Typhoid fever dynamical model with cost-e ff ective optimal control

Typhoid fever is a highly communicable and infectious disease that can be fatal and causes severe complications if unattended to timely. The infection, at times, can be more complex, challenging, and impossible to treat as antibiotics become less e ff ective. Hence, the e ff ect of limited clinical e ffi cacy of the antibiotics with corresponding relapse response to treatment on infected humans is considered in this paper by formulating a deterministic model for direct and indirect transmission mode of Typhoid infection. The basic reproduction number is analytically derived and used to implement the global sensitivity analysis. Following the sensitivity analysis result, the optimal control analysis is carried out and simulated numerically with four controls: sanitation and hygiene practice and awareness campaign control, sterilisation and disinfection control, the potency of antibiotics control and screening control. Finally, the cost-e ff ectiveness analysis for infected and susceptible humans with four cases that compared fifteen strategies is analysed. The results indicate that the sanitation and hygiene practice and awareness campaign is good to implement for single control implementation, while for double control implementation, Strategy 6, which is the combination of Strategy 1 and the potency of antibiotics administered to typhoid patients, is the best to consider. Combining Strategy 6 and screening control is the most cost-e ff ective for triple controls. Furthermore, the overall computation of cost-e ff ectiveness among all the most cost-e ff ective with all the controls combined suggests that sanitation and hygiene practice and awareness campaign is the most cost-e ff ective strategy to implement for eradicating typhoid infection in the population and for preventing susceptible populations from contracting the bacteria.


Introduction
Typhoid fever is a highly communicable bacterial infection that can be fatal and cause severe complications without prompt treatment.It is caused by salmonella typhi [1].The mode of transmission is through faecal-oral means [2], which can be environment to humans indirectly through contaminated food or water [3].According to Ref. [4], typhoid transmission can also be direct, that is, person-to-person transmission; this could be very rare.In Ref. [3], the disease is peculiar in places with poor sanitation and areas lacking potable water.Yearly, it is estimated that about 11-20 million people contract typhoid fever, out of which between 128,000 and 161,000 people die from it worldwide.
The disease is much more common in Africa and South-East Asia than in other regions of the world, making the dis-ease endemic in Africa and South-East Asia.For example, in Nigeria, typhoid incidence ranges from 3.9% to 18.6% in a population of about 100,000 [5].It is noteworthy to state that some individuals who have totally recovered from a disease like salmonella typhi and still harbour and shed the bacteria in the environment for a prolonged period without knowing such individuals are called chronic or active carriers [6].
About one in every 5 cases of Typhoid infection can be fatal if untreated, while fewer than 4 in every 100 cases are fatal with treatment.Symptoms of Typhoid fever usually begin between 6 and 30 days after exposure to the bacteria [7].The incubation period for typhoid fever is typically between 6 to 30 days, and 1 to 10 days for paratyphoid fever [8].If Typhoid fever is diagnosed early, antibiotics may be prescribed for the treatment from 7 to 14 days.However, some typhoid fever patients experience relapse response to treatment in which the symptoms of the bacteria return, though this entails further treatment with antibiotics [1].Meanwhile, after antibiotics treatment, about 2 − 5% of people who recover from typhoid fever still harbour the bacteria and continue shedding the organism for over a year; these people are called Chronic Carriers [1,7].
The mathematical modelling of infectious diseases is an essential tool for analysing the dynamic nature of the infectious disease.It helps develop control strategies to forecast appropriate control strategies [9,10].An optimal control problem requires regularising and solving problems by choosing the best way in a dynamic process, which depends on controls and is always subject to constraints [11].
Lauer et al. [12] stated that cost-effectiveness analysis (CEA) is a kind of economic evaluation and assessment geared towards efficiency to achieve the most for the available resources or, to be precise, the value of money.Several authors have studied the dynamics of typhoid disease incorporating control measures, such as [13][14][15][16][17][18][19].Meanwhile, the optimal control analysis of typhoid infection has been investigated by Refs.[13], [20], where Ref. [13] considered only indirect transmission with vaccination, hygiene practices, screening and sterilisation as controls.In contrast, Ref. [20] considered both direct and indirect transmission modes of Typhoid fever with education campaigns, sanitation, screening, and early treatment as control measures.Employing the cost-effectiveness of the optimal controls, Refs.[10,13] (with only indirect transmission) and Ref. [15] (in the presence of direct and indirect transmission).In Ref. [9], the authors examined sanitation, hygiene, and treatment as control measures.
Complementing the work of Ref. [16], we constructed a mathematical model of the type S , I s , I m , I c , T, R, and B c .We include severe and mild compartments and logistic growth in the bacteria compartment.We also introduced the incidence function of limited clinical efficacy of antibiotics with corresponding relapse response to treatment in which some treatment individuals failed to recover [13] but returned to the severe infected compartment instead of the recovery compartment because infections like pneumonia, tuberculosis, blood poisoning, gonorrhoea, and foodborne diseases like typhoid -are sometimes becoming more brutal and more challenging, and even impossible to treat as antibiotics become less effective [21].We also carry out global sensitivity analysis, optimal control and cost-effectiveness analysis for both infected cases and susceptible individuals in this study.
The rest of the paper is arranged as Section 2 is the model formulation, while Section 3 is the mathematical analysis.Section 4 is the optimal control analysis and cost-effectiveness analysis.Finally, the paper is concluded in Section 5.

Formulation of Model
For the formulation of the model, the total human population, N(t), at any time, t, is subdivided into six (6) subpopulations: Susceptible humans, S (t), Mild infected humans, I m (t), Severe infected humans, I s (t), Infected carrier humans, I c (t), Treatment humans, T (t) and Recovered humans, R(t), N(t) = S (t) + I s (t) + I m (t) + I c (t) + T (t) + R(t).B c (t) represents the bacteria-contaminated environment.Susceptible humans, S (t), are likely to be infected by typhoid fever infection when they have contact with a contaminated environment or infected humans.Severe infected humans, I s (t), are symptomatic infected individuals.Mild infected humans, I m (t), represent infected humans with mild symptoms.Infected carrier humans, I c (t), stands for asymptomatic infected individuals treated or people on the verge of total natural recovery but still carrying Salmonella Typhi [1], [8].Treatment humans, T (t), are individuals undergoing treatment assuming that they cannot infect susceptible people due to their restriction to a particular place and that they do not shed the bacteria in the environment.Recovery humans, R(t), have recovered from the disease entirely by treatment.
Figure 1 and Table 1 are detailed descriptions of the model parameters and the systematic diagram.Consequently, with the systematic diagram of Figure 1, the descriptions of the model's parameters in Table 1, and the initial conditions, S (0) > 0, K+B c , the treatment functions, T (I s ) = θ 1 I s 1+ω 1 I s and g(T ) = θ 2 T 1+ω 2 T and the model parameters are assumed to be nonnegative, the autonomous system of equations for the typhoid fever model is obtained as follows: The degree of the effect of demand for treatment 0.62 [16]

Mathematical Analysis of the Model
The basic properties of the system of equations ( 1) and its reproduction number are established in this section.

Invariant Region
The mathematical well-posedness of the model is proven in this subsection to show that the system (1) is epidemiologically meaningful.Defining and initial conditions, N(0) = N 0 and B c (0) = B c0 , we state the following theorem. and , are subset for the human population and bacteria, respectively provided µ B > α.
Proof.Applying the approach of integrating factors to equation (2) to get 3), implying that the feasible solutions of the human population are in the region, , it means that the last equation of the model ( 1) can be written as Employing the method of integrating factor to equation ( 4) yields with µ B > α.As t → ∞ in equation ( 5), B c ≤ (π 1 +π 2 +π 3 )Λ µ(µ B −α) , this exist for µ B > α.Therefore, the feasible solution for the bacteria concentration enters the region, provided µ B > α.Thus, the feasible region for the model system (1) is given by D = D H × D B c provided µ B > α and this completes the proof.
In addition, with the non-negative parameters of system (1), it is sufficient to state that the solutions of the system of equations of the model (1) are non-negative.Therefore, it is rich enough to study the dynamics of the typhoid model (1) in this region D = D H × D B c whenever µ B > α.

Disease-Free Equilibrium (DFE) and Basic Reproduction
Number, R 0 For the computation of the Basic reproduction number, R 0 , we determine the disease-free equilibrium of the system (1) that is given as , 0, 0, 0, 0, 0 .
The basic reproduction number, R 0 , determines the transmission tendency of a disease.R 0 is mathematically defined as the matrix's spectral radius, FV −1 , where F = ∂F i (E 0 ) are the transmission and transition matrices derived at disease-free equilibrium (DFE), E 0 .Here, F i is the appearance rate of new infections in compartments i, while V i is the transfer of infections from one compartment i to another (see [29] for detail).Given the DFE, E 0 = Λ µ , 0, 0, 0, 0, 0, 0 , we have , By the definition of R 0 as the spectral radius of FV −1 , we have where with A, B > 0, for µ B > α and p < 1.
The first term of equation (7) represents the reproduction number contribution for human-to-human interaction with the transmission rate, β 1 .The second term of equation ( 7) is the reproduction number contribution of the human to contaminated environment interaction with the transmission rate, β 2 .
By the approach of Next-generation used for the computation of R 0 , we state the stability of the DFE theorem as follows.
Theorem 3.2 means that the typhoid infection can be eliminated in the population with time if R 0 < 1.Otherwise, it will remain in the population for R 0 > 1.

Global Sensitivity Analysis
Global sensitivity analysis (GSA) is examined in this subsection to determine the most sensitive parameters of R 0 as multiple points entry.It determines the behaviour and degree of each parameter of R 0 .Latin Hypercube Sampling (LHS) sampling-based method with Partial Rank Correlation Coefficient (PRCC) is used to analyse GSA by generating 1000 samples from a uniform distribution of each parameter range (see [30] for details of LHS and PRCC).The PRCCs for the parameters and their corresponding p-values are presented in Table 2.The signs (+ and -) of PRCC indicate the definite qualitative relationship between the parameters and output R 0 .The parameters with positive PRCC imply that increasing them increases the value of R 0 , increasing the disease's spread.At the same time, the parameters with PRCC negative values indicate that R 0 decreases whenever their values increase, showing a reduction in the disease transmission dynamics.The most significant parameters are those with |PRCC| ≥ 0.5.From Table 2, the parameters β 1 , β 2 , µ B and σ 3 with |PRCC| ≥ 0.5 are the most significant; increasing β 1 and β 2 tends to make typhoid infection worst in the population while increasing the parameters µ B and σ 3 reduces the transmission of typhoid bacteria in the community.
Furthermore, the combined effect of some important parameters is shown in Figure 2 as a 3D plot.Figure 2 shows that increasing the shedding rate (π 3 ) of the infected carrier individuals will increase the value of R 0 .Also, the higher the recovery rate (σ 3 ) of treatment individuals, the lower the value of R 0 .The implication is that the typhoid infection will reduce in the community if the treatment rate is increased and the shedding rate from unaware infected persons is reduced; this will happen when people undergo screening tests during a typhoid outbreak, or someone close is infected with typhoid infection.Therefore, the main strategy to curtail the spread of the bacteria disease is to reduce the number of asymptomatic infected individuals, as it has a more significant tendency to shoot up the basic reproduction number

Optimal Control Analysis
With the result of the sensitivity analysis, we formulate an optimal control model version of the system (1) given as follows: with initial conditions of the system (1).
Here, u 1 (t) is the sanitation and hygiene practice and awareness campaign control, u 2 (t), the sterilisation and disinfection control, u 3 (t) as the potency of antibiotics administered to typhoid patients and u 4 (t) is the screening control for infect carrier's humans.
The objective function to be minimised is given as where the coefficient associated with the infected state variables, A, B, C and D, and the control weight coefficients, m 1 , m 2 , m 3 , m 4 , are assumed positive.The quadratic form of the control variables, 4 i=1 m i u 2 i (t) in equation (10), is due to the nonlinearity of the cost of controls as used in the literature on optimal control of infectious diseases [10,31].
The objective functional goal is to minimise the number of infected humans, bacteria concentration in the environment and the cost of implementing them.Thus, the optimal controls, U is to find a way such that where The state and the control variables of equations ( 9) and ( 10) are non-negative, as established in Subsection (3.1) and the condition in equation ( 12); this implies that the set Φ 1 is closed, convex and exists.The optimal control exists by applying Corollary 4.1 of Pages 68-69 in [32] as implemented in [33].

The Optimal Control Problem Simulations
The optimality system is simulated numerically to illustrate the dynamics of the optimal control system with time.The fourth-order Runge-Kutta method, coded in MATLAB R2007b, is used for the numerical simulations (see [28] for the fourthorder Runge-Kutta method and its stability details).Table 1 is the parameter values used for the simulations while the initial conditions and weight coefficient values are as follows; The simulations are partitioned into four (4) possible cases according to control combinations; Case A (one control implementation) • Strategy 1 (u 1 ): sanitation and hygiene practice and awareness campaign (u 1 0, u 2 , u 3 , u 4 = 0), • Strategy 2 (u 2 ): sterilisation and disinfection of the contaminated environment (u 2 0, u 1 , u 3 , u 4 = 0), • Strategy 3 (u 3 ): the potency of antibiotics administered to typhoid patients (u 3 0, u 1 , u 2 , u 4 = 0), • Strategy 4 (u 4 ): screening control (u 4 0, u 1 , u 2 , u 3 = 0).
Case D (all controls combine implementation) • Strategy 15: (u 1234 )sanitation and hygiene practice and awareness campaign + sterilisation and disinfection of the contaminated environment + potency of antibiotics administered to typhoid patients + screening control (u 1 , u 2 , u 3 , u 4 0).

The Discussion of Optimal Control Problem Simulations
Figure 3 shows the simulation results when applying a single control.It shows that implementing Strategy 2 will prevent more susceptible humans from contracting the infection compared to other single strategies, followed by Strategy 1 (u 1 ), likewise in the infected humans and bacteria in the environment.To implement Strategy 2, u 2 must be maintained at 39% for about 145 days before declining to its lower bound at day 150 (see Figure 3d).
For the double control combinations in Figure 4, it is observed that the combined double strategies with sterilisation and disinfection of contaminated environment control (that is, u 12 , u 23 , u 24 ) prevented the susceptible humans from contracting the infection and also reduced the infected human population and bacteria concentration in the environment more than other double control implementations, which shows the importance of sterilisation and disinfection of contaminated environment on typhoid infection control.The best control combinations are Strategy 9 (u 24 ), sanitation and hygiene practice and awareness campaigns, and screening control.Figure 4d shows that Strategy 9 maintains a control profile at 58% for 147 days before declining to the lower bound.
According to the simulation in Figure 5, three controls combined strategies with sterilisation and disinfection of contaminated environment control (that is, u 124 , u 234 ) have more impact than the others (that is, u 134 , u 123 ).All the control combinations, u 124 , u 234 , should be maintained at their various percentages for almost 147 days before declining to achieve these results (See Figure 5d).
Figure 6(a-c) indicates that the combined implementation of all the controls (Strategy 15) significantly impacts the susceptible human population, infected human population and the bacteria in the environment than without control.Figure 6(d) shows the control profile for applying all four controls together and should be maintained at 75% for about 145 days before declining.We further determine which of these strategies is the most cost-effective to implement by conducting a cost-effective analysis of the strategies.

Cost-effectiveness analysis
Cost-effectiveness analysis is an analysis for finding the cost and economic health results of one or more control measures.It determines the most cost-effective control strategy to eliminate the disease at a reduced cost.We consider two approaches for cost-effectiveness analysis, namely, average cost-effectiveness ratio (ACER) and incremental cost-effectiveness ratio (ICER) [31,34].Also, we consider two situations, which are to determine the strategies that avert infected cases and prevent susceptible humans from contracting the infection.

Average cost-effectiveness ratio (ACER)
The average cost-effectiveness ratio (ACER) is defined as Total cost generated by the control Total number of infections averted with the control .
Here, the total cost generated by the control strategy is evaluated using the objective function in Equation ( 9).The strategy with the least ACER is the most cost-effective, while the strategy with the highest ACER is the least cost-effective, meaning it is costlier to implement.For the single control implementation (CASE A), Strategy 1 is the most cost-effective, followed by Strategy 4, Strategy 3 and then Strategy 2 for infected cases averted (see Table 3) and susceptible cases prevented (see Table 6).For Case B, the double control implementation, Strategy 6 is the most cost-effective strategy compared to other double combined control strategies for infected cases averted (see Table 4) and susceptible cases prevented (see Table 7).At the same time, for Case C, Strategy 13 is the most cost-effective strategy for infected cases averted (see Table 5) and susceptible cases prevented (see Table 8).

Incremental cost-effectiveness ratio (ICER)
Incremental cost-effectiveness ratio (ICER) is the changes between the costs and health benefits of any two intervention strategies competing for the same limited resources.In the ICER approach, two competing control intervention strategies are compared incrementally; one is compared with the following less effective alternative strategy [23].It is calculated using the following formula: Considering two strategies, p and q, as two control intervention strategies, then ICER is computed as Change in total costs in strategies p and q Change in control benefits in strategies p and q .
The difference in disease-averted costs, as well as the costs of screening, disinfection, and prevention, can be represented by the ICER numerator.The difference in health outcome, or the difference between the total number of infections avoided or the total number of susceptible cases avoided, is the denominator of the ICER.Put another way, it can be calculated as the difference between the susceptible population or the entire infectious population with and without control.The strategy with the highest ICER value is excluded from the computation of ICERs since it is the most expensive and ineffective to implement.The total infection averted is arranged in ascending order.

Calculation of ICERs for infected cases averted 4.5.1. ICER for single control implementation for infected cases averted
For Case A, the ICERs are calculated using Table 3     The results show that the ICER of Strategy 10 is higher than other strategies; hence, it is more expensive and less effective.Therefore, Strategy 10 is eliminated, and the only strategy left is Strategy 6.The implication is that Strategy 6 (combination of sanitation and hygiene practice and awareness campaign and potency of antibiotics administered to typhoid patients) is the most cost-effective strategy to contain the bacteria disease for double control implementation, as shown graphically in Figure 8, that is, cost expended on Strategy 6 produced high result by averting highest number of infection.

ICER for triple control implementation for infected cases averted
The ICER for Case C is calculated as follows using the details in Table 5;    12) >ICER(13) implies that Strategy 12 is more costly and less expensive.Therefore, Strategy 13, the combination of hygiene practice and awareness campaign, the potency of antibiotics administered to typhoid patients, and screening control, is the most cost-effective triple combined control strategy to fight the typhoid disease for Case C, and it is shown graphically in Figure 9, which shows that expenditure on Strategy 13 produced effective result by averting highest number of infection compare to others that have no significant infection averted.From these results, Strategy 13 has higher ICER than the two other strategies, which implies that Strategy 13 is strongly dominated, costlier and less effective.Hence, Strategy 13 is eliminated from the alternative control intervention strategies list.Therefore, the ICERs of the remaining two strategies are computed in increasing order of their total infections averted, and this is shown as follows:    in Figure 10, that is, implementing Strategy 1 will have more susceptible people prevented from contracting the typhoid disease with a less cost than when compared to other strategies such as Strategy 2 that will prevent more susceptible people at a high cost.This shows that Strategy 10 has a higher ICER value than other strategies, meaning it is strongly dominated, costlier and less effective.Therefore, Strategy 10 is eliminated from competing alternative control strategies.So, we evaluate the ICERs of the remaining two strategies in increasing order of their susceptible prevented, and this is shown as follows: ICER The results show that the ICER of Strategy 7 is higher than other strategies; hence, it is more expensive and less effective.Therefore, Strategy 7 is eliminated, and the only     Strategy 13 has greater ICER than Strategy 1, meaning that Strategy 13 is costlier and less effective when compared with Strategy 1.Therefore, Strategy 1, a sanitation and hygiene practice and awareness campaign, is the most cost-effective strategy and the overall best strategy that can be implemented to prevent susceptible populations from contracting bacterial disease.

Conclusion
A mathematical model for the dynamics of typhoid fever infection with treatment relapse of the limited clinical efficacy of antibiotics is investigated in this study.The basic reproduction number, R 0 , is derived and used to investigate the sensitivity of the model parameters via Latin hypercube sampling (LHS) with a partial rank correlation coefficient (PRCC) approach.The sensitivity analysis results indicate that R 0 decreases when the maximum treatment intake over time with high clinical efficacy of antibiotics, the recovery rate for treated individuals with a high potency of antibiotics, and the bacteria decay rate increases.Meanwhile, R 0 increases when the human-to-human contact rate, environment-to-human contact rate, relapse response to treatment, the growth rate for the bacteria, and the shedding rates for severe, mild and carrier-infected individuals increase, meaning disease invasion in the population.The sensitivity results for the parameters with PRCCs (≥ 0.5), which are the bacteria decay, the treated individuals' recovery rate, the human-to-human contact rate and the environment-to-human contact rate, have more impact on R 0 .
Furthermore, the optimal control model with four control measures is formulated and analysed based on the sensitivity analysis result.The controls are sanitation and hygiene practice and awareness campaign control, sterilisation and disinfection control, the potency of antibiotics administered to control and the screening control for carrier-infected humans.Also, cost-effectiveness analysis is computed to determine the most cost-effective optimal control strategy for both infected cases so that the bacteria can be eliminated and for susceptible humans so that they can be prevented from contracting the disease.In conclusion, the deduction from the study suggests that for both infected humans and susceptible humans, Strategy 1, which is the sanitation and hygiene practice and awareness campaign, is the most cost-effective strategy to contain the typhoid fever infection and prevent susceptible population from contracting the bacteria; this conforms with the work of [13].For applying double controls (Case B), Strategy 6, combining Strategy 1 and the potency of antibiotics administered, is the most cost-effective for containing the bacteria disease and preventing susceptible humans from contracting the disease.Also, for implementing triple controls (Case C), the result dictates that Strategy 13, which combines sanitation and hygiene practice and awareness campaign, the potency of antibiotics administered, and screening control is the most cost-effective for eradicating the bacteria disease and for preventing susceptible individuals.However, the overall computation of the cost-effectiveness among the most cost-effective control strategies to be considered from each case, including Case D (all the controls), indicates that Strategy 1 is the most cost-effective control intervention to control typhoid fever bacteria and to prevent susceptible humans from contracting the disease.
However, all the combined controls (Strategy 15) may be implemented to reduce the infected cases in the community, while Strategy 13 (sanitation and hygiene practice and awareness campaign, the potency of antibiotics administered to typhoid patients and screening control) may be implemented to prevent the susceptible from contracting the infection when the cost of implementation does not matter.The study has some limitations, such as the derivation endemic equilibrium state, the model parameters uncertainty, and the control interventions' efficacy.The limitations could be considered in future research.In addition, the model in this study can be extended by considering exposed individuals.

Figure 1 .
Figure 1.The systematic diagram for typhoid fever.The broken lines indicate the shedding of the bacteria into the environment.

Theorem 3 . 1 .
All feasible solutions of the model are uniformly bounded in a proper subset D = D H × D B c , where

Figure 2 .
Figure 2. The Effect of the shedding rate from infected carrier individuals, π 3 and recovery rate for treatment individuals, σ 3 , on the basic reproduction number, R 0 .

Figure 7 .
Figure 7. Plots displaying the comparison of infection averted cases for single control implementation (Case A).

Figure 8 .
Figure 8. Plots displaying the comparison of infection-averted cases for double control implementation (Case B).

Figure 9 .
Figure 9. Plots displaying the comparison of infection-averted cases for triple control implementation (Case C).

Figure 10 .
Figure 10.Plots displaying the comparison of susceptible cases prevented for single control implementation (Case A).

Figure 11 .
Figure 11.Plots displaying the comparison of susceptible cases prevented for double control implementation (Case B).

Figure 12 .
Figure 12.Plots displaying the comparison of susceptible cases prevented for triple control implementation (Case C).

Table 1 .
Description and parameter values of the model.

Table 3 .
Case A (Single Control implementation).

Table 6 .
The most Cost-Effective Strategy of each case in ascending order of Total Infections Averted.

Table 7 .
Case A (Single Control Implementation) for susceptible cases prevented.

Table 9 .
Case C (Triple Control implementation) for Susceptible cases prevented.
= 4.3414 × 10 7 − 3.9035 × 10 7 2.3912 × 10 7 − 2.1452 × 10 6 =0.2012..6.Calculation of ICERs for susceptible humans prevented 4.6.1.ICER for single control implementation susceptible human prevented For Case A, the ICERs are calculated using Table7as follows: ICER than Strategy 1, meaning that Strategy 15 is costlier and less effective.Hence, it is eliminated from the list of alternative control strategies.Therefore, Strategy 1, a hygiene practice and awareness campaign, is the most cost-effective strategy to contain typhoid bacteria.Furthermore, it is noteworthy that Strategy 6 is omitted, which is the combination of sanitation and hygiene practice and awareness campaign, and potency of antibiotics administered to typhoid patients since both Strategies 1 and 6 have the same total infections averted.However, with the cost-minimisation technique, Strategy 1 is the most cost-effective strategy.4 = 5.3383 × 10 8 − 2.7246 × 10 5 1.8734 × 10 6 − 5.5059 × 10 5 =403.3516.Since the ICER of Strategy 4 is higher than that of Strategy 1, it implies that Strategy 4 is costlier and less effective; Strategy 4 is thereby eliminated from the list of alternative control interventions.Hence, the remaining Strategy 1 (the sanitation and hygiene practice and awareness campaign) is the most cost-effective optimal control strategy in preventing the susceptible populations from contracting typhoid infection for single control implementation, Case A, and this is displayed

Table 10 .
The most Cost-Effective Strategy of each Case in ascending Order of Total susceptibility prevented.
effective combined control strategy to prevent the susceptible populations from contracting the bacteria disease for triple control implementation.It is displayed in Figure12, that is, Strategy 13 implementation will cost less in preventing more susceptible people from contracting the typhoid disease compared to other Strategies 11, 12 and 14 that have a high cost of implementation.