizing the concentration of the bacteria population in the water.Three sets of results are presented using the parameters from Table 2. Figures 8 and 9 correspond with the values on column A and column B respectively.Using values C, we varied k to see its effect on the system.
The differences in these parameters lie in the growth and loss rates of the bacteria, that is, ' ' and ' ' n r respectively.
Consequently, as shown in Figure 11, keeping 5 10  k will be easier to control.The system remains controlled for more than four days before it goes out of hand.The conclusion is that, the system would effectively be controlled when the concentration of bacteria that yields 50% chance of catching cholera (k) is not too high.The value of 10 5 is adequate for an effective control of the system.Here the first value has and the second value has .

DISCUSSION
In this work, two models were examined.We presented a cholera model with a logistic term for its growth in the expected habitat and their interaction with the susceptible population.The second model was an optimal control model that included two time-dependent control functions with one minimizing the contact between the susceptible and the bacteria and the other, the population of the bacteria in the water.The mathematical model formulated really provided insight into the dynamics of cholera in Nigeria.The  results actually substantiated the minimum condition for the development of epidemic and endemic cholera stated: that if the rate of exposure to contaminated food and drink ( a ) and the contribution of the infected ( e ) to the bacteria population is less than the concentration of the bacteria that yields 50% chance of catching cholera ( k ), 0 R will be asymptotically stable.Therefore, if we would attain a cholera-free community, we should keep the susceptible population minimal, always below the threshold condition. can be controlled if immigration is checked.In Nigeria, prior to 1970, the susceptible could have been below the threshold.However, in the late 1960's immigrants from the Asian continent came into the country, and that increased the susceptible pool and few infective that came in triggered an epidemic in 1970 (Lawoyin et al., 1999).Now that cholera has become endemic in Nigeria, we should avoid situations that will encourage the growth of V. cholerae in our community by ensuring sanitary condition perhaps through environmental sanitation exercise, and also providing portable drinking water.

Figure 1 .
Figure 1.The relationship between the state variables: the Susceptible, Infected and the pathogenic bacteria (SIB).

Figures 2
Figures 2 to 4 represents the plots from values A, while Figures 5 to 7 represents those from values B. The state variables (SIB) are

u
being the systemic combination of the organism population and the cost of reducing contact of infected and the susceptible.The parameters here are as defined in Equations (1) -(3).The control functions 1 u and 2 u represent the reduction of contact between infected persons and the susceptible, and the treatment of water to reduce the growth of the organism respectively.In Equations (6) to (8) above, the organism population grows at a density -dependent rate coefficients and are balancing cost factors due to the size and importance of the parts making up the objective functional.Hence we are interested in finding an optimal control pair

u
is bounded below by 1 a and bounded above by 1 b and 2 u bounded below by 2 a and above by 2 b , we must constrain the values of the control and obtain the characterization

Figure 10 .
Figure 10.The effect on The SIB-Populations.

Figure 11 .
Figure 11.The effect from varying K on SIB(where k is the conc. of V. cholerae that yields 50% of catching cholera).

Table 1 .
Parameters for the model.

Table 2 .
Parameters description