ON OPTIMAL CONTROL AND COST-EFFECTIVENESS ANALYSIS FOR TYPHOID FEVER MODEL

Typhoid fever is a disease of a major concern in the developing world because it adversely affects on health and finance of a large chunk of people in this part of the world. This paper is aim to develop an extend and improve the optimal control model of typhoid transmission dynamics that can select the best cost-effective strategy for some interventions. Thus, an optimal control model for typhoid, incorporating control functions representing measures of personal hygiene and sanitation, diagnosis and treatment, and vaccination, was formulated. The corresponding optimality system was characterized via the Pontryagin’s maximum principle. The optimality system was numerically simulated for all possible strategies using Runge-Kutta method of order four. For cost-effectiveness analysis, the method of incremental cost-effectiveness ratio (ICER) was employed. The results show that the model is able to select the most cost-effective strategy for any given set of parameter values and initial conditions.

Mathematical models are veritable tools for studying the dynamics of infectious diseases. See, for example, Anderson and May (1991). Optimal control techniques have been used to determine best control strategies for infectious diseases such as malaria, Ebola, Influenza, tuberculosis, hepatitis B, tungiasis, to mention a few. See [Khamis et al. (2018) Athithan and Gosh (2016); Tchuenche et al. (2011)]. Mathematical models for typhoid transmission dynamics are scanty (Tilahun et al., 2017). Tilahum et al. (2017) presented a deterministic mathematical model to investigate the dynamics of typhoid fever with optimal control strategies. However we noticed a flaw in the associated system of differential equations emanating from their model descriptions. Thus the current study improves and extended the Tilahun et al. (2017) by incorporating the dynamics of vaccinated individuals. Further, this paper extended and improved optimal control model for typhoid transmission dynamics that can select the best strategy for some interventions, analytically characterize and numerically explore the corresponding optimality system.
The paper is organized as follows. Brief introduction on Typhoid fever was presented in section 1, the basic Typhoid fever model is presented and an optimal control model is designed in section 2, analysis of the optimal control model is done in section 3, and numerical simulations are performed and the results are presented in section 4. Cost-effectiveness analysis is carried out in section 5. Discussion of results and the conclusive remarks are passed in section 6 The rate of shedding salmonella in foods and waters by carriers σ 2 The rate of shedding salmonella in foods and waters by infectives μ The death rate of salmonella bacteria The flow of all epidemiological and demographic processes involved is described as follows. Recruitment into the susceptible class which is either by birth or immigration occurs at the rate of Λ. The recovered individuals lose partial immunity to typhoid fever to become susceptible at the rate of . The force of infection in the model is = [ + ] , where is ingestion rate, is the concentration of Salmonella bacteria in foods or waters, and [ + ] is the probability of individuals in consuming foods or drinks contaminated with typhoid causing bacteria. Death occurs naturally at the rate of . is the probability that an infected person becomes a carrier after infection. Carriers become symptomatic at the rate of and acquire natural immunity at the rate of . The symptomatically infected persons acquire natural immunity at the rate of . Typhoid-related mortality occurs at the rate of . Carriers and symptomatically infected individuals discharge Salmonella at the rates of σ 1 and σ 2 respectively. The net death rate of the pathogen is given by .  From the above descriptions and flow diagram, Tilahun et al (2017) presented to the following system of ordinary differential equations:

Modified Model Equation
The parameters 1 and 2 as defined in Tilahun et al (2017) model and captured in the model equations are flaws as the carriers and asymptotically infected individuals cannot themselves become bacteria as captured in their model (see Table1 and Equation (2) and Equation (3)  Thus with descriptions and flow diagram (Table1 and Figure 1), we modify the model equations by Tilahun et al (2017) and present the following system of ordinary differential equations:

Basic Properties
We obtain the invariant region in which the model solution is bounded. All the associated parameters and state variables are non-negatives for ≥ 0. Consider the biological feasible region Ζ = {( , , , ) ∈ ℝ 4 : ≤ Λ } Lemma 1: The closed set Ζ is positively and attracting with respect to the system of equations (6) -(9). Proof: Adding equations (6) -(9) gives the rate of change of the total population: = Λ − − (11) It is clear from equation (11) that Thus, by a standard comparison theorem (Lakshmikantham et al, 1989) can be used to show that Thus the region Ω ispositively-invariant. However if N(t) ≤ Λ , then either the solution enters Ω in finite time, or N(t) approaches Λ asymptotically. Hence the region Ζ attracts all solutions in ℝ 4 .
Therefore, it is sufficient to consider the dynamics of the flow generated by equations (6) -(9) in Ζ, where the usual existence, uniqueness, continuation results hold for the system (6) -(9), that is the system is mathematically and epidemiological well-posed in Ζ.

Optimal Control Model
In this section, we modify and extend the existing optimal control model of Tilahun et al (2017) by incorporating the compartment of vaccinated individuals , so that = + + + + The efficacy of sanitation measure at killing the pathogen is and we define the parameter as the rate at which the symptomatically infected persons acquire immunity. 1 , 2 1 are weight constants; and are control variables. All other parameters retain their descriptions as in the existing model which are depicted in Table 1 above. Therefore, from our modified model (6) -(10), the extended optimal control equations for typhoid dynamics are presented as follows: where = [ + ] The objective function is given by ( 1 , 2 , 3 ) = ∫ [ 1 1 ( ) + 2 2 ( ) + 3 3 ( ) + 1 2 ( 1 1 2 + 2 2 2 + 3 3 2 ) + 1 ( ) + 2 ( )] 0 (18) where 1 , 2 3 represent the costs of hygiene and sanitation, vaccine and drugs per person respectively. 1 , 2 , 3 represent the costs of implementation of control and 1 2 represent average losses of wages due to a typhoid related death and illness respectively. = ( 1 , 2 , 3 ) is a st of Lebesgue measurable functions.

Cost-Effectiveness Analysis
In this section, the method of incremental cost-effective ratio (ICER) is used to compare cost-effectiveness of two strategies. The cost objective functional is used to evaluate the total costs associated with all possible strategies over the period. The numbers of infections averted and the total costs of the corresponding strategies are shown in Table 3.  Table 3 shows that vaccination as a single intervention imposes the highest cost, followed by hygiene and sanitation, and treatment. It is also observed that treatment alone produces cyclical effects on the dynamics of typhoid fever. Figure 6 shows that double intervention of hygiene and sanitation, and vaccination is not able to eradicate the disease from the population.
However, Figures 5 shows that double intervention of hygiene and sanitation, and treatment has the capability of eradicating the typhoid disease. Similarly, Figures 7 shows that double intervention of treatment and vaccination has the capability of eradicating the typhoid disease, with a higher cost compared to hygiene and sanitation, and treatment. In the same vein, triple intervention of hygiene and sanitation, treatment and vaccination produces the same impact and imposes the same cost as the double intervention of hygiene and sanitation, and treatment as shown Figure 8. Based on the data employed, the findings show that a double intervention of hygiene and sanitation, and treatment as a strategy; and the combination of three controls as a strategy are the most costeffective strategies. Number of Symptomatic cases cases of typhoid fever No control u1,u2,u3