Approach of the value of a rent when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions Ratio Mathematica Volume 41, 2021, pp. 255-282 255 A dynamic model of typhoid fever with optimal control analysis Chinwendu Emilian Madubueze* Reuben Iortyer Gweryina† Kazeem Austin Tijani‡ Abstract In this study, a deterministic mathematical model of Typhoid fever dynamics with control strategies; vaccination, hygiene practice, sterilization and screening is studied. The model is first analyzed for stability in terms of the control reproduction number, Rc with constant controls. The disease-free equilibrium and endemic equilibrium of the model exist and is shown to be stable whenever Rc < 1 and Rc > 1, respectively. The model by investigation shows a forward bifurcation and the sensitivity analysis conducted revealed the most biological parameters to be targeted by policy health makers for curtailing the spread of the disease. The optimal control problem is obtained through application of Pontryagin maximum principle with respect to the above-mentioned control strategies. Simulations of the optimal control system and sensitivity of the constant control system confirms that hygiene practice with sterilization could be the best strategy in controlling the disease. Keywords: Typhoid fever; global stability; sensitivity analysis; optimal control; bifurcation. 2010 AMS subject classification§: 92B05, 34D23, 49K15, 37N25, 34D05. * Department of Mathematics, Joseph Sarwuan Tarka University, Makurdi, Nigeria; ce.madubueze@gmail.com. † Department of Mathematics, Joseph Sarwuan Tarka University, Makurdi, Nigeria; gweryina.reuben@uam.edu.ng. ‡ Department of Mathematics, Joseph Sarwuan Tarka University, Makurdi, Nigeria; kazeemtijani1987@yahoo.com. § Received on September 11, 2021. Accepted on December 21, 2021. Published on December 31, 2021. doi: 10.23755/rm.v41i0.657. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors.This paper is published under the CC-BY licence agreement. C. E. Madubueze, R.I. Gweryina, and K. A. Tijani 256 1. Introduction Typhoid fever is a life-threatening infection that is usually caused by Salmonella enteric serovar Typhi (S. Typhi) and Salmonella enteric serovar Paratyphi (S. Paratyphi, that is Paratyphi A, B, and, uncommonly is S. Paratyphi C) [1]. Typhoid fever has been a public health challenge globally. However, the disease is endemic in most developing countries in Africa and South-East Asia where potable clean water, sanitation and hygiene are either grossly inadequate or non-existent. The transmission of S. Typhi and S. Paratyphi occur through the consumption of contaminated food or water resulting from inadequate environmental sanitation and hygiene practices [2]. People that are clinically ill from typhoid fever and those who have recovered from it pass out the bacteria in their stools (carriers) and urine [3]. A chronic carrier sheds Salmonella Typhoid more than 12 months after onset of illness. Human beings are the only known reservoir of Typhoid and the mode of transmission happens through food and water contaminated by acutely ill or chronic carriers of the bacteria [4]. Vaccine can be taken to prevent Typhoid fever but does not provide long- term immunity [5]. On the other hand, educating travelers moving to typhoid endemic regions on the importance of sanitation and hygiene precautions as well as vaccination will help immensely to preventing the rapid spread of Typhoid disease [4]. Mathematical models of infectious diseases are used to test and compare various intervention strategies especially when there are limited resources [6]. In controlling Typhoid fever, several mathematical models have been formulated. For instance, Mushayabas [7] considered the impact of education campaigns and treatment on the dynamics of Typhoid fever and Abboubakar and Racke [8] carried out a human and bacteria model without considering hygiene practice and individuals protected through vaccination in the population, while Karunditu et al. [9], Peter et al. [10], Nyerere et al. [11], Peter et al. [12], Edward and Nyerere [13], Kgosimore and Kelatlehegile [14] and Aji et al. [15] considered only human population without factoring in the bacteria concentration in the contaminated food and or water. Tilalum et al. [16], Okolo and Abu [17], Peter et al. [18], Abboubakar and Racke [19] and Awoke [20] studied the optimal control of typhoid transmission with control measures. None of the aforementioned works studied the combined control measures such as vaccination, hygiene practice, screening of carriers and sterilization of the bacteria in the environment as autonomous or non-autonomous system of equations. This study will bridge these gaps and form a novel contribution to the existing body of knowledge on the subject matter. Peter et al. [10] forms the motivation of this work. They considered Protected, Susceptible, Infected, Treated and Recovery model without the bacteria concentration and the effect of screening of infected carriers and A dynamic model of typhoid fever with optimal control analysis 257 hygiene practice on transmission dynamics of Typhoid fever. They assumed that the protected class belongs to only individuals that have been vaccinated before entrance into Typhoid endemic population and also optimal control and numerical simulation were not considered in their work. Modifying the work of Peter et al. [10], we consider Protected, Susceptible, Infected individuals, Carriers, Recovery and Bacteria concentration model in which some susceptible individuals are protected through vaccination and population practices hygiene which reduces the transmission rate. Hygiene practices which include safe water, sanitation and personal hygiene are crucial in preventing and controlling the spread of Typhoid. In addition, the screening and treating of carriers who are silent spreaders of the disease due to their asymptomatic nature and the sterilization of bacteria concentration in the immediate environment are also important in elimination of typhoid fever in the population. This work will be the first to consider sterilization of the bacteria concentration as a control measure for typhoid fever. Also, the sensitivity analysis for the prediction of appropriate intervention strategies for the control of typhoid fever spread and the optimal control analysis are carried out in this work. Therefore, a modified version of the work of Peter et al. [10] is formulated in Section 2 and a comprehensive mathematical analysis of the model in Section 3. The sensitivity analysis and optimal control strategies of the Typhoid fever dynamics are considered in Section 4 while the numerical simulations and discussion are given in Section 5. Section 6 is the conclusion of the work. 2. Model description and formulation In this section, the work of Peter et al. [10] is modified by considering human population (infected carriers) as well as bacteria concentration. The human population at any time, 𝑡 is subdivided into five subpopulations namely; protected population, 𝑃(𝑡), susceptible population, 𝑆(𝑡), infected population, 𝐼(𝑡), carrier population, 𝐼𝑐(𝑡) and recovered individuals, 𝑅(𝑡). The bacteria concentration is represented by 𝐵𝑐(𝑡). In this study, the protected population, 𝑃(𝑡), are susceptible individuals that are vaccinated and individuals coming in from the population that is not at risk of Typhoid fever into typhoid fever endemic population. Infected population, 𝐼(𝑡), are infected individuals that are showing symptoms of the disease and are capable of spreading the bacteria in the environment while carrier population, 𝐼𝑐(𝑡), represents asymptomatic infected individuals that are treated but still carrying the Salmonella Typhi. Recovered individuals, 𝑅(𝑡), are individuals who have recovered from the disease by treatment or natural immunity. The protected population, 𝑃(𝑡), of the proportion, 𝛼 ∈ (0,1) is increased by birth or immigration at a rate, 𝛬, and also from susceptible individuals that are protected through vaccination at a rate, ƞ. The protected population loses C. E. Madubueze, R.I. Gweryina, and K. A. Tijani 258 immunity when the vaccine wanes at a rate, 𝛾. The susceptible population is increased at a rate, (1 − 𝛼)𝛬, of the unprotected population through birth or immigration and also from recovered population,𝑅(𝑡), after losing their temporary immunity at a rate, ɸ. Susceptible population contract typhoid disease through food, water or environment contaminated by Salmonella bacteria as a result of inadequate hygiene practice measure at a rate, (1 − 𝑝)𝜆 and progress to infected population. Here, λ = 𝛽𝐵𝑐 𝐾+𝐵𝑐 is the force of infection, 𝛽 is the ingestion or consumption rate of the contaminated food, water or environment, 𝐾 is the carrying capacity of the bacteria in food, water or environment and 𝑝 ∈ (0,1) is the hygiene practice control measure. Infected individuals progress to carrier class at a rate, 𝜎 while some infected individuals recovered fully by treatment at rate 𝜏1 or they die of the disease (bacteria) at a rate, d. Carrier class, 𝐼𝑐(𝑡), recovered by natural immunity at a rate, 𝜏2 or by early treatment when they are screened at a rate, Ψ with 𝜃 as the treatment rate. The natural death rate, 𝜇 is assumed for all the human population. For the bacteria concentration, 𝐵𝑐(𝑡), in the environment, they increased through the shedding from Carriers and symptomatic population, 𝐼𝑐(𝑡) and 𝐼(𝑡) at the rates, 𝜋1 and 𝜋2 respectively. The shedding rates, 𝜋1 and 𝜋2 are reduced by 𝑝, the level of hygiene practice the infected populations, 𝐼𝑐(𝑡) and 𝐼(𝑡), observed. The bacteria decays in the environment at a rate, 𝜇Bc. We assume that there is no human to human transmission but rather human aids in shedding the bacteria in the environment or contaminating the environment; neither there is immigration of infectious humans. Also, disease induced death does not occur in carrier class since they are asymptomatic, that is before the bacteria can cause death, it must have progressed to symptomatic stage. The systematic diagram of model is given in Figure 1. A dynamic model of typhoid fever with optimal control analysis 259 Figure 1. The systematic diagram for Typhoid fever model. The system of differential equation is derived using the Figure 1 as follows. 𝑑𝑃 𝑑𝑡 = αɅ + ƞS – (γ + µ)P 𝑑𝑆 𝑑𝑡 = (1 − α)Ʌ + γP + ɸR – (ƞ + µ + (1 − p)λ)S 𝑑𝐼 𝑑𝑡 = (1 − p)λS – (σ + 𝜏1 + µ + 𝑑)I 𝑑𝐼𝑐 𝑑𝑡 = σ𝐼 – (𝜏2 + Ѱ𝜃 + 𝜇)𝐼𝐶 𝑑𝑅 𝑑𝑡 = 𝜏1I + (𝜏2 + Ѱ𝜃)𝐼𝐶 – (µ + ɸ)R 𝑑𝐵𝑐 𝑑𝑡 = 𝜋2(1 − 𝑝)I + 𝜋1(1 − 𝑝)𝐼𝐶 − µ𝐵𝐵𝑐 } (1) with initial conditions, 𝑃(0) > 0, 𝑆(0) > 0, 𝐼𝐶(0) ≥ 0,I(0) ≥ 0, 𝑅(0) ≥ 0, 𝐵𝐶(0) ≥ 0, where λ= 𝛽𝐵𝑐 (𝐾+𝐵𝑐) and the model parameters are assumed to be nonnegative. C. E. Madubueze, R.I. Gweryina, and K. A. Tijani 260 3. Mathematical Analysis of the Model 3.1 Invariant Region Invariant region is a region where the model solutions are uniformly bounded. Theorem 1. All feasible solutions of the model are uniformly bounded in a proper subset 𝐷 = 𝐷𝐻 𝑋 𝐷𝐵𝑐, where DH = {(P,S, I, 𝐼𝐶,R) ∈ Ɍ+ 5 : N(t) ≤ Ʌ µ } is a subset for human population and 𝐷𝐵𝑐 = {𝐵𝑐 ∈ ℝ+: 𝐵𝑐 ≤ [(𝜋2 + 𝜋1)(1−𝑝)]Ʌ µµ𝐵 } is a subset for bacteria concentration in environment. Proof. The total human population, 𝑁(𝑡) is given by 𝑁= 𝑃 + 𝑆 + 𝐼 + 𝐼𝑐 + 𝑅 with initial conditions 𝑁(0) = 𝑁0 and 𝐵𝑐(0) = 𝐵𝑐0 for the bacteria in the environment. This implies that from equation (1) that 𝑑𝑁 𝑑𝑡 = 𝛬 − 𝜇𝑁 − 𝑑𝐼. In the absence of disease-induced death rate, that is, 𝑑 = 0, we have 𝑑𝑁𝐻 𝑑𝑡 ≤ 𝛬 − 𝜇𝑁 which by method of integrating factor and the initial condition, 𝑁(0) = 𝑁0 gives 𝑁(𝑡) ≤ Ʌ µ + (𝑁0 − Ʌ µ )𝑒−µ𝑡. (2) As 𝑡 → ∞ in equation (2), we have 𝑁(𝑡) ≤ Ʌ µ . This means that the feasible solutions of the model for the human population are in the region, DH = {(P,S, I, 𝐼𝐶,R) ∈ Ɍ+ 5 : N(t) ≤ Ʌ µ }. For bacteria concentration since 𝑁(𝑡) ≤ Ʌ µ , it means that 𝐼 ≤ Ʌ µ and 𝐼𝐶 ≤ Ʌ µ , we have from the last equation of (1) that 𝑑𝐵𝑐 𝑑𝑡 = 𝜋2(1 − 𝑝)𝑁 + 𝜋1(1 − 𝑝)𝑁 − µ𝐵𝐵𝑐 ≤ 𝜋2(1 − 𝑝) Ʌ µ + 𝜋1(1 − 𝑝) Ʌ µ − µ𝐵𝐵𝑐 . (3) Solving equation (3) with 𝐵𝑐(0) = 𝐵𝑐0 as the initial condition yields 𝐵𝑐 ≤ (𝜋2 +𝜋1)(1−𝑝)Ʌ µµ𝐵 + (𝐵𝑐0 − (𝜋2 +𝜋1)(1−𝑝)Ʌ µµ𝐵 )𝑒−µ𝐵𝑡. (4) A dynamic model of typhoid fever with optimal control analysis 261 As t → ∞ in equation (4), we have 𝐵𝑐 ≤ (𝜋2 +𝜋1)(1−𝑝)Ʌ µµ𝐵 . Therefore, the feasible solution of the bacterial population enters the region 𝐷𝐵𝑐 = {𝐵𝑐 ∈ ℝ+: 𝐵𝑐 ≤ [(𝜋2 + 𝜋1)(1−𝑝)]Ʌ µµ𝐵 }. This completes the proof. Theorem 1 implies that the model is well posed mathematically and epidemiologically. Therefore, it is sufficient enough to study the dynamics of the model (1) in the region 𝐷 = 𝐷𝐻 × 𝐷𝐵𝑐. 3.2 Positivity of the Solutions Theorem 2. Let D = {𝑃,𝑆,𝐼, 𝐼𝐶,𝑅, 𝐵𝑐} ∈ ℝ+ 6 be solution set such that 𝑃(0) = 𝑃0, 𝑆(0) = 𝑆0, 𝐼𝐶(0) = 𝐼𝐶0, I(0) = 𝐼0, 𝑅(0) = 𝑅0 and 𝐵𝐶(0) = 𝐵𝐶0 are positive, then the elements of the solution set 𝐷 are all positive for 𝑡 ≥ 0. Proof. From the first equation of the model equations (1), we have 𝑑𝑃 𝑑𝑡 = 𝛼Λ + 𝜂𝑆 − (𝛾 + µ)P ≥ −(𝛾 + µ)P. (5) Integrating equation (5) with initial conditions 𝑃(0) = 𝑃0 yields 𝑃(𝑡) ≥ 𝑃0𝑒 – (γ + µ)t ≥ 0 . In a similar way, the rest of the equations of the model equation (1) with initial conditions, 𝑆(0) = 𝑆0, 𝐼(0) = 𝐼0𝐼𝐶 (0) = 𝐼𝑐0,𝑅(0) = 𝑅0 and 𝐵𝑐(0) = 𝐵𝑐0 give 𝑆(𝑡) ≥ 𝑆0exp(∫ – (ƞ + µ + (1 − 𝑝)λ) 𝑡 0 )𝑑𝑢 ≥ 0, 𝐼(𝑡) ≥ 𝐼0exp{−(σ + 𝜏1 + µ + 𝑑)t} ≥ 0, 𝐼𝐶 (𝑡) ≥ 𝐼𝑐0exp{− (𝜏2 + Ѱ𝜃 + 𝜇)𝑡} ≥ 0, 𝑅(𝑡) ≥ 𝑅0exp{–(µ + ɸ)t} ≥0, 𝐵𝑐(𝑡) ≥ 𝐵𝑐0exp(− µ𝐵𝑡) ≥ 0. Therefore, the solution set {𝑃(𝑡), 𝑆(𝑡), 𝐼(𝑡),𝐼𝑐(𝑡), 𝑅(𝑡), 𝐵𝑐(𝑡)}, of the system (1) is positive for all 𝑡 ≥ 0 since exponential functions and their initial conditions are positive. 3.3 Disease-free equilibrium point and Control Reproduction Number We compute the control reproduction number,𝑅𝑐, which is define as the average number of secondary cases reproduced when an infected person is introduced into a population where control measures like vaccination, screening, sanitation and hygiene are in place. C. E. Madubueze, R.I. Gweryina, and K. A. Tijani 262 In obtaining this, we apply the next-generation matrix approach [21] at the disease-free disease (DFE). The disease-free equilibrium (DFE) is obtained by equating the right hand side of the equation (1) to zero and solve simultaneously for the disease-free equilibrium, 𝐸0 = (𝑃 0,𝑆0, 𝐼0, 𝐼𝐶 0,𝑅0,𝐵𝐶 0). We have DFE, 𝐸0 = ( Ʌ(𝛼µ+ƞ) µ(𝛾+ƞ+µ) , Ʌ(𝛾+𝜇(1−𝛼)) µ(𝛾+ƞ+µ) ,0,0,0,0). By the principle of next-generation matrix approach, we have 𝐹 = ( 0 0 a𝛽𝑆0 𝐾 0 0 0 0 0 0 ), 𝑉 = ( k3 0 0 −𝜎 k4 0 − 𝜋2(1 − 𝑝) −𝜋1(1 − 𝑝) µ𝐵 ), (6) where 𝑎 = (1 − p), 𝑘1 = (γ + µ),𝑘2 = (ƞ + µ), 𝑘3 = (σ + 𝜏1 + µ + 𝑑), 𝑘4 = (𝜏2 + Ѱ𝜃 + 𝜇), 𝑘5 = (µ + ɸ). (7) Solving for the maximum eigenvalue of the matrix, 𝐹𝑉−1, we have 𝑅𝑐 = 𝑎𝛽S0[(𝜎𝜋1+𝜋2𝐾4)(1−𝑝)] 𝐾µ𝐵𝐾3𝐾4 . (8) With the definition of equation (7), we have 𝑅𝑐 = 𝛽𝛬(1−p)(𝛾+(1−𝛼)𝜇)[(𝜎𝜋1+𝜋2(𝜏2+ Ѱ𝜃+𝜇))(1−𝑝)] 𝜇𝐾𝜇𝐵(𝛾+ƞ+µ)(σ+𝜏1+ µ+𝑑)(𝜏2+ Ѱ𝜃+𝜇) . (9) The control reproduction number, 𝑅𝑐, can be written as 𝑅𝑐=𝑅𝐼 + 𝑅𝐼𝐶 , (10) where 𝑅𝐼 = 𝛽(1−p)2𝜋2𝑆0 𝐾µ𝐵(σ+𝜏1+ µ+𝑑) , 𝑅𝐼𝑐 = 𝛽(1−p)2𝜎𝜋1𝑆0 𝐾µ𝐵(𝜏2 + Ѱ𝜃+𝜇)(σ+𝜏1+ µ+𝑑) (11) denote the reproduction numbers which the infected population and carrier population contributed respectively through their shedding in the environment. 3.4 Local stability of the disease-free equilibrium, 𝑬𝟎 A dynamic model of typhoid fever with optimal control analysis 263 Theorem 3. If 𝐸0 is the DFE of the model, then 𝐸0 is locally asymptomatically stable if 𝑅𝑐 < 1, otherwise it is unstable if 𝑅𝑐 > 1. Proof. In proving this theorem, the Jacobian matrix of equation (1) at the disease-free equilibrium, 𝐸0 is given as 𝐽(𝐸0) = ( –k1 ƞ 0 0 0 0 γ −k2 0 0 ɸ − a𝛽𝑆0 𝐾 0 0 – k3 0 0 a𝛽𝑆0 𝐾 0 0 σ – k4 0 0 0 0 𝜏1 (𝜏2 + Ѱ𝜃) – k5 0 0 0 𝜋2(1 − 𝑝) 𝜋1(1 − 𝑝) 0 −µ𝐵 ) . (12) The eigenvalues of the Jacobian matrix (12) are −k5 and the solutions of the polynomial 𝜆5 + 𝐴𝜆4 + 𝐵𝜆3 + 𝐶𝜆2 + 𝐷𝜆 + 𝐸 = 0 (13) where A = k1 + k2 + k3 + k4 + μB, 𝐵 = (k1 + k2 )(k3 + k4 + μB) + k4(k3 + μB) + 𝜇(k2 + γ) + k3μB(1 − 𝑅𝐼), 𝐶 = 𝜇(k4 + μB)(k2 + γ) + k1k2k3 + k3μB(k1 + k2 )(1 − 𝑅𝐼) + k3k4μB(1 − 𝑅𝐶) + k4(k1 + k2 )(k3 + μB), 𝐷 = 𝜇k3μB(k2 + γ)(1 − 𝑅𝐼) + 𝜇k4(k3 + μB)(k2 + γ) + k3k4μB(k1 + k2 )(1 − 𝑅𝐶), 𝐸 = 𝜇(k2 + γ)k3k4μB(1 − 𝑅𝐶). Using the theorem in Heffernan et al. [22], the roots of the polynomial (13) have negative real part if 𝐴,𝐵,𝐶,𝐷,𝐸 > 0. With the definition of 𝑅𝑐 in equation (10), we have 𝐴,𝐵,𝐶,𝐷,𝐸 > 0 if 𝑅𝑐 < 1. Therefore, the Jacobian Matrix (12) has negative real eigenvalues if 𝑅𝐶 < 1. Hence, the disease-free equilibrium, 𝐸0, is locally asymptotically stable if 𝑅𝐶 < 1. This ends the proof. 3.5 Global stability of disease-free equilibrium Theorem 4. The disease-free equilibrium, 𝐸0, is globally asymptotically stable if 𝑅𝑐 < 1. Proof. We construct a Lyapunov function using the infected classes only and this is given by C. E. Madubueze, R.I. Gweryina, and K. A. Tijani 264 𝐿 = [𝜎𝜋1(1−𝑝)+𝜋2(1−𝑝)k4] µ𝐵k4k3 𝐼 + 𝜋1(1−𝑝) µ𝐵k4 𝐼𝐶 + 1 µ𝐵 𝐵𝑐 . (14) Differentiating (14) with respect to time, 𝑡, along the solutions of the model (1) gives 𝑑𝐿 𝑑𝑡 = ( 𝜎𝜋1(1−𝑝)+𝜋2(1−𝑝)k4 µ𝐵k4k3 )( 𝑎𝛽𝐵𝑐 (𝐾+𝐵𝑐) S – k3I) + 𝜋1(1−𝑝) µ𝐵k4 (σ𝐼 – k4𝐼𝐶) + 1 µ𝐵 (𝜋2(1 − 𝑝)I + 𝜋1(1 − 𝑝)𝐼𝐶 − µ𝐵𝐵𝑐). (15) Expanding and simplifying (15) yields 𝑑𝐿 𝑑𝑡 = ( 𝑎𝛽𝐵𝑐 (𝐾+𝐵𝑐) ( 𝜎𝜋1(1−𝑝)+𝜋2(1−𝑝)k4 µ𝐵k4k3 )𝑆 − 1)𝐵𝑐 = ( 𝑅𝑐𝐾𝑆 (𝐾+𝐵𝑐) 𝑆0 − 1)𝐵𝑐 . Since 𝑆 ≤ 𝑆0 and 𝐾 (𝐾+𝐵𝑐) ≤ 1, we have 𝑑𝐿 𝑑𝑡 ≤ 𝐵𝑐(𝑅𝑐 − 1). Clearly, 𝑑𝐿 𝑑𝑡 ≤ 0 if 𝑅𝑐 ≤ 1. If 𝐵𝑐 = 0, 𝑑𝐿 𝑑𝑡 = 0. By virtue of LaSalle’s Invariance Principle, the disease-free equilibrium,𝐸0, is globally asymptotically stable (GAS) whenever 𝑅𝑐 < 1. 3.6 Endemic Equilibrium State The Endemic equilibrium state 𝐸∗ is a state where the disease is present in the population. At 𝑑𝑃 𝑑𝑡 = 𝑑𝑆 𝑑𝑡 = 𝑑𝐼 𝑑𝑡 = 𝑑𝐼𝑐 𝑑𝑡 = 𝑑𝑅 𝑑𝑡 = 𝑑𝐵𝑐 𝑑𝑡 = 0 , we obtain after solving simultaneously that 𝐼∗ = k4𝑘5(Rc−1) 𝐵 , 𝐼𝑐 ∗ = 𝜎𝑘5(Rc−1) B , 𝑅∗ = (𝜏1k4+𝜎 (𝜏2 +Ѱ𝜃))(Rc−1) B , 𝐵𝑐 ∗ = [𝜋2(1−𝑝)k4+𝜎𝜋1(1−𝑝)]𝑘5(Rc−1) B𝜇𝐵 , 𝑆∗ = 𝛬(k1−αμ)𝐵+k1[𝜎(𝜏2 +Ѱ𝜃)ɸ+ɸk4τ1−k3k4k5](Rc−1) 𝐵(k1k2−ƞ𝛄) , 𝑃∗ = 𝛼𝛬(B(k1k2−ƞ𝛄))+ƞ[𝛬(k1−αμ)𝐵+k1[𝜎(𝜏2 +Ѱ𝜃)ɸ+ɸk4τ1−k3k4k5](Rc−1)] k1B(k1k2−ƞ𝛄) . Then, 𝑃∗,𝑆∗, 𝐼∗, 𝐼𝑐 ∗,𝑅∗, 𝐵𝑐 ∗ are all positive if and only if Rc > 1, which established that the endemic equilibrium state, 𝐸∗ = (𝑃∗,𝑆∗, 𝐼∗, 𝐼𝑐 ∗,𝑅∗, 𝐵𝑐 ∗) exists for Rc > 1. A dynamic model of typhoid fever with optimal control analysis 265 3.7 Global Stability of Endemic Equilibrium The global stability of endemic equilibrium, 𝐸∗, is established in the absence of disease induced death. Theorem 4.The endemic equilibrium, 𝐸∗, is globally asymptotically stable if Rc > 1 and 𝑑 = 0. Proof. We construct the Lyapunov function given by 𝐿 = 1 2 [(𝑃 − 𝑃∗ ) + (𝑆 − 𝑆∗ ) + (𝐼 − 𝐼∗ ) + (𝐼𝑐 − 𝐼𝑐 ∗) + (𝑅 − 𝑅∗ )]2 + (𝐵𝑐 − 𝐵𝑐 ∗ − 𝐵𝑐 ∗ ln 𝐵𝑐 𝐵𝑐 ∗). Taking the derivative of 𝐿 along the solutions of equation (1) yields 𝐿′ = [(𝑃 − 𝑃∗) + (𝑆 − 𝑆∗) + (𝐼 − 𝐼∗) + (𝐼𝑐 − 𝐼𝑐 ∗) + (𝑅 − 𝑅∗)] 𝑑 𝑑𝑡 (𝑃 + 𝑆 + 𝐼 + 𝐼𝑐 + 𝑅) +(1 − 𝐵𝑐 ∗ 𝐵𝑐 ) 𝑑𝐵𝑐 𝑑𝑡 ̇ which upon substitution gives 𝐿′ = [(𝑃 − 𝑃∗) + (𝑆 − 𝑆∗) + (𝐼 − 𝐼∗) + (𝐼𝑐 − 𝐼𝑐 ∗) + (𝑅 − 𝑅∗)](𝛬 −𝜇(𝑃 + 𝑆 + 𝐼 + 𝐼𝑐 + 𝑅) − 𝑑𝐼) +(1 − 𝐵𝑐 ∗ 𝐵𝑐 )(𝜋2(1 − 𝑝)I + 𝜋1(1 − 𝑝)𝐼𝐶 − µ𝐵𝐵𝑐). ̇ (16) Substituting at endemic equilibrium, 𝛬 = 𝜇(𝑃∗ + 𝑆∗ + 𝐼∗ + 𝐼𝑐 ∗ + 𝑅∗) + 𝑑𝐼∗, µ 𝐵 = 𝜋2(1−𝑝)𝐼 ∗ 𝐵𝑐 ∗ + 𝜋1(1−𝑝)𝐼𝑐 ∗ 𝐵𝑐 ∗ in equation (16) and simplify, we have 𝐿′ = −𝜇[(𝑃 − 𝑃∗) + (𝑆 − 𝑆∗) + (𝐼 − 𝐼∗) + (𝐼𝑐 − 𝐼𝑐 ∗) + (𝑅 − 𝑅∗)]2 + 𝜋2(1 − 𝑝)𝐼∗ [1 + 𝐼 𝐼∗ − 𝐵𝑐 𝐵𝑐 ∗ − 𝐵𝑐 ∗𝐼 𝐼∗𝐵𝑐 ] + 𝜋1(1 − 𝑝)𝐼𝑐 ∗ [1 + 𝐼𝑐 𝐼𝑐 ∗ − 𝐵𝑐 𝐵𝑐 ∗ − 𝐵𝑐 ∗𝐼 𝐼∗𝐵𝑐 ] − 𝑑(𝐼 − 𝐼∗)[(𝑃 − 𝑃 ∗) + (𝑆 − 𝑆∗) + (𝐼 − 𝐼∗) + (𝐼𝑐 − 𝐼𝑐 ∗) + (𝑅 − 𝑅∗)]. Using the hypothesis that 𝑑 = 0, we have 𝐿′ = −𝜇[(𝑃 − 𝑃∗) + (𝑆 − 𝑆∗) + (𝐼 − 𝐼∗) + (𝐼𝑐 − 𝐼𝑐 ∗) + (𝑅 − 𝑅∗)]2 + (𝜋2(1 − 𝑝)𝐼 ∗ + 𝜋1(1 − 𝑝)𝐼𝑐 ∗)[2 − 𝐵𝑐 𝐵𝑐 ∗ − 𝐵𝑐 ∗ 𝐵𝑐 ] with 𝐼𝑐 𝐼𝑐 ∗ ≤ 1, 𝐼 𝐼∗ ≤ 1. This implies that 𝐿′ ≤ 0 since 2 − 𝐵𝑐 𝐵𝑐 ∗ − 𝐵𝑐 ∗ 𝐵𝑐 ≤ 0, by arithmetic and geometric theorem and 𝐿 = 0 if 𝑃 = 𝑃∗ , 𝑆 = 𝑆∗ , 𝐼 = 𝐼∗ , 𝐼𝑐 = C. E. Madubueze, R.I. Gweryina, and K. A. Tijani 266 𝐼𝑐 ∗ , 𝑅 = 𝑅∗ and 𝐵𝑐 = 𝐵𝑐 ∗. This means that the endemic equilibrium, 𝐸∗, is globally asymptotically stable (GAS) whenever 𝑅𝑐 > 1 and 𝑑 = 0 according to LaSalle’s Invariance Principle. 3.8 Local Stability of Endemic Equilibrium State Due to the mathematical complexity of the stability of endemic equilibrium, the Centre manifold theory approach is used to establish the local stability of endemic equilibrium by proving the existence of a forward bifurcation of the system. A forward bifurcation means that the endemic equilibrium is locally asymptotically stable if 𝑅𝑐 > 1 but near unity. Theorem 5. The endemic equilibrium is locally asymptotically stable whenever if 𝑅𝑐 > 1 but near unity. Proof. Using the approach of Centre manifold theory by Castillo-Chavez and Song [23], let 𝛽 = 𝛽∗ be a bifurcation parameter at 𝑅𝑐 = 1 so that 𝛽 = 𝛽∗ = 𝐾µ𝐵𝑘3𝑘4 𝑎S0[(𝜎𝜋1+𝜋2𝑘4)(1−𝑝)] . This implies that the Jacobian matrix of equation (12) has negative eigenvalues and a simple zero eigenvalue. The left and right eigenvectors associated with the Jacobian matrix (12) are 𝑤 = (𝑤1, 𝑤2, 𝑤3, 𝑤4, 𝑤5, 𝑤6) and 𝑣 = (𝑣1, 𝑣2, 𝑣3, 𝑣4, 𝑣5, 𝑣6) respectively where 𝑤1 = ƞ(k3k4k5−ɸ𝜏1k4−𝜎ɸ(𝜏2 + Ѱ𝜃))𝑤3 k4k5(ƞ𝛾−𝑘1k2) ,𝑤2 = 𝑘1(k3k4k5−ɸ𝜏1k4−𝜎ɸ(𝜏2 + Ѱ𝜃))𝑤3 k4k5(ƞ𝛾−𝑘1k2) ,𝑤4 = 𝜎𝑤3 k4 ,𝑤5 = (𝜏1k4+σ(𝜏2 + Ѱ𝜃) )𝑤3 k4k5 ,𝑤6 = k3𝑤3 𝑐 ,𝑣1 = 𝑣2 = 𝑣5 = 0,𝑣4 = 𝜋1(1−𝑝)c𝑣3 k4µ𝐵 ,𝑣6 = 𝑐𝑣3 µ𝐵 ,𝑤3,𝑣3 > 0,𝑐 = a𝛽𝑆0 𝐾 . Representing the state variables 𝑃 = 𝑥1, 𝑆 = 𝑥2, 𝐼 = 𝑥3, 𝐼𝑐 = 𝑥4, 𝑅 = 𝑥5, 𝐵𝑐 = 𝑥6 so that the system (1) becomes 𝑑𝑋 𝑑𝑡 = 𝐹 = (𝑓1,𝑓2,𝑓3,𝑓4,𝑓5,𝑓6) 𝑇 with 𝑓𝑖 = 𝑓𝑖(𝑥1,𝑥2,𝑥3,𝑥4,𝑥5,𝑥6), we have the non-zero second order partial derivatives at 𝐸0 given as 𝜕2𝑓3(𝐸0) 𝜕𝑥2𝜕𝑥6 = (1−𝑝)𝛽∗ 𝑘 , 𝜕2𝑓3(𝐸0) 𝜕𝑥6 2 = − 2(1−𝑝)𝛽∗𝑆0 𝑘2 , 𝜕2𝑓6(𝐸0) 𝜕𝑥6𝜕𝛽 ∗ = (1−𝑝)𝑆0 𝑘 . We compute the coefficients, 𝑚 and 𝑛 as follows 𝑚 = 𝑣3 (𝑤2𝑤6 𝜕2𝑓3 𝜕𝑥2𝜕𝑥6 + 𝑤6 2 𝜕 2𝑓3 𝜕𝑥6 2) and 𝑛 = 𝑤3𝑣3 𝜕2𝑓3 𝜕𝑥3𝜕𝛽 ∗ > 0. Upon substituting, we have 𝑚 = − k3(1−𝑝)𝛽 ∗𝑣3𝑤3 2 𝑐𝑘 ( 𝑘1(k3k4k5−ɸ𝜏1k4−𝜎ɸ(𝜏2 + Ѱ𝜃)) k4k5(𝑘1k2−ƞ𝛾) + 2k3𝑆0 𝑐𝑘 ) < 0, A dynamic model of typhoid fever with optimal control analysis 267 𝑛 = 𝑤3k3𝑣3(1−𝑝)𝑆0 𝑐𝑘 > 0 , which implies that a forward bifurcation exists. Thus, the endemic equilibrium is locally asymptotically stable if 𝑅𝑐 > 1 but near unity. This is shown graphically in Figure 2. Figure 2. Forward bifurcation for typhoid model. 4. Sensitivity Analysis and Optimal Control Analysis 4.1 Sensitivity Analysis of the Model Parameters Sensitivity analysis is used to examine the connection between uncertain parameters of a mathematical model and a property of the observable output [24]. It is used to determine the model parameters that have a great impact on reproduction number, 𝑅𝑐 for the purpose of targeting such by intervention strategies [25]. In carrying out the sensitivity analysis, we adopted normalized forward index method as used by Rodrigues et al. [25] and this is given by 𝑆𝑌 𝑅𝑐 = 𝜕𝑅𝑐 𝜕𝑌 × 𝑌 𝑅𝑐 , where 𝑌 is the parameters reflecting in the control reproduction number, 𝑅𝑐. The sensitivity indices of 𝑅𝑐 are given in Table 2 using the parameter values in Table 1. 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0 2 4 6 8 10 12 14 16 18 20 R c In fe c te d I n d iv id u a ls , I* ( t) Stable DFE Stable EE Unstable DFE C. E. Madubueze, R.I. Gweryina, and K. A. Tijani 268 Table 1. Parameter values of the model with their sources Table 2. Sensitivity index of the parameters values Parameter Index sign Index sign Sensitivity index values Parameter Index sign Sensitivity index values 𝛾 + 0.66520217 𝑑 − 0.53789731 𝛼 − 0.03617419 π1 + 0.161676646 𝜎 − 0.08282213 π2 + 0.838323354 𝜏1 − 0.01629918 μB − 0.999999999 𝜏2 − 0.00027716 Ѱ − 0.138579983 𝑝 − 1.99999999 𝜃 − 0.138579983 Ƞ − 0.67891735 𝛽 + 1.00000000 From Table 2, the parameters with positive indices (𝜋1,𝜋2,𝛾 ) indicate that they have impact on expanding the disease in the population if their values are increasing because the control reproduction number increases as their values increase. Also, the parameters in which their sensitivity indices are negative have influence in reducing the burden of the bacteria in the population as their values increase because the control reproduction number decreases as their values increase, which will lead to reducing the endemicity of the bacteria in the population. Parameter Value Source Parameter Value Source Λ 100 [16] 𝜇 0.0247 [16] 𝛾 0.33 [26] 𝑑 0.066 [30] 𝛼 0.5 [27] π1 0.9 [30] ɸ 0.000904 [28] π2 0.8 [16] 𝜎 0.03-0.05 [28] μB 0.0345 [31] 𝜏1 0.002 [16] Ѱ 0.75 [11] 𝜏2 0.0003 [16] 𝜃 0.2 [16] p Ƞ 0.3 0.75 Assumed [11] 𝛽 𝐾 0.9 500000 [16] [29] A dynamic model of typhoid fever with optimal control analysis 269 (a) (b) (c) (d) Figure 3. Simulations for the impact of model parameters on control reproduction number. According to the phase plane (Figure 3a), the value of 𝑅𝑐 decreases drastically as 𝜌 and 𝜇𝐵 increases. Also, the value of 𝑅𝑐 decreases sharply in Figure 3b as 𝜌 increases, but the change of 𝜂 has a significantly lower impact on 𝑅𝑐. The phase planes in Figures 3c and 3d illustrate similar results, which shows that 𝑅𝑐 is much sensitive to 𝜇𝐵 than to 𝜂 and 𝜓, respectively. Therefore, from all cases, 𝜇𝐵 has shown to be a superior force in reducing the burden of Typhoid fever. However, the combination of 𝜌 and 𝜇𝐵 has proven to be the best control strategy as compared to the rest. 4.2 Optimal Control Analysis The optimal control model is formulated from system (1) when the constant parameters, ƞ,p,Ψ,µ𝐵 are time dependent that is ƞ(t), p(t), Ѱ(t) and µ𝐵(t) C. E. Madubueze, R.I. Gweryina, and K. A. Tijani 270 where ƞ(t) is vaccination control, p(t) is hygiene practice control, Ѱ(t) is the screening control and µ𝐵(t) is the sterilization control. The objective function to be minimized is given as 𝐽(ƞ(t), 𝑝(t), Ѱ(t), µ𝐵(t)) = ∫ (𝐼 + 𝐼𝐶 + Bc + 𝑚1ƞ 2(t) 2 + 𝑚2𝑝 2(t) 2 + 𝑡𝑓 0 𝑚3Ѱ 2(𝑡) 2 + 𝑚4𝜇𝐵 2(t) 2 ) (18) subject to equation (1) with ƞ = ƞ(𝑡),p = p(t),Ψ = Ψ(t),µ𝐵 = µ𝐵(𝑡). (19) The coefficients, 𝑎,𝑏,𝑐,are the weight constants for the infected, carriers and the bacteria concentration respectively whereas 𝑚𝑖, i = 1,2,3,4 are cost of implementing these control measures. We assume a quadratic expression for the costs based on literature. The control measures, ƞ(t), 𝑝(t), Ѱ(t), µ𝐵(t) are Lebesgue measurable with 0 ≤ ƞ(t) < 0.9,0 ≤ 𝑝(t) < 1,0 ≤ Ѱ(t) < 1, 0 ≤ µ𝐵(t) < 1 for 0 ≤ 𝑡 ≤ 𝑡𝑓. We aimed to minimize the number of infectives, carriers, bacteria concentration and their costs of implementations, that is, 𝐽(ƞ(t)∗, 𝑝(t)∗,Ѱ(t)∗,µ𝐵(t) ∗) = 𝑚𝑖𝑛 𝐽(ƞ(t), 𝑝(t), Ѱ(t), µ𝐵(t)). The optimal control pair is obtained using Pontryagin maximum principle [32]. This principle converts equations (18) and (1) with (19) into a problem of minimizing pointwise a Hamiltonian H with respect to ƞ(t), 𝑝(t), Ѱ(t), µ𝐵(t) such that; 𝐻(𝑃,𝑆,𝐼, 𝐼𝐶,𝑅,𝐵𝐶) = 𝑑𝐽 𝑑𝑡 + 𝜆1 𝑑𝑃 𝑑𝑡 + 𝜆2 𝑑𝑆 𝑑𝑡 + 𝜆3 𝑑𝐼 𝑑𝑡 + 𝜆4 𝑑𝐼𝑐 𝑑𝑡 + 𝜆5 𝑑𝑅 𝑑𝑡 + 𝜆6 𝑑𝐵𝐶 𝑑𝑡 . Thus, 𝐻(𝑃,𝑆,𝐼, 𝐼𝐶,𝑅,𝐵𝐶) = (𝑎𝐼 + 𝑏𝐼𝐶 + 𝑚1ƞ(t) 2 2 + 𝑚2𝑝(t) 2 2 + 𝑚3Ѱ(t) 2 2 + 𝑚4µ𝐵(t) 2 2 ) + 𝜆1 𝑑𝑃 𝑑𝑡 + 𝜆2 𝑑𝑠 𝑑𝑡 + 𝜆3 𝑑𝐼 𝑑𝑡 + 𝜆4 𝑑𝐼𝐶 𝑑𝑡 + 𝜆5 𝑑𝑅 𝑑𝑡 + 𝜆6 𝑑𝐵𝐶 𝑑𝑡 , where 𝜆1,𝜆2,𝜆3,𝜆4,𝜆5 and𝜆6 are the adjoint variable functions. So, 𝐻 = (𝑎𝐼 + 𝑏𝐼𝑐 + 𝑐𝐵𝑐 + 𝑚1ƞ(t) 2 2 + 𝑚2𝑝(t) 2 2 + 𝑚3Ѱ(t) 2 2 + 𝑚4µ𝐵(t) 2 2 ) + 𝜆1(αɅ + ƞ(t)S – (γ + µ)P) + 𝜆2 ((1 − α)Ʌ + γP + ɸR – (ƞ(t) + µ + (1 − p(t)) 𝛽𝐵𝑐 (𝐾+𝐵𝑐) )S) + 𝜆3 ((1 − p(t)) 𝛽𝐵𝑐 (𝐾+𝐵𝑐) S – (σ + 𝜏1 + µ + 𝑑)I ) + A dynamic model of typhoid fever with optimal control analysis 271 𝜆4(σ𝐼 – (𝜏2 + Ѱ(t)𝜃 + 𝜇)𝐼𝑐) + 𝜆5(𝜏1I + (𝜏2 + Ѱ(t)𝜃)𝐼𝑐 – (µ + ɸ)R ) + 𝜆6(𝜋2(1 − 𝑝(t))I + 𝜋1(1 − 𝑝(t))𝐼𝑐 − µ𝐵(t)𝐵𝑐). (20) Theorem 5. Given an optimal control ƞ(t)∗, 𝑝(t)∗,Ѱ(t)∗,µ𝐵(t) ∗ and corresponding state variables 𝑃,𝑆,𝐼,𝐼𝐶,𝑅,𝑎𝑛𝑑 𝐵𝐶 that minimize the objective function 𝐽(ƞ(t), 𝑝(t), Ѱ(t), µ𝐵(t)) over U, there exist adjoint functions 𝜆1,…𝜆6 satisfying; 𝑑𝜆1 𝑑𝑡 = (𝜆1 − 𝜆2)𝛾 + μ𝜆2, 𝑑𝜆2 𝑑𝑡 = (𝜆2 − 𝜆1)ƞ(t)+ μ𝜆2 + (𝜆2 − 𝜆3)(1 − p(𝑡)) 𝛽𝐵𝑐 (𝐾+𝐵𝑐) , 𝑑𝜆3 𝑑𝑡 = −𝑎 + (𝜆3 − 𝜆4)σ + (𝜆3 − 𝜆5)𝜏1 + 𝜆3(µ + 𝑑) − 𝜋2(1 − 𝑝(t))𝜆6 , 𝑑𝜆4 𝑑𝑡 = −𝑏 + (𝜆4 − 𝜆5)(𝜏2 + Ѱ(t)𝜃) + 𝜆4𝜇 − 𝜋1(1 − 𝑝(t))𝜆6, 𝑑𝜆5 𝑑𝑡 = (𝜆5 − 𝜆4)ɸ + 𝜆5𝜇 , 𝑑𝜆6 𝑑𝑡 = −𝑐 + (𝜆2 − 𝜆3)(1 − p(t)) 𝛽 (𝐾+𝐵𝑐) (1 − 𝐵𝑐 (𝐾+𝐵𝑐) ) + 𝜆6µ𝐵(t). } (21) with the transversality condition, 𝜆𝑖(𝑡𝑓) = 0,for 𝑖 = 1(1)6 and the controls ƞ∗(t),𝑝∗(t),Ѱ∗(t)and 𝜇𝐵 ∗(t) satisfying the optimality condition; ƞ∗ = 𝑚𝑎𝑥{0,𝑚𝑖𝑛 (1, (𝜆2−𝜆1)𝑆 𝑚1 )} , p∗ = {0,𝑚𝑖𝑛 (1, (𝜆3−𝜆2)𝛽𝐵𝑐𝑆+𝜆6𝜋2𝐼(𝐾+𝐵𝑐)+𝜆6𝜋1𝐼𝑐(𝐾+𝐵𝑐) 𝑚2(𝐾+𝐵𝑐) )} , Ѱ∗ = 𝑚𝑎𝑥{0,𝑚𝑖𝑛 (1, (𝜆4−𝜆5)𝜃𝐼𝑐 𝑚3 )} , µ𝐵 ∗ = 𝑚𝑎𝑥{0,𝑚𝑖𝑛 (1, 𝜆6𝐵𝑐 𝑚4 )} . } (22) Proof. Using Pontryagin maximum principle, we obtained the adjoint equation and tranversality conditions by differentiating the Hamiltonian function with respect to state variables 𝑃,𝑆,𝐼,𝐼𝐶,𝑅 and 𝐵𝐶 respectively which is evaluated at the optimal control functions ƞ(t), 𝑝(t), Ѱ(t), µ𝐵(t). So, the adjoint system (21) is obtained using the following derivatives 𝑑𝜆1 𝑑𝑡 = − 𝜕𝐻 𝜕𝑃 , 𝑑𝜆2 𝑑𝑡 = − 𝜕𝐻 𝜕𝑆 , 𝑑𝜆3 𝑑𝑡 = − 𝜕𝐻 𝜕𝐼 , 𝑑𝜆4 𝑑𝑡 = − 𝜕𝐻 𝜕𝐼𝐶 , 𝑑𝜆5 𝑑𝑡 = − 𝜕𝐻 𝜕𝑅 , 𝑑𝜆6 𝑑𝑡 = − 𝜕𝐻 𝜕𝐵𝑐 while the interior of the control set of equation (22) is obtained by solving for ƞ(t), 𝑝(t), Ѱ(t), µ𝐵(t) in the respective equations C. E. Madubueze, R.I. Gweryina, and K. A. Tijani 272 𝜕𝐻 𝜕ƞ(t) = 0, 𝜕𝐻 𝜕p(t) = 0, 𝜕𝐻 𝜕Ѱ(t) = 0 , 𝜕𝐻 𝜕µ𝐵(t) = 0. This completes the proof. The optimality system involves equation (1) with (19), equations (21) and (22). 5. Numerical Simulations and Discussion The numerical simulations of the optimality system involving equations (1) with (19), (21) and (22) are implemented using Runge-Kutta method with the aid of MATLAB R2007b. The simulations are carried out to examine the impact of the control measures on Typhoid fever. The parameter values used for the simulations are in Table 2 while the initial c onditions are from Mushanyu et al. (2018) as follows; S(0) = 10000,I(0) = 10,Ic(0) = 10,R(0) = 0,Bc(0) = 100000. P(0) = 100 is assumed. The weight constants for simulation are given as m1 = 9 × 10 −1,m2 = 5 × 105,m3 = 7 × 10 2 and m4 = 4 × 10 6. (a) Optimal and constant control. The importance of time-dependent control measures is considered in Figure 3. With optimal control, a typhoid-free population is attained within 200 days compared with constant control which shows the endemicity of the typhoid in the population. This is achieved when 𝑢1 is at the upper bound for 150 days and 𝑢2, 𝑢3 and 𝑢4 are below a bound of 0.3 for 175 days before they decline to their final time. This implies that control measures should be implemented in time to achieve a typhoid free population. Figure 4. Solutions of Typhoid model for the infected state variables with and without control measures with control profile. 0 50 100 150 200 0 500 1000 1500 Time (Days) In fe c te d I n d iv id u a ls , I( t) A Optimal Constant 0 50 100 150 200 0 200 400 600 800 Time (Days) In fe c te d C a rr ie rs , I c (t ) B Optimal Constant 0 50 100 150 200 0 5 10 x 10 4 Time (Days) B a c te ri a C o n c e n tr a ti o n , B (t ) C Optimal Constant 0 50 100 150 200 0 0.5 1 Time (Days) C o n tr o l p ro fi le D 1 2 3 4 A dynamic model of typhoid fever with optimal control analysis 273 (b) Vaccination and hygiene practices. We minimize the objective function for vaccination and hygiene practices (𝑢1,𝑢2 ≠ 0,𝑢3 = 𝑢4 = 0) to assess their effect on the disease. The number of infected individuals and bacteria concentration are reduced when compared to without control (See Figure 5). This is obtained when 𝑢1 is at its upper bound for all the time 200 days and 𝑢2 attains a bound of 0.9 and decline after 5 days (Figure 5D). However, typhoid disease still remains in the population. Figure 5. Solutions of Typhoid model for the infected state variables without and with vaccination (𝑢1) and hygiene practices (𝑢2) control measures only. W/o means without. (c) Vaccination and screening. We minimize the objective function for vaccination and screening (𝑢1,𝑢3 ≠ 0,𝑢2 = 𝑢4 = 0). They reduced the number of infected persons and bacteria concentration but not as in case (b) (see Figures 5 and 6) as the number of carriers reduces in Figure 6B than Figure 5B. This may be as a result of screening in the combined control measures. This is achieved when the control, 𝑢1, is maintain at the upper bound for all time (200 days) while 𝑢3 decline after attaining upper bound for 110 days (Figure 6D). 0 50 100 150 200 0 2000 4000 6000 Time (Days) In fe ct ed I nd iv id ua ls , I( t) A u 1 , u 2 W/o control 0 50 100 150 200 0 500 1000 1500 2000 Time (Days) In fe ct ed C ar rie rs , I c( t) B u 1 , u 2 W/o control 0 50 100 150 200 1 2 3 4 5 x 10 5 Time (Days) B ac te ria C on ce nt ra tio n, B (t ) C u 1 , u 2 W/o control 0 50 100 150 200 0 0.5 1 Time (Days) C on tr ol p ro fil e D u 1 u 2 C. E. Madubueze, R.I. Gweryina, and K. A. Tijani 274 Figure 6. Solutions of Typhoid model for the infected state variables without and with vaccination (𝑢1) and screening (𝑢3) control measures only. W/o means without. (d) Vaccination and sterilization. We minimize the objective function for vaccination and sterilization (𝑢1,𝑢4 ≠ 0,𝑢2 = 𝑢3 = 0). The simultaneous implementation of 𝑢1 and 𝑢4 reduced the number of infected persons and bacteria concentration to zero after 70 days and 30 days respectively while the number of carriers in the population is almost zero as at 200 days. The control, 𝑢1, maintains an upper bound for 200 days while 𝑢4 attains a bound of 0.2 for 190 days before decline to its final time. 0 50 100 150 200 0 2000 4000 6000 Time (Days) In fe ct ed I nd iv id ua ls , I( t) A u 1 , u 3 W/o control 0 50 100 150 200 0 500 1000 1500 2000 Time (Days) In fe ct ed C ar rie rs , I c( t) B u 1 , u 3 W/o control 0 50 100 150 200 1 2 3 4 5 x 10 5 Time (Days) B ac te ria C on ce nt ra tio n, B (t ) C 0 50 100 150 200 0 0.5 1 Time (Days) C on tr ol p ro fil e D u 1 u 3 u 1 , u 3 W/o control A dynamic model of typhoid fever with optimal control analysis 275 Figure 7. Solutions of Typhoid model for the infected state variables without and with vaccination (𝑢1) and sterilization (𝑢4) control measures only. Here, W/o means without. (e) Hygiene practices and screening. We minimize the objective function for hygiene practices and screening (𝑢2,𝑢3 ≠ 0,𝑢1 = 𝑢4 = 0). The observed effect is similar to case (c) except that 𝑢3 attains an upper bound and declines after 70 days while 𝑢2 of a bound of 0.55 and declines immediately to final time. The disease still remains endemic in the population. 0 50 100 150 200 0 2000 4000 6000 Time (Days) In fe ct ed I nd iv id ua ls , I( t) A u 1 , u 4 W/o control 0 50 100 150 200 0 500 1000 1500 2000 Time (Days) In fe ct ed C ar rie rs , I c( t) B u 1 , u 4 W/o control 0 50 100 150 200 0 2 4 6 x 10 5 Time (Days) B ac te ria C on ce nt ra tio n, B (t ) C u 1 , u 4 W/o control 0 50 100 150 200 0 0.5 1 Time (Days) C on tr ol p ro fil e D u 1 u 4 C. E. Madubueze, R.I. Gweryina, and K. A. Tijani 276 Figure 8. Solutions of Typhoid model for the infected state variables without and with hygiene practices (𝑢2) and screening (𝑢3) control measures only. Here, W/o means without. (f) Hygiene practices and sterilization. We minimize the objective function for hygiene practices and sterilization as control measures (𝑢2,𝑢4 ≠ 0,𝑢1 = 𝑢3 = 0). The combined implementation of 𝑢2 and 𝑢4 reduces the number of infected persons and bacteria concentration to zero after 110 days and 50 days respectively while there is still some infected carriers in the population after 200 days. The hygiene practice 𝑢2, initially increases from 0.18 to 0.28 bound within 8 days and declines after 120 days while 𝑢4 attains a bound of 0.2 for 195 days before declining to its final time. 0 50 100 150 200 0 2000 4000 6000 Time (Days) In fe ct ed I nd iv id ua ls , I( t) A u 2 , u 3 W/o control 0 50 100 150 200 0 500 1000 1500 2000 Time (Days) In fe ct ed C ar rie rs , I c( t) B u 2 , u 3 W/o control 0 50 100 150 200 1 2 3 4 5 x 10 5 Time (Days) B ac te ria C on ce nt ra tio n, B (t ) C u 2 , u 3 W/o control 0 50 100 150 200 0 0.5 1 Time (Days) C on tr ol p ro fil e D u 2 u 3 A dynamic model of typhoid fever with optimal control analysis 277 Figure 9. Solutions of Typhoid model for the infected state variables without and with hygiene practices (𝑢2) and sterilization (𝑢4) control measures only. Here, W/o means without. (g) Screening and sterilization. We minimize the objective function for screening and sterilization as control measures (𝑢3,𝑢4 ≠ 0,𝑢1 = 𝑢2 = 0). The simultaneous implementation of 𝑢3 and 𝑢4 behaves similar as cases (e) and (f). Here, the number of infected persons, carriers and bacteria concentration reduce to zero after 75 days, 100 days and 45 days respectively. This is achieved when 𝑢3 and 𝑢4 are at bound 0.28 for 170 days and 0.19 for 190 days respectively before declining to their final time. 0 50 100 150 200 0 2000 4000 6000 Time (Days) In fe ct ed I nd iv id ua ls , I( t) A u 2 , u 4 W/o control 0 50 100 150 200 0 500 1000 1500 2000 Time (Days) In fe ct ed C ar rie rs , I c( t) B u 2 , u 4 W/o control 0 50 100 150 200 0 2 4 6 x 10 5 Time (Days) B ac te ria C on ce nt ra tio n, B (t ) C u 2 , u 4 W/o control 0 50 100 150 200 0 0.1 0.2 0.3 0.4 Time (Days) C on tr ol p ro fil e D u 2 u 4 C. E. Madubueze, R.I. Gweryina, and K. A. Tijani 278 Figure 10. Solutions of Typhoid model for the infected state variables without and with screening (𝑢3) and sterilization (𝑢4) control measures only. Here, W/o means without. (h) Three combine control measures We minimize the objective function for three control measures that is 𝑢1,𝑢2,𝑢3,≠ 0,𝑢4 = 0 (123), 𝑢1,𝑢2,𝑢4,≠ 0,𝑢3 = 0 (124), 𝑢1,𝑢3,𝑢4,≠ 0,𝑢2 = 0 (134) and 𝑢2,𝑢3,𝑢4,≠ 0,𝑢1 = 0 (234). We notice from Figure (10) that bacteria clearance reduces the number of infected populations (𝐼(𝑡),𝐼𝑐(𝑡)) and bacteria concentration. However, the combine implementation of vaccination, screening and sterilization gives a better result compared to 𝑢1,𝑢2,𝑢4, and 𝑢1,𝑢3,𝑢4, as it achieves a typhoid-free population in shortest period of time than others. 0 50 100 150 200 0 2000 4000 6000 Time (Days) In fe ct ed I nd iv id ua ls , I( t) A u 3 , u 4 W/o control 0 50 100 150 200 0 500 1000 1500 2000 Time (Days) In fe ct ed C ar rie rs , I c( t) B u 3 , u 4 W/o control 0 50 100 150 200 0 2 4 6 x 10 5 Time (Days) B ac te ria C on ce nt ra tio n, B (t ) C u 3 , u 4 W/o control 0 50 100 150 200 0 0.1 0.2 0.3 0.4 Time (Days) C on tr ol p ro fil e D u 3 u 4 A dynamic model of typhoid fever with optimal control analysis 279 Figure 11. Solutions of Typhoid model for the infected state variables with optimal control. Here, 123 means 𝑢1,𝑢2,𝑢3 combine, 124 means 𝑢1,𝑢2,𝑢4 combine, 134 means 𝑢1,𝑢3,𝑢4 combine, 234 means 𝑢2,𝑢3,𝑢4 combine. 7. Conclusion In this study, the mathematical model of Typhoid fever dynamics with protected human population and bacteria concentration is examined. The control measures such as vaccination, hygiene practice and screening are taken into consideration. The disease-free and endemic equilibrium states are both locally and globally stable whenever 𝑅𝑐 < 1 and 𝑅𝑐 > 1 respectively. The local stability of endemic equilibrium state is established using Centre manifold theorem in order to show existence of forward bifurcation while the global stability is done when disease-related death rate is neglected. The sensitivity analysis of the control reproduction number is carried out and the result indicates that the typhoid fever disease will be controlled in the population if susceptible people are vaccinated with high practice of personal hygiene as well as screening of the carriers are screened and also the bacteria in the environment is disinfect or sterilization. The optimal control analysis is carried out for time-dependent control functions to form non-autonomous system. The Pontryagin maximum principle is used to establish the optimality conditions for the system. This is solved numerically to establish that optimal control implementation achieved infection- free population on time compare to constant control. Considering when there is limited resources to implement all the controls together, screening and bacteria sterilization should be adopted for two combined controls, while vaccination, 0 50 100 150 200 0 1000 2000 3000 Time (Days) In fe ct ed In di vi du al s, I( t) A 123 124 134 234 0 50 100 150 200 0 200 400 600 Time (Days) In fe ct ed C ar rie rs , I c( t) B 123 124 134 234 0 50 100 150 200 0 5 10 15 x 10 4 Time (Days) B ac te ria C on ce nt ra tio n, B (t) C 123 124 134 234 C. E. Madubueze, R.I. Gweryina, and K. A. Tijani 280 screening and bacteria sterilization should be implemented together for three combine controls. However, the combined implementation of all controls is more effective in eradicating the disease from the environment. It is therefore recommended that these preventive measures (vaccination, hygiene practice, screening and sterilization) should be adopted by the policy makers to eliminate the typhoid bacteria from the population. References [1] WHO. Typhoid. www.whoints/news-room/fact-sheets/detail/typhoid, 2020. Accessed on 10th May, 2021. [2] M. M. Gibani, M. Voysey, C. Jin, C. Jones, H. Thomaides-Brears, …, A. J. Pollard. The impact of vaccination and prior exposure on stool shedding of salmonella typhi and salmonella paratyphi in 6 controlled human infection studies. Clinical Infectious Diseases, 68(8): 1265-1273, 2019. [3] CDC. Typhoid fever. www.nc.cdc.gov/travel/diseases/typhoid, 2020. Accessed on 11th September, 2021. [4] A. C. Bradley and S. Eli. Typhoid and paratyphoid fever in travellers. Lancet Infect Dis., 5(10):623-8, 2005. [5] S. Cairncross, C. Hunt, S. Boisson, K. Bostoen, V. Curtis, I. C. H. Fung and W. Schmidt. Water, sanitation and hygiene interventions and the prevention of Diarrhoea. Int J Epidemiol, Suppl 1(Suppl 1):193-205, 2010. [6] I. Bakach, M. R. Just, M. Gambhir and I. C. Fung. Typhoid transmission: A historical perspective on mathematical model development. Trans R Soc Med Hyg., 109(11): 679-89, 2015. [7] S. Mushayabasa. A simple epidemiological model for typhoid with saturated incidence rate and treatment effect. International Journal Mathematical and Computational Sciences, 6(6): 688-695, 2012. [8] H. Abboubakar and R. Racke. Mathematical modelling and optimal control of typhoid fever. Konstanzer Schriften in Mathematik, Universitat Konstanz, Konstanzer Online-Publikatins-System (KOPS), Nr. 384, Dezember 2019. [9] J. W. Karunditu, G. Kimathi and S. Osman. Mathematical modeling of typhoid fever disease incorporating unprotected humans in the spread dynamics. Journal of Advances in Mathematical and Computer Science, 32(3):1-11, 2019. [10] O. J. Peter, M. O. Ibrahim, A. Oluwaseun and R. Musa. Mathematical model for the control of typhoid fever. IOSR Journal of Mathematics,13(4): 60- 66, 2017. http://www.whoints/news-room/fact-sheets/detail/typhoid http://www.nc.cdc.gov/travel/diseases/typhoid A dynamic model of typhoid fever with optimal control analysis 281 [11] N. Nyerere., S. C. Mpeshe and S. Edward. Modeling the impact of screening and treatment on the dynamics of typhoid fever. World Journal of Modelling and Simulation. 14(4): 298-306, 2018. [12] O. J. Peter, A. Afolabi, F. A. Oguntolu, C. Y. Ishola and A. A. Victor. Solution of a deterministic mathematical model of typhoid fever by variational iteration method. Science World Journal, 13(2): 64-68, 2018. [13] S. Edward and N. Nyerere. A Mathematical model for the dynamics of cholera with control measure. Applied and Computational Mathematics, 4(2): 53-63, 2015. [14] M. Kgosimore and G. R. Kelatlehegile. Mathematical analysis of typhoid infection with treatment. Journal of Mathematical Sciences: Advancse and Applications, 40: 75-91, 2016. [15] B. S. Aji, D. Aldila and B. D. Handari. Modeling the impact of limited treatment resources in the success of typhoid intervention. AIP Conference Proceedings, International Conference on Science and Applied Science, 2202(1):020040, 2019. Doi:10.1063/1.5141653 [16] G. T. Tilahum. O. D. Makinde and D. Malonza. Modelling and optimal control of typhoid fever disease with cost-effective strategies. Computational and Mathematical Methods in Medicine, Article ID 2324518, 2017. [17] P. N. Okolo and O. Abu. On optimal control and cost-effectiveness analysis for typhoid fever model, FUDMA Journal of Sciences (FJS), 4(3): 437 – 445, 2020. [18] O. J. Peter, M. O. Ibrahim, H. O. Edogbanya, F. A. Oguntolu, K. Oshinubi, A. A. Ibrahim, T. A. Ayoola and J. O. Lawal. Direct and indirect transmission of typhoid fever model with optimal control. Results in Physics, 27, 104463, 2021. [19] H. Abboubakar and R. Racke. Mathematical modelling forecasting and optimal control of typhoid fever transmission dynamics. Chaos, Solitons and Fractals, 149, 111074, 2021. [20] T. D. Awoke. Optimal control strategy for the transmission dynamics of typhoid fever. American Journal of Applied Mathematics, 7(2):37 – 49, 2019. [21] J. H. Jones. Notes on 𝑅0. Department of Anthropological Sciences, Standford University, May 1, 2007. [22] J. M. Heffernan, R. J. Smith and L. M. Wahl. Perspectives on the basic reproductive ratio. Journal of the Royal Society Interface. 2(4): 281-293, 2005. [23] C. Castillo-Chavez and B. Song. Dynamical model of tuberculosis and their applications. Mathematical Bioscience and Engineering, 1(2): 361-404, 2004. C. E. Madubueze, R.I. Gweryina, and K. A. Tijani 282 [24] A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Salsana and S. Tarantola. Global sensitivity analysis - the primer. Wiley, Chichester, 2008. [25] H. S. Rodrigues, M T. T. Monteiro and D. F. M. Torres. Sensitivity analysis in a dengue epidemiological model. Conference Papers in Mathematics, 2013: Articles ID: 721406, 2013. doi: 10.1155/2013/721406 [26] D. T. Lauria, B. Maskery, C. Poulous and D. Whittington. An optimization model for reducing typhoid cases in developing countries without increasing public spending. Vaccine, 27(10): 1609-1621, 2009. [27] S. Mushayabasa. Impact of vaccine on controlling typhoid fever in Kassena-Nankana district of Upper East Region of Ghana: Insights from a mathematical model. Journal of Modern Mathematics and Statistics. 5(2): 54- 59, 2011. [28] I. A. Adetunde. Mathematical models for the dynamics of typhoid fever in Kassena-Nankana district of Upper East Region of Ghana. Journal of Modern Mathematics and Statistics. 2(2): 45-49, 2008. [29] M. Gosh, P. Chandra, P. Sinha. and J. B. Shukla. Modelling the spread of bacterial infectious disease with environmental effect in a logistically growing human population. Nonlinear Analysis Real World Applications, 7(3): 341–363, 2006. [30] J. Mushanyu, F. Nyabadza, G. Muchatibaya, P. Mafuta and G. Nhawu. Assessing the potential impact of limited public health resources on the spread and control of typhoid. Journal of Mathematical Biology, 77(3):647-670, 2018. [31] J. M. Mutual, F. B. Wang and N. K. Vaidya. Modeling malaria and typhoid co-infection dynamics. Mathematical Bioscience, 264(1): 128-144, 2015. [32] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko. The mathematical theory of optimal processes, John Wiley &Sons, Lonon, UK, 1962.