Typhoid fever is a life-threatening infection that originated from the bacterium Salmonella Typhi and is a major cause of illness and mortality in regions of the world with limited access to treated water and sanitation [1,2]. Recent statistics show that an average of 15 million cases and 145 000 typhoid-related deaths occur annually worldwide and its more endemic in many developing countries, and despite recent interventions in sanitation coverage, the disease remains a significant public health problem [1,3]. People who are infected with Salmonella Typhi, often referred to as "typhoid carriers," shed the bacteria in their feces (stool) and, to a lesser extent, in their urine. These individuals may have symptoms of typhoid fever or be asymptomatic carriers. The transmission of the disease is primarily a result of poor sanitation and lack of clean drinking water, but can also be transmitted via person-to-person on unclean surfaces [1]. Symptoms of typhoid fever include headache, weakness, loss of appetite, prolonged fever, nausea and constipation, or sometimes diarrhea [1].

In developing countries, the public health goal of preventing and controlling typhoid disease through sanitation and adequate medical care is challenging. Besides that, delays in diagnosis and treatment occur due to barriers to medical care, such as difficulty accessing medical facilities caused by delayed referral, the distance to hospital, and the cost of getting healthcare. The socio-economic class (SEC) of individuals has been shown to influence the dynamics of some infectious diseases [4,5,6,17]. Since typhoid fever is linked with poor sanitation and unclean water, individuals in the SEC are expected to be more exposed to typhoid fever compared to the individuals in a higher SEC [17]. In this work, we analyze the influence of control measures on the dynamics of typhoid fever disease for multiple socio-community. Specifically, we look at a case when the community consists of two socio-economic classes (i.e, lower SEC and higher SEC).

Several control measures have been implemented in fighting typhoid fever. Some of the effective ones include sanitation, vaccination, and treatment [1]. Medically, each of these three control measures (sanitation, vaccination, and treatment) are independent and hence can be applied simultaneously. Others include accurate diagnosis at the right time and treatment of typhoid disease in the community to prevent hospitalization complications. Prevention of the disease through improving sanitation and accessibility to safe water and food, health education to increase public awareness and induce healthy behavioral changes [3,7,8,9].

A mathematical model is a powerful approach to learning how disease spreads among both human and animal populations. Mathematical models help us to understand the transmission dynamics quantitatively and allow us to check hypotheses to understand their importance [10]. Interpreting a mathematical model for an infectious disease requires some assumptions about the spreading infection mechanism [11]. The basic compartmental mathematical models to describe the spread of disease are introduced [12,13]. This approach may test and compare different disease interventions as a strategy to prevent and control the disease. The origin of a mathematical model for typhoid disease could be related to Branko Cvjetanovi´c, an assistant professor at the Zagreb School of Medicine, who implemented the medical trial of the first typhoid vaccine that was funded by the US Public Health Service and WHO in the 1950s [16]. He was involved in research about the vaccines for diphtheria, pertussis, tetanus, cholera, and typhoid. In the 1973s, Cvjetanovi´c noted that no controlled trials had been run to demonstrate the extent to which typhoid transmission can control the spread through various sanitation strategies. Moreover, some researchers developed a non-autonomous mathematical model [14,15] to study typhoid transmission by considering the effect of seasonal conditions and some time-dependent parameters.

The epidemiology of some infectious disease has been studied extensively using mathematical models [17,18,19,20,21,22,23,24,25,26,32,35]. In this study, we utilize a mathematical model to ascertain the influence of control measures in reducing typhoid fever in a diverse socio-economic community.

A multiple socio-economic community with total human inhabitant N is considered. We assume that the community is made up of two socio-economic classes whose sub-population is $N_i$. Suppose there is a typhoid fever outbreak within the two socio-economic classes of the community. Assumed that each of these socio-economic community ($N_i$) engages three control measures (vaccination, treatment, and sanitation) in fighting the disease. Based on these assumptions, the formulation of mathematical model requires that the total population ($N_i$) for each socio-economic class is partitioned into susceptible population $S_i\left(t\right)$, vaccinated population $V_i\left(t\right)$, infected population $I_i\left(t\right)$, treated population $T_i\left(t\right)$ and recovered population $R_i\left(t\right)$. The variable $P_i\left(t\right)$ represented the pathogen in the environment for each SEC i. Epidemiologically, the transmission of typhoid fever disease is either through contact with infected humans or through exposure to the bacteria causing the illness. Recruitment of individuals into each of the susceptible class $S_i\left(t\right)$ occur at a rate ${\mu }_i{N}_i\left(t\right)$. Individuals in each $S_i\left(t\right)$ moves to $V_i\left(t\right)$ as they get vaccinated at a rate ${\varphi }_i$. Direct transmission from $I_i\left(t\right)$ to $S_i\left(t\right)$ and $V_i\left(t\right)$ occur at a rate ${\beta }_i$ while the indirect transmission from $P_i\left(t\right)$ to $S_i\left(t\right)$ and $V_i\left(t\right)$ occur at a rate ${\alpha }_i$. Note that the vaccinated individuals have less chances of being infected because they are vaccinated. This is captured in the model by assuming that the efficacy of the vaccine is ${\epsilon }_i$ for SEC i. Each infected class $I_i\left(t\right)$ get treated at a rate ${\sigma }_i$. The treated class $T_i\left(t\right)$ recovers at a rate ${\rho }_i$. The $I_i\left(t\right)$ who did not get treatment can recover naturally at a rate ${\gamma }_i$. Note that we discourage not getting treatment because typhoid fever can be very fatal and there are available treatment for the disease. Natural death occur at each of the SEC $N_i\left(t\right)$ at a rate ${\mu }_i$. Each of the recovered class $R_i\left(t\right)$ can loss immunity and become susceptible again at a rate ${\phi }_i$. Susceptible individuals move from $S_i\left(t\right)$ to $S_j\left(t\right)$ at a rate $k_{ij}$ whereas infected individuals move from $I_i\left(t\right)$ to $I_j\left(t\right)$ at a rate $b_{ij}$. Infected individuals $I_i\left(t\right)$ shed pathogens into the environment $P_i\left(t\right)$ at a rate ${\delta }_i$ and the pathogen $P_i\left(t\right)$ decay at a rate $\xi$. Sanitation enhances pathogens $P_i\left(t\right)$ decay at a rate ${\theta }_i$. Based on these explanations, we obtained the typhoid fever control model given by The variables and parameters meanings can be found in Table 1 and Table 2 respectively.

Let the initial conditions of the multiple control model be assumed as:

In this section, we present the dynamical system analysis of the multiple control model 1. The analysis will improve our understanding of typhoid fever disease dynamics. Mathematically, there exists a unique disease-free equilibrium (DFE) for the multiple control model 1 where $S_1^0=\frac{k_{21}N}{{\phi }_1{k}_{12}+{\phi }_2{k}_{21}},S_2^0=\frac{k_{12}N}{{\phi }_1{k}_{12}+{\phi }_2{k}_{21}},V_1^0=\frac{{\varphi }_1{S}_1^0}{{\mu }_1},V_2^0=\frac{{\varphi }_2{S}_2^0}{{\mu }_2},{\phi }_1=\frac{{\mu }_1+{\varphi }_1}{{\mu }_1}$ and ${\phi }_2=\frac{{\mu }_2+{\varphi }_2}{{\mu }_2}$.

The basic reproduction number for the multiple control model 1 can be referred to as the expected number of new infections of typhoid fever produced when an infected individual is brought into contact with the population susceptible to typhoid fever in the presence of vaccination and sanitation. Mathematically, the basic reproduction number of model 1, using the next generation matrix approach [20] is where ${\mathcal{R}}_{11}=\frac{({\beta }_1{S}_1^0+(1-{\epsilon }_1){\beta }_1{V}_1^0){\psi }_2}{{\varrho }_1{\varrho }_2+{\varrho }_1{b}_{21}+{\varrho }_2{b}_{12}}+\frac{({\alpha }_1{S}_1^0+(1-{\epsilon }_1){\alpha }_1{V}_1^0){\delta }_1{\psi }_2}{(\xi +{\theta }_1)({\varrho }_1{\varrho }_2+{\varrho }_1{b}_{21}+{\varrho }_2{b}_{12})}$, ${\mathcal{R}}_{14}=\frac{({\beta }_1{S}_1^0+(1-{\epsilon }_1){\beta }_1{V}_1^0)b_{21}}{{\varrho }_1{\varrho }_2+{\varrho }_1{b}_{21}+{\varrho }_2{b}_{12}}+\frac{({\alpha }_1{S}_1^0+(1-{\epsilon }_1){\alpha }_1{V}_1^0){\delta }_1{b}_{21}}{(\xi +{\theta }_1)({\varrho }_1{\varrho }_2+{\varrho }_1{b}_{21}+{\varrho }_2{b}_{12})}$, ${\mathcal{R}}_{41}=\frac{({\beta }_2{S}_2^0+(1-{\epsilon }_2){\beta }_2{V}_2^0)b_{12}}{{\varrho }_1{\varrho }_2+{\varrho }_1{b}_{21}+{\varrho }_2{b}_{12}}+\frac{({\alpha }_2{S}_2^0+(1-{\epsilon }_2){\alpha }_2{V}_2^0){\delta }_2{b}_{12}}{(\xi +{\theta }_2)({\varrho }_1{\varrho }_2+{\varrho }_1{b}_{21}+{\varrho }_2{b}_{12})}$,

${\mathcal{R}}_{44}=\frac{({\beta }_2{S}_2^0+(1-{\epsilon }_2){\beta }_2{V}_2^0){\psi }_1}{{\varrho }_1{\varrho }_2+{\varrho }_1{b}_{21}+{\varrho }_2{b}_{12}}+\frac{({\alpha }_2{S}_2^0+(1-{\epsilon }_2){\alpha }_2{V}_2^0){\delta }_2{\psi }_2}{(\xi +{\theta }_2)({\varrho }_1{\varrho }_2+{\varrho }_1{b}_{21}+{\varrho }_2{b}_{12})}$, ${\varrho }_1={\mu }_1+{\gamma }_1+{\sigma }_1$, ${\varrho }_2={\mu }_2+{\gamma }_2+{\sigma }_2$, ${\psi }_1={\varrho }_1+b_{12}$ and ${\psi }_2={\varrho }_2+b_{21}$.

Epidemiologically, when ${\mathcal{R}}_0<1$, the disease can be eradicated from the two socio-economic classes. This can be shown by proving that the disease free equilibrium is stable when ${\mathcal{R}}_0<1$[20,21,22]. This implies that the control measures ensure that the basic reproduction number is less than unity so that the disease will not be established in any of the socio-economic class in the community. On the contrary, if the control measures are not effectual in decreasing ${\mathcal{R}}_0$ below unity, a typhoid fever outbreak is likely to occur. The outbreak may persist or remain endemic in either or both socio-economic classes of the population [18,20,21,22]. Further investigation on the influence of the control measures on typhoid fever disease dynamics is considered via numerical illustrations in the subsequent section.

Numerical illustrations are presented here to analyze the influence of the control measures on typhoid fever disease dynamics for the diverse socio-economic community. The parameter values used in the numerical illustrations are given in Table 3.

Vaccination is one of the effective control measures for minimizing typhoid fever [1]. Figure 1 illustrates the influence of vaccination rate in the population. We observe from the figure that increasing vaccination rates lead to a decrease in infected humans in both socio-economic classes. We observe that the infected populace is greater in the lower SEC 1 in the presence of vaccination. Hence, to achieve disease eradication, this lower SEC 1 should be the main target of vaccination.

Vaccine efficacy is a major factor in vaccination that determines the percentage reduction of the disease in a vaccinated group. Figure 2 illustrates the impact of vaccine efficacy ${\epsilon }_i$ on the dynamics of typhoid fever. The figure shows that an increase in vaccine efficacy decreases typhoid fever infected humans in the entire community. Hence, considering a vaccination with a very high efficacy (say 99% as we have in Figure 2) will results in faster disease eradication in the two socio-economic classes if the vaccine is applied uniformly in the entire populace.

Typhoid fever can be treated with appropriate antibiotic medicine [1]. Treatment of infected individuals is one of the effective control measures for reducing typhoid fever infections. Figure 3 is a graphical illustration of the effect of treatment rate in decreasing typhoid fever. The figure illustrates that an increase in treatment rate results in to decrease in typhoid fever in both socio-economic classes. Based on this, effective treatment of infected humans is recommended in the entire population.

Contaminated food and environment are one of the major routes of contracting typhoid fever [1]. So, to reduce typhoid fever infections, sanitation should be maintained in society. Figure 4 is a graphical representation of the impact of sanitation ${\theta }_i$ on the dynamics of typhoid fever. From the figure, we observe that an increase in sanitation results in a decrease in typhoid fever infected humans. The effects of sanitation are less in the higher SEC 2. A possible explanation for this could be because the higher SEC 2 have a certain level of sanitation in their environment, so introducing what is already in existence in their environment will not lead to major results unlike in the lower SEC 1 that have limited access to sanitation. Based on these, the lower SEC 1 should be the target of sanitation for maximum results.

Multiple control measures in this study are the situation when different possible control measures are introduced simultaneously in fighting a particular disease. In this study, we have discussed three possible control measures that can be used in fighting typhoid fever outbreaks. Figure 5 describes the effects of introducing these three control measures in fighting typhoid fever. The figure shows that using multiple control measures has maximum influence in decreasing the infected population (in both socio-economic classes) when compared with any of the single control measures. Therefore, whenever a typhoid fever outbreak occurs, multiple control measures should be considered for the faster eradication of the disease.

Qualitative and numerical analysis of our model showed that implementing multiple control measures as aforementioned plays major roles in reducing the influence of Typhoid fever among humanity. Here, we intend to carryout optimal control analysis to determine the most effective control strategy for minimizing the number of humans affected by Typhoid fever along different socioeconomic classes. To minimize the cost of implementing the controls. we assume that the control parameters ${\varphi }_i$, ${\sigma }_i$ and ${\theta }_i$ denoting vaccination, treatment and sanitation, respectively are measurable functions of time and then we formulate an appropriate optimal control function that minimizes the cost of implementing the controls subject to the model (1). For simplicity, we write the control strategies as control functions given as ${\varphi }_i=u_i\left(t\right)$, ${\sigma }_i=v_i\left(t\right)$ and ${\theta }_i=w_i$ which are bounded, lebesgue integrable functions. Given the above, we now write the optimal control model as subject to the initial conditions $S_1\left(0\right)=S_1^0$, $V_1\left(0\right)=V_1^0$, $I_1=I_1^0$, $T_1\left(0\right)=T_1^0$, $R_1\left(0\right)=R_1^0$, $S_2\left(0\right)=S_2^0$, $V_2\left(0\right)=V_2^0$, $I_2\left(0\right)=I_2^0$, $T_2\left(0\right)=T_2^0$, $R_2\left(0\right)=R_2^0$.

This implies that the optimal control model is said to be optimal if it minimizes the objective functional subject to the model (5), where the coefficients $A_i$, $B_i$, $C_i$, $d_i$, $e_i$ and $f_i$ are cost balancing coefficients that transform the integral into money expended over time T. Here, $A_i$, is the direct cost associated with reducing the number of susceptibility to disease in each SEC, $B_i$ is the direct cost associated with reducing the number of infected humans in each SEC and $C_i$ is the direct cost associated with reducing the number of bacteria in the environment, while $d_i$, $e_i$ and $f_i$ are relative costs for enforcing the control strategies $u_i$, $v_i$, $w_i$. The goal is to minimise the number of humans susceptible to Typhoid fever among different SECs, minimize the infectious humans in all SECs and minimize the bacteria that causes Typhoid. In doing this, we anticipate nonlinear costs arising from these controls and so we consider quadratic functions for measuring the control costs [19,27,28,29,30,31].

The goal is to determine an optimal control $u_i$, $v_i$ and $w_i$ such that where $\mathrm{\Omega }=\{u_i\left(t\right),w_i\left(t\right),w_i\left(t\right)|0\le u_i\left(t\right),v_i\left(t\right),w_i\left(t\right)\le 1$ are measurable}.

The Pontryagins Maximum Principle [36] introduces adjoint functions that enable us to combine the state system to the objective functional. With the Pontryagins principle we can convert the problem of minimizing the objective functional to the state system into a problem that involves minimizing a Hamiltonian H, with respect to $u_i\left(t\right)$, $v_i\left(t\right)$ and $w_i\left(t\right)$. From the idea above we now have the Hamiltonian for the objective functional and the state system given as where ${\lambda }_{S1}$, ${\lambda }_{V1}$, ${\lambda }_{I1}$, ${\lambda }_{T1}$, ${\lambda }_{R1}$, ${\lambda }_{P1}$, ${\lambda }_{S2}$, ${\lambda }_{V2}$, ${\lambda }_{I2}$, ${\lambda }_{T2}$, ${\lambda }_{R2}$, and ${\lambda }_{P2}$ are associated adjoint for the states $S_i$, $V_i$, $I_i$, $T_i$, $R_i$ and $P_i$, respectively. Given an optimal control triple ($u_i^{*}\left(t\right)$, $v_i^{*}\left(t\right)$, $w_i^{*}\left(t\right)$) together with corresponding states ($S_i^{*}$, $V_i^{*}$, $I_i^{*}$, $T_i^{*}$, $R_i^{*}$, $P_i^{*}$) that minimizes $\mathcal{J}(u_i,v_i,w_i)$ over $\mathrm{\Omega }$, there exits adjoint variables ${\lambda }_{Si}$, ${\lambda }_{Vi}$, ${\lambda }_{Ii}$, ${\lambda }_{Ti}$, ${\lambda }_{Ri}$, and ${\lambda }_{Pi}$$(i=1,2)$ that satisfies together with the transversality conditions ${\lambda }_k\left(t_f\right)=0$, for $k=S_i,V_i,I_i,T_i,R_i$ and $P_i$.

Note that we get the differential equation (8) which governs the adjoint variables by differentiating the appropriate Hamiltonian function (8) with respect to the corresponding state as follows:

Now, consider the optimality conditions So for the control triplet $u_i^{*}$, $v_i^{*}$ and $w_i^{*}$ to satisfy the optimality condition we have;

For $u_i$ we have The solving for $u_i$ using the optimality condition (10), we have and subsequently taking bounds into consideration, we have

Solving for $v_i$ using the optimality condition, we have and subsequently taking bounds into consideration, we have Similarly for $w_i$, we have

The results obtained above, shows that the optimal triple $(u_i^{*},v_i^{*},w_i^{*})$ has the tendency of reducing the impact of typhoid fever in any human population, if there is application of the disease control measures at a minimum cost. The optimal control triple is parameter dependent, hence further analysis is to be carried out to determine the exact effect of these optimal control parameters in eradicating these diseases and its magnitude of reduction taking into consideration cost of application. This is done using numerical simulation to give pictorial view of this impact using published data in similar peer reviewed works.