MODELLING AND OPTIMAL CONTROL ANALYSIS OF TYPHOID FEVER

In this paper, we formulate a deterministic mathematical model to describe the transmission dynamics of typhoid fever by incorporating some control strategies. In order to study the impact of these control strategies on the dynamics of typhoid fever, the model captures vaccination and educational campaign as control variables. We show that the model is mathematically and epidemiologically well positioned in a biologically feasible region in human populations. We carry out a detailed analysis to determine the basic reproduction number R0 necessary for the control of the disease. The optimal control strategies are used to minimize the infected carriers and infected individuals and the adverse side effects of one or more of the control strategies. We derive a control problem and the conditions for optimal control of the disease using Pontryagin’s Maximum Principle and it was shown that an optimal control exists for the proposed model. The optimality system is solved numerically, the numerical simulation of the model shows 6667 MODELLING AND OPTIMAL CONTROL ANALYSIS OF TYPHOID FEVER that possible optimal control strategies become more effective in the control and containment of typhoid fever when vaccination and educational campaign are combined optimally would reduce the spread of the disease.


INTRODUCTION
Typhoid fever is caused by the bacteria Salmonella typhi. Typhoid fever infects 21 million people and kills 200,000 worldwide every year. Asymptomatic carriers are believed to play an essential role in the evolution and global transmission of Typhoid fever, and their presence greatly hinders the eradication of typhoid fever using treatment and vaccination. Typhoid fever is becoming an increasingly common illness worldwide, increasing resistance to various antibiotics is making antibiotic treatment less effective. Anderson and May [1], Hyman [2].
Typhoid fever has continued to be a health problem in developing countries where there is poor sanitation, poor standard of personal hygiene and prevalence of contaminated food. It is endemic in many parts of the developing world, illness do occur around the world in the span of a day.
Typhoid fever treatment is anchored on the blood culture condition of the patients. If the species is sensitive, the oral antibiotic is used. When dealing with large populations, as in the case of Typhoid fever, compartmental mathematical models are used. In the deterministic model, individuals in the population are assigned to different subgroups, each representing a specific stage of the epidemic. Several mathematical models have been developed on the transmission dynamics of typhoid fever these includes, (Adetunde [4]; Lauria et al., [5]; Kalajdzievska [6]; Mushayabasa [7]; Cvjetanovic et al., [8]; Moffact [9]; Pitzer et al., [10]; Date et al., [11]; Muhammad, et al., [12]; Watson and Edmunds [13]; Nthiiri [14]; Moatlhod and Gosaamang [15]; Mushayabasa [16]; Tilahun et al., [17] Peter and Ibrahim [18]). All of the above studies reveal an important result for typhoid fever dynamics by considering different situation, but we have identified that till now there is no studies that has been done to investigate the typhoid fever 6668 dynamics with the application of educational campaign and vaccination as control strategies. In view of the above, we incorporate two control strategies to the proposed model which are educational campaign and vaccination to control the spread of the disease. Many studies have examined optimal control in a good number of models of epidemic diseases [20][21][22][23][24][25]. . When the awareness parameter is set to zero, that is 0 =  , this implies that, there is no awareness so that the entire population in the susceptible class are unaware of typhoid fever and when awareness parameter is equal to unity that is, 1 =  , the 6669 MODELLING AND OPTIMAL CONTROL ANALYSIS OF TYPHOID FEVER entire population in the susceptible class are fully aware of typhoid fever, that is, they are aware of what causes the disease, the mode of transmission and how to avoid contracting the disease.

Model formulation
The illustration above is governed by the following set of differential equations.

Positivity of Solution
Theorem 1.
Let the initial conditions under consideration be given as  (1) , .
By separating the variable in (2) and then integrate At the initial time, t=0 and on substituting into (3) ) 0 ( By repeating the same process for other variables in (1) respectively,

The Basic Reproduction Number, ∘
The size of the basic reproduction number ∘ can be computed by using the popular technique known as the next generation matrix approach. This approach was formulated by Diekmann et al.
[26] but modified by Driessche and Watmough [27] and Peter et al. [28] by constructing n n matrix from the system of equations of the model an considering only the infective classes.
The inverse of the matrix V is obtained as: The product of matrices and −1 is the basic reproduction number which is the highest eigenvector for system (1) is given as

EXTENSION OF THE BASIC MODEL INTO OPTIMAL CONTROL SYSTEM
In this section, the basic model of typhoid fever is generalized by incorporating two control interventions. These are educational campaign and vaccination 1 u and 2 u respectively.
The objective functional is defined as The goal here is to minimize the total number of carriers and infected individuals and the cost associated with the use of educational campaign and vaccination on ] , In formulation the optimal problem we consider The goal here is to minimize the total number of infected individuals and the cost associated with

Theorem 2
Suppose the objective functional dt

Proof
To prove the existence of an optimal control pair we use the result in [29]. The control and the state variables are non-negative values and are non-empty. In the minimization problem, the necessary convexity of the objective functional in 1 u is satisfied. The control variable 1 u , 2 u ∈ U is also convex and closed by definition. The optimal system is bounded which determines compactness needed for the existence of the optimal control. Furthermore, the integrand in the objective functional which is 2 2

Necessary conditions of the control
By using Pontryagin's Maximum Principle. Pontryagin's et.al, [30], we give the minimized pointwise Hamiltonian as follows which converts system (1) and (5) objective function into an optimal problem, minimizing pointwise Hamiltonian H with respect to u 1 and u 2 . 6674 AYOOLA, EDOGBANYA, PETER, OGUNTOLU, OSHINUBI, OLAOSEBIKAN

Theorem 3
There exist an optimal control u 1 * and u 2 * corresponding solution S(t), I c (t), I(t) and R(t) which minimizes ) , ( 2 1 u u C over U. Furthermore, there exist adjoint variables ) , ( We now differentiate the Hamiltonian withrespect to each state variables ( ) ( ) To find the optimal * 1 u and * 2 u , we use the following partial differential equations

NUMERICAL SIMULATIONS OF THE OPTIMAL CONTROL ANALYSIS
Here, we provide the numerical simulations of the model in (5)       rapidly. Figure 7 shows the control profile, that is, the effectiveness of the two controls when combined together.
In all, each of the interventions is capable of influencing typhoid outbreak but the combination of the two strategies that is, vaccination and educational campaign are more efficient in limiting the spread and propagation of the disease and should be implemented in every typhoid prone community.

CONCLUSION
We showed that the model is mathematically and epidemiologically well positioned in a biologically feasible region in human populations. We also carried out a detailed analysis to determine the basic reproduction number R 0 necessary for the control of the disease. The optimal control strategies are used to minimize the infected carriers and infected individuals and the adverse side effects of one or more of the control strategies. In order to achieve control of the disease. The study concluded that possible optimal control strategies become more effective in the control and elimination of typhoid fever when vaccination and educational campaign are combined.
It is therefore recommended that any measure directed towards achieving typhoid fever-free society should include vaccination and educational campaign.

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