Optimal control and cost-effectiveness analysis for the human melioidosis model

Abstract

In this work, we formulated and investigated an optimal control problem of the melioidosis epidemic to explain the effectiveness of time-dependent control functions in controlling the spread of the epidemic. The basic reproduction number $(R_{0c})$ with control measures is obtained, using the next-generation matrix approach and the impact of the controls on $R_{0c}$ is illustrated numerically. The optimal control problem is analyzed using Pontryagin's maximum principle to derive the optimality system. The optimality system is simulated using the forward-backward sweep method based on the fourth-order Runge-Kutta method in the MATLAB program to illustrate the impact of all the possible combinations of the control interventions on the transmission dynamics of the disease. The numerical results indicate that among strategies considered, strategy C is shown to be the most effective in reducing the number of infectious classes compared to both strategy A and strategy B. Furthermore, we carried out a cost-effectiveness analysis to determine the most cost-effective strategy and the result indicated that the strategy B (treatment control strategy) should be recommended to mitigate the spread and impact of the disease regarding the costs of the strategies.

1. Introduction

Melioidosis is an emerging infectious disease caused by the environmental gram-negative bacterium called Burkholderia pseudomallei [4], [14], [30]. This bacterium is commonly found in moist soil and surface water in tropical and subtropical regions, particularly in Southeast Asia, Northern Australia and sub-Saharan Africa [10], [21], [28]. The disease can affect both humans and animals. Humans acquire infection through ingestion, inoculation and inhalation of the bacteria from contaminated soil or water [10], [11], [28]. Human-to-human transmission is usually uncommon, with only a small number of suspected cases documented to date, which occurred through contact with the reproductive fluid of an infected human or breastfeeding from an infected mother [7], [9], [14], [36], [41]. According to recent studies, B. pseudomallei causes approximately 165,000 human infections and 89,000 (54%) fatalities per year throughout the world [14], [27], [47].

Melioidosis is a serious infection in humans and it can cause a wide range of symptoms, from mild flu-like illness to severe sepsis and pneumonia. The symptoms may include fever, headache, muscle aches, joint pain, cough, chest pain, difficulty breathing, and skin lesions. In severe cases, the infection can lead to septic shock, multiple organ failure, and death [9], [13], [14], [22].

Subsequent studies have indicated the effectiveness of a two-stage antibiotic treatment for melioidosis, consisting of an initial phase (the acute phase) of intravenous antibiotics followed by a subsequent phase (the eradication phase) of oral drug. In the acute phase, the current antibiotic options are ceftazidime and carbapenem, and the treatment with these antibiotics should continue for a duration longer than 2 weeks. While oral trimethoprim-sulfamethoxazole (TMP-SMX) is the antibiotic choice during the subsequent eradication phase for 13-26 weeks [10], [12], [16], [41]. However, B. pseudomallei species are intrinsically resistant to several classes of antimicrobial agents and some isolates are also resistant to the two-phase antibiotics [16], [39]. In addition, the disease infection relapse varies from 13% to 23% in patients [39]. The reasons for infection relapse are antimicrobial resistance (severity of B. pseudomallei), inadequate or incomplete antimicrobial agents and improper eradication therapy among others [23], [39]. To date, there is no vaccine for human melioidosis [29]. Prevention measures involve avoiding contact with contaminated soil or water, wearing protective clothing and footwear when working in high-risk environments, treating drinking water and practising good personal hygiene [8], [41].

Mathematical modeling has shown its significant role in the study of epidemiology by providing more insight into the underlying mechanisms for preventing and controlling infectious diseases [19], [20]. Nowadays, optimal control theory plays a critical role in showcasing the effectiveness of various optimal control interventions for a given mathematical model through the incorporation of suitable control measures [26], [40]. A few researchers have proposed autonomous compartmental mathematical models to address the transmission dynamics of the disease, see literature [29], [45]. The authors in [29] presented a susceptible-exposed-infected-recovered-susceptible (SEIRS) deterministic model to show key factors of the disease incidence pattern in Thailand. The authors in [45] proposed a susceptible-exposed-infected-recovered (SEIR) model of the melioidosis epidemic to address the dynamics behavior of the disease in the human population. Also, the author in [44] considered a SEIR compartmental model to achieve the effectiveness of hygiene care and treatment control factors on the transmission process of melioidosis. Recently, the authors in [14] developed a susceptible-exposed-asymptomatic infectious-symptomatic infectious-recovered-susceptible (SEAIRS) deterministic model that describes the transmission dynamics of human melioidosis with an asymptomatic class. They presented a detailed stability analysis of the steady states of their model qualitatively and quantitatively. They also demonstrated the impact of model parameters on disease dynamics. They further carried out numerical experiments to support their theoretical results.

In this paper, we propose an optimal control problem for the dynamics of human melioidosis, which is a continuation of the research started in [14]. This work aims to demonstrate the effectiveness and cost-effectiveness of optimal control interventions in controlling the spreading of the epidemic.

The rest of the paper is divided into the following Sections: we describe the basic model from the study [14] in Section 2. The formulation of an optimal control problem and characterization of the optimal control problem are described in Section 3. Numerical simulations of different control interventions and their cost-effectiveness analysis are carried out in Section 4. The concluding results are given in Section 5.

2. The autonomous model

The state system is an autonomous system of non-linear ordinary differential equations from the model formulated in [14]. The model divides the total human (host) population at time t, into five distinct compartments: susceptible class ($S(t)$), latently-infected class ($E(t)$), asymptomatic class (infectious individuals without symptoms and can transmit the disease) ($A(t)$), symptomatic class (infectious individuals that are showing symptoms and can transmit the disease to others) ($I(t)$), and recovered class ($R(t)$), where, the total population is given by $N(t)=S(t)+E(t)+A(t)+I(t)+R(t)$. The concentration of B. pseudomallei in the environment at time t, is denoted by $B_m(t)$. The susceptible class is assumed to increase due to newborns at a constant recruitment rate given by Π. Susceptible humans may be infected through either contact with bacterial class at the rate ${\beta }_1\frac{B_m}{C+B_m}$, or contact with infectious classes $(A(t)\mathrm{\&}I(t))$ at the rate ${\beta }_2(I+\sigma A)$, where ${\beta }_1$ and ${\beta }_2$, respectively, are transmission rates due to the pathogen and infectious individuals, C is the constant pathogen concentration in the environment that yields 50% chance of catching the disease, and σ is the reduction rate in infectivity of A with, $0<\sigma <1$. The number of susceptible humans infected by B. pseudomallei due to the contaminated environment per unit time is given by ${\beta }_1\frac{B_m}{C+B_m}S$, while the number of susceptible humans infected by B. pseudomallei due to $A(t)$ and $I(t)$ per unit time is given by ${\beta }_2(I+\sigma A)S$. Hence, the number of susceptible humans infected by melioidosis per unit of time is given by $({\beta }_1\frac{B_m}{C+B_m}+{\beta }_2(I+\sigma A))S$ and joins E. The rate at which individuals in E progress to I is denoted by θ, while the individuals in E progress to A at the rate of ($1-\theta$). The shedding rate of B. pseudomallei into the environment due to A and I is given by η. The respective human and bacterial natural death rates are represented by μ and ${\mu }_b$. The disease-induced death rate is denoted as δ. The recovery rate from A is denoted by ${\gamma }_1$ while from I is denoted by ${\gamma }_2$. The individuals in R progress to S at the rate of α. Moreover, the description, values and sources of the model parameters are given in Table 1. The autonomous system of the model from [14] is given as

$$\{\begin{array}{cc}\hfill \frac{dS}{dt}& =\mathrm{\Pi }+\alpha R-(\lambda +\mu )S,\hfill \\ \hfill \frac{dE}{dt}& =\lambda S-(\rho +\mu )E,\hfill \\ \hfill \frac{dA}{dt}& =(1-\theta )\rho E-(\delta +{\gamma }_1+\mu )A,\hfill \\ \hfill \frac{dI}{dt}& =\theta \rho E-(\delta +{\gamma }_2+\mu )I,\hfill \\ \hfill \frac{dR}{dt}& ={\gamma }_{1}A+{\gamma }_{2}I-(\alpha +\mu )R,\hfill \\ \hfill \frac{d{B}_m}{dt}& =\eta (A+I)-{\mu }_b{B}_m,\hfill \end{array}$$ (1)

where, λ = ${\beta }_1\frac{B_m}{C+B_m}+{\beta }_2(I+\sigma A)$, with the initial conditions;

$$S(0)=S_0>0,E(0)=E_0>0,A(0)=A_0>0,I(0)=I_0>0,R(0)=R_0\ge 0,B_m(0)=B_{m,0}>0.$$

Table 1.

Values and description of parameters of the model.

Parameter	Description	Value	Unit	Source
Π	Human recruitment rate	μ × N0	Humans day−1	[14], [45]
β1	Human transmission rate due to pathogen	0.0999	Day−1	Assumed
β2	Human transmission rate due to A & I	0.0004	(Humans day)−1	Assumed
θ	Probability of progress of E to I	0.5125	Dimensionless	Assumed
μ	Natural death rate of humans	$\frac{1}{65\times 365}$	Day−1	[45]
δ	Disease-induced death rate of A & I	0.0732	Day−1	[14], [45]
γ1	Recovery rate from A	0.0248	Day−1	[14], [29]
γ2	Recovery rate from I	0.0157	Day−1	[14], [29]
ρ	The progression rate of E to A & I	0.088	Day−1	[14], [44]
σ	Reduction rate of infectivity A	0.0493	Dimensionless	Assumed
α	Disease waning immunity	0.0726	Day−1	Assumed
η	Rate at which bacteria increase by A & I	0.13	$\frac{\text{No. of}\mathit{\text{B. pseudomallei}}\text{cell}}{\text{Humans day}}$	[44]
μb	Natural death rate of bacteria	0.0185	Day−1	[14], [44]
C	Concentration of B. pseudomallei	5000	No. of B. pseudomallei cell	Assumed

For the model equation (1), stability analysis and sensitivity analysis were carried out, and the non-negativity and boundedness of the solution components of such system was established in [14]. Also, the basic reproduction number ($R_0$) of the model is given as (see, in [14])

$$R_0=\mathrm{\Pi }\rho \left(\frac{{\beta }_1\eta (\theta {ϵ}_1+(1-\theta ){\epsilon }_2)+C{\mu }_b{\beta }_2(\theta {ϵ}_1+\sigma (1-\theta ){\epsilon }_2)}{(\rho +\mu )C\mu {\mu }_b{ϵ}_1{ϵ}_2}\right)=R_{0h}+R_{0b}$$

where,

$$R_{0h}=\frac{\mathrm{\Pi }\rho {\beta }_2[\theta (\delta +{\gamma }_1+\mu )+\sigma (1-\theta )(\delta +{\gamma }_2+\mu )]}{\mu (\rho +\mu )(\delta +{\gamma }_1+\mu )(\delta +{\gamma }_2+\mu )},$$

is the basic reproduction number due to human-to-human transmission, and

$$R_{0b}=\frac{\mathrm{\Pi }\rho {\beta }_1[\theta (\delta +{\gamma }_1+\mu )+(1-\theta )(\delta +{\gamma }_2+\mu )]}{C\mu {\mu }_b(\rho +\mu )(\delta +{\gamma }_1+\mu )(\delta +{\gamma }_2+\mu )},$$

is the basic reproduction number due to environment-to-human transmission. $R_0$ measures the average number of secondary infectious individuals generated by a single infected person (asymptomatic or symptomatic) in completely susceptible humans.

By using the approach given in [14] and the parameter values from Table 1, the most influencing parameter of the model are human recruitment rate (Π) followed by the transmission rate due to infectious humans $({\beta }_2)$, the transmission rate due to the environment $({\beta }_1)$ mortality rate of bacteria $({\mu }_b)$ and recovery rates $({\gamma }_2\mathrm{\&}{\gamma }_1)$, respectively. This indicates that a personal preventive intervention for susceptible humans and control intervention of the transmission rate $({\beta }_1)$ will sufficiently diminish the spread of the disease. Also, control strategies that increase the natural mortality rates of bacteria or a strategy that increases the recovery rates of infectious humans would be effective in diminishing the spread of the epidemic.

This paper transforms an autonomous mathematical model without controls into a non-autonomous model with controls by incorporating two bounded control functions of time, which affect the value of $R_0$ and the effect of these control functions on it is demonstrated in section 3.1. Therefore, to use optimal efforts on interventions to eliminate the disease from the infected population, an optimal control problem of melioidosis with cost-effectiveness analysis is discussed in this study.

3. Analysis of the optimal control model

In this section, we formulate an optimal control problem to investigate the effect of two time-dependent preventive and control measures on the transmission dynamics of melioidosis. These time-dependent measures are introduced at a specified time $t\in [0,T]$ as follows, where T is the final time;

• (i):$u_1(t)$ the preventive measure applied on susceptible class: using appropriate personal protective equipment (wearing rubber boots, rubber gloves or waterproof dressings to cover wounds or skin) and using treated water for drinking; this is to minimize the acquisition of the infection in the population.
• (ii):$u_2(t)$ the treatment control measure implemented on infectious classes; this is to reduce the number of infectious individuals in the population. The disease infection can be treated by intravenous antibiotics including ceftazidime, imipenem, or meropenem for a minimum of 10-14 days, followed by oral trimethoprim-sulfamethoxazole (TMP-SMX) drug for 3-6 months [16], [29], [39].

The melioidosis non-autonomous model with the time-dependent measures is formulated as;

$$\{\begin{array}{cc}\hfill \frac{dS}{dt}& =\mathrm{\Pi }+\alpha R-((1-u_1){\beta }_1\frac{B_m}{C+B_m}+{\beta }_2(I+\sigma A)+\mu )S,\hfill \\ \hfill \frac{dE}{dt}& =(1-u_1){\beta }_1\frac{B_m}{C+B_m}S+{\beta }_2(I+\sigma A)S-(\rho +\mu )E,\hfill \\ \hfill \frac{dA}{dt}& =(1-\theta )\rho E-(\delta +{\gamma }_1+\xi u_2+\mu )A,\hfill \\ \hfill \frac{dI}{dt}& =\theta \rho E-(\delta +{\gamma }_2+\xi u_2+\mu )I,\hfill \\ \hfill \frac{dR}{dt}& =({\gamma }_1+\xi u_2)A+({\gamma }_2+\xi u_2)I-(\alpha +\mu )R,\hfill \\ \hfill \frac{d{B}_m}{dt}& =\eta (A+I)-{\mu }_b{B}_m.\hfill \end{array}$$ (2)

The constant coefficient ξ is the control rate of treatment. The incidence rate of infection due environment-to-human interaction diminished by the factors $(1-u_1(t))$ and infectious classes is minimized at the rate of $\xi u_2(t)$. Next, we need to formulate an appropriate objective function for the state system (2). This study is aimed to reduce the size of infectious classes and the associated cost of the interventions over the specified time interval. Thus, the form of the objective functional, J, to be minimized is given by:

$$J(u_1,u_2)=\underset{0}{\overset{T}{\int }}(w_{1}A+w_{2}I+\frac{w_3}{2}{u}_1^2+\frac{w_4}{2}{u}_2^2)dt$$ (3)

subject to the model equation (2). We consider the quadratic forms of the cost control functions in the objective functional $J(u_1,u_2)$. The nonlinear nature of the cost of intervention arises based on the fact that there is no linear relationship between the effects of interventions and their costs in the infected population. See for more details in some related literature [3], [24], [33], [35], [42]. $w_k$ are the positive balancing weight constants, for k = 1, 2. The terms $w_{1}A$ and $w_{2}I$, respectively, in the objective cost functional J are the costs associated with asymptomatic infectious individuals and symptomatic infectious individuals that need to be minimized. The expressions $\frac{w_3}{2}{u}_1^2$ and $\frac{w_4}{2}{u}_2^2$ represent the cost associated with prevention and treatment control measures, respectively. Since the formulated optimal control problem is aimed to minimize the number of infectious individuals under minimum cost, we seek the optimal control pair $u_1^{⁎}(t)$ and $u_2^{⁎}(t))$ such that

$$J(u_1^{⁎},u_2^{⁎})=\underset{u_k\in U}{\min }\{J(u_1,u_2)\},\text{where}k=1,2,$$ (4)

where, U is a non-empty bounded admissible control set described by

$$U=\{(u_1(t),u_2(t)):0\le u_k\le 1,\text{each}{u}_k\text{is Lebesgue measurable},\text{for}k=1,2,t\in [0,T]\}.$$

3.1. The effects of the control variables ($u_1\mathrm{\&}{u}_2$) on basic reproduction number of the system (2)

This section presents the effects of the control variables, $u_1$ and $u_2$, on the basic reproduction number (denoted by $R_{0c}$) of the non-autonomous system (2). To obtain $R_{0c}$ of the system (2), the next-generation matrix approach [46] is used as follows. The associated Jacobian matrices of the new infection terms F and the rate of transfer of individuals to the compartments V are given by:

$$F=\left(\begin{array}{cccc}\hfill 0\hfill & \hfill \frac{{\beta }_2\sigma \mathrm{\Pi }}{\mu }\hfill & \hfill \frac{{\beta }_2\mathrm{\Pi }}{\mu }\hfill & \hfill (1-u_1)\frac{{\beta }_1\mathrm{\Pi }}{C\mu }\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\text{and}V=\left(\begin{array}{cccc}\hfill \rho +\mu \hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -(1-\theta )\rho \hfill & \hfill (\delta +{\gamma }_1+\xi u_2+\mu )\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\theta \rho \hfill & \hfill 0\hfill & \hfill (\delta +{\gamma }_2+\xi u_2+\mu )\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -\eta \hfill & \hfill -\eta \hfill & \hfill {\mu }_b\hfill \end{array}\right).$$

Thus, $R_{0c}$ of the system (2) given by

$$R_{0c}=\rho (F{V}^{-1})=\mathrm{\Pi }\rho \left(\frac{(1-u_1){\beta }_1\eta (\theta {ϵ}_1+(1-\theta ){\epsilon }_2)+C{\mu }_b{\beta }_2(\theta {ϵ}_1+\sigma (1-\theta ){\epsilon }_2)}{(\rho +\mu )C\mu {\mu }_b{ϵ}_1{ϵ}_2}\right),$$

where, ${ϵ}_1=\delta +{\gamma }_1+\xi u_2+\mu ,{ϵ}_2=\delta +{\gamma }_2+\xi u_2+\mu$.

The variation of $R_{0c}$ with respect to the controls is depicted in surface plot (see, Fig. 1) for $0\le u_1\le 1,0\le u_2\le 1$. A similar approach can be found in [25]. The Fig. 1 indicates that the highest value of $R_{0c}$ is achieved when both control variables are at their minimum value, $u_1=u_2=0$. This value, $R_{0c}=2.0326$, corresponds to the basic reproduction number $(R_0)$ of autonomous system (1). Another higher value of $R_{0c}$ ($1.29914>1$) is obtained when $u_1=1$ and $u_2=0$, which suggests that implementing the preventive measure $u_1$ alone is not effective in eradicating the melioidosis epidemic. On the other hand, when both controls, $u_1$ and $u_2$ are set to their maximum values (i.e. full efforts), the minimum value of $R_{0c}$ is achieved, which is 0.13239 as depicted in Fig. 1. This indicates that the combination of both control measures is highly effective in reducing $R_{0c}$. The results suggest that the control strategy involving the use of both $u_1$ and $u_2$ can lead to the complete elimination of the disease from the infected population. Moreover, when only $u_2$ is used at full effort ($u_2=1$) while $u_1$ is absent ($u_1=0$), a lower value of $R_{0c}$ is observed, which is 0.219461. This value is less than unity, suggesting that the treatment control measure $u_2$ alone is also effective for eradicating the disease. Consequently, a strategy of the combination of both control measures ($u_1$ and $u_2$) as well as a strategy involving treatment control only would be effective in reducing the spread of the disease within the community. Furthermore, the effectiveness and cost-effectiveness of these control functions are discussed in the numerical simulation section.

Figure 1.

Plot showing the effects of u₁ & u₂ on R_0c for 0 ≤ u₁ ≤ 1,0 ≤ u₂ ≤ 1. The parameter values given in Table 1 are used.

3.2. Existence of an optimal control

Theorem 3.1

Given the objective functional J, defined on the control set U, and subject to the non-autonomous system (2) , then there exists an optimal control pair $u=(u_1^{⁎},u_2^{⁎})$ such that the equation (4) holds, provided that the following conditions are hold [15], [17], [32], [34] :

• (i) The control set is closed and convex.
• (ii) The right hand-side of the state system (2) is bounded by a linear function in the state and control variables.
• (iii) The integrand of the objective functional in the equation (3) is convex with respect to the controls.
• (iv)
  There exist constants ${\alpha }_1,{\alpha }_2>0$ and ${\alpha }_3>1$ such that the integrand of the objective functional is bounded below by

  $${\alpha }_1{(\sum _{k=1}^2|u_k|)}^{\frac{{\alpha }_3}{2}}-{\alpha }_2.$$

Proof

We need to verify the four hypotheses stated by Theorem 3.1.

(i) Given the control set $U_0=\{(u_1,u_2):0\le u_1\le 1,0\le u_2\le 1\}$. Then, $U_0$ is closed by definition. Further, let a and b be any two arbitrary points in $U_0$, such that $a=(a_1,a_2)$ and $b=(b_1,b_2)$. From the concept of convex set [43], it follows that

$$\varphi a_k+(1-\varphi )b_k\in [0,1],\forall \varphi \in [0,1],k=1,2.$$

This implies, $\varphi a+(1-\varphi )b\in U_0$. Thus, $U_0$ is convex set.

(ii) Let $x=(S,E,A,I,R,B_m)$ be the state variables of the model, $u=(u_1,u_2)\in U_0$ and $f(t,x,u)$ be the right-hand side of the system (2) given by

$$f(t,x,u)=\left[\begin{array}{c}\hfill \mathrm{\Pi }+\alpha R-((1-u_1){\beta }_1\frac{B_m}{C+B_m}+{\beta }_2(I+\sigma A)+\mu )S\hfill \\ \hfill (1-u_1){\beta }_1\frac{B_m}{C+B_m}S+{\beta }_2(I+\sigma A)S-(\rho +\mu )E\hfill \\ \hfill (1-\theta )\rho E-(\delta +{\gamma }_1+\xi u_2+\mu )A\hfill \\ \hfill \theta \rho E-(\delta +{\gamma }_2+\xi u_2+\mu )I\hfill \\ \hfill ({\gamma }_1+\xi u_2)A+({\gamma }_2+\xi u_2)I-(\alpha +\mu )R\hfill \\ \hfill \eta (A+I)-{\mu }_b{B}_m\hfill \end{array}\right].$$ (5)

Then, the equation (5) can be written as $f(t,x,u)=f_1(t,x)+f_2(t,x)u$, where

$$f_1(t,x)=\left[\begin{array}{c}\hfill \mathrm{\Pi }+\alpha R-({\beta }_1\frac{B_m}{C+B_m}+{\beta }_2(I+\sigma A)+\mu )S\hfill \\ \hfill ({\beta }_1\frac{B_m}{C+B_m}+{\beta }_2(I+\sigma A))S-(\rho +\mu )E\hfill \\ \hfill (1-\theta )\rho E-(\delta +{\gamma }_1+\mu )A\hfill \\ \hfill \theta \rho E-(\delta +{\gamma }_2+\mu )I\hfill \\ \hfill {\gamma }_{1}A+{\gamma }_{2}I-(\alpha +\mu )R\hfill \\ \hfill \eta (A+I)-{\mu }_b{B}_m\hfill \end{array}\right],f_2(t,x)=\left[\begin{array}{cc}\hfill {\beta }_1\frac{B_m}{C+B_m}S\hfill & \hfill 0\hfill \\ \hfill -{\beta }_1\frac{B_m}{C+B_m}S\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -\xi A\hfill \\ \hfill 0\hfill & \hfill -\xi I\hfill \\ \hfill 0\hfill & \hfill \xi (A+I)\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right].$$

From Euclidean norm of a matrix in [6], [38], we obtain,

$$\parallel f(t,x,u)\parallel \le \parallel f_1(t,x)\parallel +\parallel f_2(t,x)\parallel \parallel u\parallel \le \text{max}\{C_0,D\}(1+\parallel u\parallel ),$$

where $C_0$ and D are positive constants given by

$$C_0=\frac{\mathrm{\Pi }}{\mu }\sqrt{c_1(1+\mathrm{\Pi }+{\mathrm{\Pi }}^2)},\text{and}D=\frac{\mathrm{\Pi }}{\mu }\sqrt{\text{max}\{d_1,d_2,d_3\}(1+\mathrm{\Pi }+{\mathrm{\Pi }}^2)}.$$

With

$$c_1=2({\beta }_1^2+3{\xi }^2),d_1={\beta }_1^2+{(\alpha +\mu )}^2+{(\rho -\rho \theta )}^2+{\rho }^2{\theta }^2+{({\gamma }_1+{\gamma }_2)}^2+4\eta ,d_2=\frac{2{\beta }_1{\beta }_2(1+\sigma )}{\mu },d_3=\frac{{\beta }_2^2{(1+\sigma )}^2}{{\mu }^2}.$$

(iii) Let $y=(A,I)$ and $a,b\in U_0$, such that $a=(a_1,a_2)$ and $b=(b_1,b_2)$. The integrand of the objective functional in the equation (3) is the Lagrangian of the form denoted by $L(t,y,u)$ defined as

$$L(t,y,u)=L_1(t,y)+L_2(t,u),$$ (6)

where, $L_1(t,y)=w_{1}A+w_{2}I,L_2(t,u)=\frac{w_3}{2}{u}_1^2+\frac{w_4}{2}{u}_2^2$. It suffices to show that the function $L_2(t,u)$ is convex on the control set $U_0$. Thus, we need to show that

$$L(t,y,\varphi a+(1-\varphi )b)\le \varphi L(t,y,a)+(1-\varphi )L(t,y,b),\forall \varphi \in [0,1].$$

The equation (6) gives,

$$L(t,y,\varphi a+(1-\varphi )b)=L_1(t,y)+L_2(t,\varphi a+(1-\varphi )b),=L_1(t,y)+\frac{1}{2}\sum _{k=1}^2{w}_k{(\varphi a_k+(1-\varphi )b_k)}^2,=L_1(t,y)+\frac{1}{2}{\varphi }^2\sum _{k=1}^2{w}_k{a}_k^2+(1-\varphi )\varphi \sum _{k=1}^2{w}_k{a}_k{b}_k+\frac{1}{2}(1-{\varphi }^2)\sum _{k=1}^2{w}_k{a}_k^2,$$

and

$$\varphi L(t,y,a)+L(t,y,(1-\varphi )b)=\varphi (L_1(t,y)+L_2(t,a))+(1-\varphi )(L_1(t,y)+L_2(t,b)),=L_1(t,y)+\frac{1}{2}\varphi \sum _{k=1}^2{w}_k{a}_k^2+\frac{1}{2}(1-\varphi )\sum _{k=1}^2{w}_k{b}_k^2.$$

Since ${\varphi }^2\le \varphi \forall \varphi \in [0,1]$, it follows that

$$L(t,y,\varphi a+(1-\varphi )b)-(\varphi L(t,y,a)+(1-\varphi )L(t,y,b))=\frac{1}{2}({\varphi }^2-\varphi )\sum _{k=1}^2{w}_k{(a_k-b_k)}^2\le 0.$$

As a result, the function $L_2(t,u)$ is convex on $U_0$.

(iv) Using the equation (6), the last hypothesis is shown as follows:

$$L(t,y,u)=L_1(t,y)+\frac{1}{2}\sum _{k=1}^2{w}_k{u}_k^2\ge \frac{1}{2}\sum _{k=1}^2{w}_k{u}_k^2\ge {\alpha }_1{(\sum _{k=1}^2|u_k{|}^2)}^{\frac{{\alpha }_3}{2}}-{\alpha }_2,$$

where, ${\alpha }_1=\frac{1}{2}\text{min}\{w_3,w_4\},{\alpha }_2>0$, and, ${\alpha }_3=2$.

3.3. Characterization of the optimal controls

To formulate an optimality system we need to generate the necessary conditions that the optimal control doublet and state must satisfy. Such conditions are generated from the Pontryagin's maximum principle (PMP) [37]. This principle converts the system (2) together with (3) into a problem of minimizing point-wise, with respect to controls $u_1$ and $u_2$, a Hamiltonian function, H, which is defined by

$$H=w_{1}A+w_{2}I+\frac{1}{2}\sum _{k=1}^2{w}_k{u}_k^2+{\lambda }_1[\mathrm{\Pi }+\alpha R-((1-u_1){\beta }_1\frac{B_m}{C+B_m}+{\beta }_2(I+\sigma A)+\mu )S]+{\lambda }_2[(1-u_1){\beta }_1\frac{B_m}{C+B_m}S+{\beta }_2(I+\sigma A)S-(\rho +\mu )E]+{\lambda }_3[(1-\theta )\rho E-(\delta +{\gamma }_1+\xi u_2+\mu )A]+{\lambda }_4[\theta \rho E-(\delta +{\gamma }_2+\xi u_2+\mu )I]+{\lambda }_5[({\gamma }_1+\xi u_2)A+({\gamma }_2+\xi u_2)I-(\alpha +\mu )R]+{\lambda }_6[\eta (A+I)-{\mu }_b{B}_m],$$ (7)

where, ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5$ and ${\lambda }_6$ are the adjoint variables associated with their respective state variables. The PMP [37] and the existence of the optimal control from [see Theorem 4.1, [17]] can be used to obtain the following theorem.

Theorem 3.2

If${\mathrm{\Phi }}^{⁎}=(S^{⁎},E^{⁎},A^{⁎},I^{⁎},R^{⁎},B_m^{⁎})$and$u^{⁎}=(u_1^{⁎},u_2^{⁎})$are optimal state and optimal control solutions for the optimal problem(3), respectively, then, there exist six adjoint variables${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5$and${\lambda }_6$that satisfy the adjoint system given by

$$\frac{d{\lambda }_1}{dt}=({\lambda }_1-{\lambda }_2)((1-u_1){\beta }_1\frac{B_m}{C+B_m}+{\beta }_2(I+\sigma A))+{\lambda }_1\mu ,\frac{d{\lambda }_2}{dt}={\lambda }_2(\rho +\mu )-{\lambda }_3(1-\theta )\rho -{\lambda }_4\theta \rho ,\frac{d{\lambda }_3}{dt}=-w_1+({\lambda }_1-{\lambda }_2){\beta }_2\sigma S+{\lambda }_3(\delta +{\gamma }_1+\mu +\xi u_2)-{\lambda }_5({\gamma }_1+\xi u_2)-{\lambda }_6\eta ,\frac{d{\lambda }_4}{dt}=-w_2+({\lambda }_1-{\lambda }_2){\beta }_{2}S+{\lambda }_4(\delta +{\gamma }_2+\mu +\xi u_2)-{\lambda }_5({\gamma }_2+\xi u_2)-{\lambda }_6\eta ,\frac{d{\lambda }_5}{dt}=-{\lambda }_1\alpha +{\lambda }_5(\alpha +\mu ),\frac{d{\lambda }_6}{dt}=({\lambda }_1-{\lambda }_2)(1-u_1)\frac{{\beta }_{1}C}{{(C+B_m)}^2}S+{\lambda }_6{\mu }_b,$$ (8)

with transversality conditions,

$${\lambda }_k(T)=0,k=1,2,...,6.$$

Furthermore, the optimal controls$u_1^{⁎}$and$u_2^{⁎}$are characterized by

$$u_1^{⁎}=\mathit{\text{min}}\{1,\mathit{\text{max}}\{\frac{({\lambda }_2-{\lambda }_1){\beta }_{1}S\frac{B_m}{C+B_m}}{w_3},0\left\}\right\},u_2^{⁎}=\mathit{\text{min}}\{1,\mathit{\text{max}}\{\frac{\xi A({\lambda }_3-{\lambda }_5)+\xi I({\lambda }_4-{\lambda }_5)}{w_4},0\left\}\right\}.$$ (9)

Proof

We apply PMP in order to obtain the adjoint relations, the transversality conditions and the optimal control doublet. By taking partial derivatives of the formulated Hamiltonian in the equation (7) with respect to the state variables $S,E,A,I,R$ and $B_m$, as follows; $\frac{d{\lambda }_1}{dt}=-\frac{\partial H}{\partial S},\frac{d{\lambda }_2}{dt}=-\frac{\partial H}{\partial E},\frac{d{\lambda }_3}{dt}=-\frac{\partial H}{\partial A}=,\frac{d{\lambda }_4}{dt}=-\frac{\partial H}{\partial I},\frac{d{\lambda }_5}{dt}=-\frac{\partial H}{\partial R},\frac{d{\lambda }_6}{dt}=-\frac{\partial H}{\partial B_m}$, ${\lambda }_k(T)=0,k=1,2,...,6$, yields the adjoint system given in the equation (8).

Finally, the characterization of optimal controls can be derived from H  in (7) by using the optimal condition for each control measure $u_i$, where $0<u_i<1$, for $i=1,2$. Thus,

$$\frac{\partial H}{\partial u_i}=0,\text{for}i=1,2(\mathbf{\text{optimal condition}})$$ (10)

Solving the equation (10) for optimal controls $u_1^{⁎}$ and $u_2^{⁎}$ we obtain the following characterization.

$$u_1^{⁎}=\frac{({\lambda }_2-{\lambda }_1){\beta }_{1}S\frac{B_m}{C+B_m}}{w_3},u_2^{⁎}=\frac{\xi A({\lambda }_3-{\lambda }_5)+\xi I({\lambda }_4-{\lambda }_5)}{w_4}.$$

Since the two control measures have lower bounds zero and upper bounds 1, we have

$$u_1^{⁎}\in u\Rightarrow u_1^{⁎}=\{\begin{array}{cc}0,\hfill & \text{if}{\mathrm{\Psi }}_1\le 0,\hfill \\ {\mathrm{\Psi }}_1,\hfill & \text{if}0<{\mathrm{\Psi }}_1<1,\hfill \\ 1,\hfill & \text{if}{\mathrm{\Psi }}_1\ge 1.\hfill \end{array}\text{and}{u}_2^{⁎}\in u\Rightarrow u_2^{⁎}=\{\begin{array}{cc}0,\hfill & \text{if}{\mathrm{\Psi }}_2\le 0,\hfill \\ {\mathrm{\Psi }}_2,\hfill & \text{if}0<{\mathrm{\Psi }}_2<1,\hfill \\ 1,\hfill & \text{if}{\mathrm{\Psi }}_2\ge 1,\hfill \end{array}$$

where, ${\mathrm{\Psi }}_1=\frac{({\lambda }_2-{\lambda }_1){\beta }_{1}S\frac{B_m}{C+B_m}}{w_3},{\mathrm{\Psi }}_2=\frac{\xi A({\lambda }_3-{\lambda }_5)+\xi I({\lambda }_4-{\lambda }_5)}{w_4}$. Therefore, in compact notation the optimal controls $u_1^{⁎}$ and $u_2^{⁎}$ given in the equation (9) are characterized by

$$u_1^{⁎}=\text{min}\{1,\text{max}\{{\mathrm{\Psi }}_1,0\}\},u_2^{⁎}=\text{min}\{1,\text{max}\{{\mathrm{\Psi }}_2,0\}\}.$$

4. Numerical results and cost-effectiveness analysis

4.1. Numerical methods

We compare the numerical results of the autonomous model (1) and the controlled model (2) to evaluate the effectiveness of the control strategies in controlling the spread of the disease. To perform this, we used an iterative method so-called Forward-Backward Sweep method (FBSM) based on the fourth-order Runge-Kutta Method (RKM-4) in the Matlab program, described in detail in a book by Lenhart and Workman [26]. The process begins with an initial guess for the control variables, we solve the state equations over the interval [0, 250] using forward RKM-4. Then, we apply the backward RKM-4 to solve the adjoint equations by using the current iteration solution of (2). The control values are updated by averaging the previous value and the new value from the control characterization (9), and the process is repeated until the required convergence occurs. The parameters values used in the simulations are given in Table 1.

For those parameters values the basic reproduction number is obtained as $R_0\approx 2.0326>1$, the unique positive endemic equilibrium ${\epsilon }_{12}^{⁎}=(235.5204,1.6735,0.8452,0.7296,0.4297,13.833)$, with initial conditions of state variables $S(0)=452.09,E(0)=48,A(0)=I(0)=10,R(0)=0,Bm(0)=200$. In addition, the weight constants values and the control rate of treatment are chosen as; $w_1=10,w_2=10,w_3=8,w_4=10,\xi =0.75$. It is important to note that, the choice of treatment control rate ($\xi =0.75$) is based on the melioidosis infection relapse cases, which varies from 13% to 23% in patients [23], [39] (see details in the introduction section of the paper).

4.2. Numerical results

To illustrate the impact of different optimal control strategies for melioidosis epidemic, we implemented all the three possible optimal strategies of alternative combinations of the control measures, $u_1$ and $u_2$:

• •Strategy A: Preventive measure only ($u_1\ne 0,u_2=0$),
• •Strategy B: Treatment of infectious individuals (asymptomatic and symptomatic) ($u_2\ne 0,u_1=0$),
• •Strategy C: Combination of both preventive measure and treatment control $(u_1\ne 0,u_2\ne 0)$.

4.2.1. Strategy A: Preventive measure only ($u_1\ne 0,u_2=0$)

This strategy implements the preventive measure $u_1$ only. As shown, in Fig. 2 (a), the number of susceptible individuals decreases without strategy A compared to the number of susceptible individuals with the strategy. On the other hand, in Fig. 2 (b) the size of latently infected individuals decreases rapidly when the strategy is implemented throughout the intervention period, while in the absence of the strategy the number of latently infected individuals increases rapidly to peak in the first 35 days and then diminishes slowly in rest of time interval. In Fig. 2 (c), it can be seen that the size of infectious individuals (asymptomatic + symptomatic) increases more sharply without preventive measures than the number of infectious individuals with the preventive measure in the first 50 days, while the size of infectious individuals (asymptomatic + symptomatic) decreases more rapidly with preventive measure than the number of infectious individuals without preventive measure in the rest of interventions days. A similar situation is observed in Fig. 2 (d) for Burkholderia pseudomallei population. Further, in Fig. 5 (a), the control profile for the “strategy A” shows that the preventive effort $u_1$ should be implemented at maximum coverage (100%) for the initial 236.8 days. After that period, preventive effort should be gradually reduced to the lower bound for the rest of the simulation time.

Figure 2.

Simulations of the model system (2) showing the effects of the “Strategy A”; (a) dynamics of susceptible class with and without optimal control u₁, (b) dynamics of latently infected class with and without optimal control u₁, (c) dynamics of infectious classes (A + I) with and without optimal control u₁ and (d) dynamics of the bacterial population with and without optimal control u₁.

Figure 5.

Simulations showing the control profiles of the three strategies (A,B & C).

4.2.2. Strategy B: Treatment control only ($u_2\ne 0,u_1=0$)

This strategy is applied to the optimal use of treatment control $u_2$ for infectious individuals. In Fig. 3(a), we observed that the susceptible human population gradually increases in size when control measure $u_2$ is implemented, in contrast, the number of susceptible individuals decreases rapidly in the absence of control measure. From Figs. 3 (b) - 3 (d) we observed that the number of latently infected humans, infectious individuals (asymptomatic + symptomatic) and the size of pathogen population decreased dramatically when the optimal control is implemented throughout the simulation period. This illustrates that the number of infective classes ($E,A,\mathrm{\&}I$), and $B_m$ could be vanished when the strategy is implemented. In general, from Figs. 3 (a) - 3 (d), we observed that this strategy is more effective in reducing the spread of disease in the infected population than strategy A. Furthermore, the control profile of this strategy in Fig. 5 (b) shows that the treatment control $u_2$ should be maintained at the upper bound (100%) for the first 84 days and subsequently it should be gradually reduced to zero (lower bound) for the rest of the simulation time.

Figure 3.

Simulations of the model system (2) showing the effect of the “Strategy B”; (a) dynamics of susceptible class with and without optimal control u₂, (b) dynamics of latently infected class with and without optimal control u₂, (c) dynamics of infectious classes (A & I) with and without optimal control u₂ and (d) dynamics of bacterial population with and without optimal control u₂.

4.2.3. Strategy C: Combination of both preventive and treatment measures ($u_1\ne 0,u_2\ne 0$)

This strategy implements both intervention measures $u_1(t)$ and $u_1(t)$. In Fig. 4 (a), we noted that the population's number of susceptible individuals gradually rises when the optimal strategy C is present. Conversely, in the absence of the strategy, the number of susceptible individuals declines rapidly over the simulation period. In Fig. 4 (b) - 4 (d), we noted that the size of latently infected humans, infectious individuals (asymptomatic + symptomatic), and the size of the pathogen population diminished more rapidly when the strategy C implemented. The size of latently infected and infectious individuals increase sharply to a peak point in the first 40 days and then decrease slowly in the remaining simulation period in the absence of optimal control as shown in Figs. 4 (b) and 4 (c). The same situation is observed in Fig. 4 (d) for the pathogen population. Further, in Fig. 5 (c), the control profile for the “strategy C” indicates the following:

• (i)The preventive effort should be maintained at the upper bound (100%) for the first 126 days.
• (ii)The treatment control should be kept at the upper bound (100%) for the first 28 days.

After the specified durations, both controls should be gradually reduced to the lower bound for the rest of the simulation time.

Figure 4.

Simulations of the model system (2) showing the effect of the “Strategy C”; (a) dynamics of susceptible class with and without combined optimal controls u₁ & u₂, (b) dynamics of latently infected class with and without combined optimal controls u₁ & u₂, (c) dynamics of infectious classes (A & I) with and without combined optimal controls u₁ & u₂ and (d) dynamics of bacterial population with and without combined optimal controls u₁ & u₂.

4.2.4. Comparison of the three control strategies

In this section, we compare the effectiveness of the three strategies A, B and C, to determine the most effective strategy, typically for minimizing the infectious classes. Further, the comparison of the simulations of the model with and without the strategies for populations of susceptible humans, latently infected individuals, infectious individuals ($A+I$) and bacteria are demonstrated in Figs. 6(a) - 6(d). From Fig. 6(a), it can be seen that strategy C is the most effective in reducing the size of the susceptible class getting infected by the disease, followed by strategies B and A, respectively. While in Fig. 6(b) we observed that strategy C is the most effective to reduce the size of the latently infected class compared to strategies A and B. Likewise, Fig. 6(c) shows that strategy C has the highest number of infectious averted humans ($A+I$) followed by strategy B and then strategy A. Moreover, strategy C is most effective in reducing the growth of bacterial population in the environment as indicated in Fig. 6(d).

Figure 6.

Simulations of the model system (2) showing the comparison of the three optimal control Strategies (A,B&C) with the autonomous system (1); (a) the effectiveness of all three strategies with the autonomous system (1) for susceptible humans, (b) the effectiveness of all three strategies with the autonomous system (1) for latently infected individuals, (c) the effectiveness of all three strategies with the autonomous system (1) for infectious individuals (A&I) and (d) the effectiveness of all three strategies with the autonomous system (1) for the bacterial population.

In view of the numerical simulations of the three control strategies, the implementation of the control strategies B and C dramatically minimizes the size of infectious classes in the population as well as the concentration of bacteria in the environment as illustrated in Figs. 3(c) & 3 (d) and in Figs. 4(c) & 4 (d). Therefore, these control strategies target-fully reduce disease transmission in the human population. While, the numerical results indicate that strategy C is more effective in diminishing the number of infectious classes compared to strategy B regardless of the costs of the optimal control strategies as confirmed in Fig. 6. These results agree with the results obtained in section 3.1. Furthermore, the cost profiles for each of the three strategies are depicted in Figs. 7(a), 7(b) and 7(c). To determine the most cost-effective strategy among the three control strategies, we carry out a cost-effectiveness analysis in the next section.

Figure 7.

Simulations showing the cost profiles of the three Strategies (A,B & C).

4.3. Cost-effectiveness analysis

The cost-effective analysis is one of most economical evaluation tool used to measure mainly the costs of alternative interventions and therefore, it is used to compare and assess how the greatest health benefits can be generated and to rank the implemented strategies in-terms of their cost [5].

In this section two approaches, namely the average cost-effectiveness ratio (ACER) and the incremental cost-effectiveness ratio (ICER) are performed to obtain the most cost-effective strategy among the three alternative strategies (A, B, and C), using the ideas in [1], [2], [15], [18], [31].

4.3.1. Average cost-effectiveness ratio (ACER)

The average cost-effectiveness ratio (ACER) of an optimal control strategy S is given by the formula:

$$\text{ACER(S)}=\frac{\text{Total cost produced by the strategy S}}{\text{Total number of infection averted by the strategy S}}$$ (11)

It is concerned with a single optimal strategy and measures the strategy against baseline option. The total number of infection averted is given as the difference between the total number of infectious individuals over the simulation period without control and the number of total infectious individuals with control. A strategy with the least ACER value is the most cost-effective measure.

The total number of symptomatic infection averted during the intervention period is approximated from;

$$T_I=\underset{0}{\overset{250}{\int }}I(t)dt-\underset{0}{\overset{250}{\int }}{I}^{⁎}(t)dt$$ (12)

where, $T_I$ total number of symptomatic infections averted,

$$\underset{0}{\overset{250}{\int }}I(t)dt$$

represents the total symptomatic infectious cases without control over [0,250], and $I^{⁎}(t)$ is the optimal solution associated to the susceptible infections. In the same manner, the total number of asymptomatic infection averted during the intervention period is approximated from;

$$T_A=\underset{0}{\overset{250}{\int }}A(t)dt-\underset{0}{\overset{250}{\int }}{A}^{⁎}(t)dt$$ (13)

It is important to note that the total number of infections averted for the strategies is calculated using the MATLAB program, based on the above equations (12) and (13). The total number of infections averted for each strategy is determined by taking the average value of both the total number of symptomatic infections averted and the total number of asymptomatic infections averted by the strategy. These values are provided in the second column of Table 2.

Table 2.

Total infection averted, total cost produced by control strategies, ACER values and ICER values.

Strategy	Infection averted	Total Cost ($)	ACER values	ICER values
Strategy A	880.3	972.9036	1.1	1.1
Strategy B	2868.0042	934.3206	0.33	-0.0194
Strategy C	2940.8231	1219.6	0.4	3.9

While the total cost associated to the intervention is approximated from

$$T_C=\frac{1}{2}\underset{0}{\overset{250}{\int }}(w_3{u}_1^2+w_4{u}_2^2)dt.$$

Specifically, the total cost produced by the strategy A, strategy B, and strategy C are, respectively, approximated as:

$$T_{Ca}=\frac{1}{2}\underset{0}{\overset{250}{\int }}\left(w_3{u}_1^2\right)dt,T_{Cb}=\frac{1}{2}\underset{0}{\overset{250}{\int }}\left(w_4{u}_2^2\right)dt,\text{and}{T}_{Cc}=\frac{1}{2}\underset{0}{\overset{250}{\int }}(w_3{u}_1^2+w_4{u}_2^2)dt,\text{using MATLAB}.$$

The ACER for each of the strategies is calculated using the equation (11) and the values are given in fourth column of Table 2. From the Table 2, we conclude that the strategy B is the most cost effective of all the three possible strategies under consideration for this particular investigation.

4.3.2. The incremental cost-effectiveness ratio (ICER)

The incremental cost-effectiveness ratio is given by the formula:

$$\text{ICER}=\frac{\text{Difference in total intervation costs between strategies}}{\text{Difference in the total number of infection averted between strategies}}$$ (14)

The ICER's numerator includes (if applicable) the differences in the cost of interventions, costs of disease averted or costs of prevented cases among others. While the denominator given in the equation (14) determines the differences in health outcomes which include the total number of infections averted or the number of susceptibility cases prevented. To implement the ICER, we simulate the model using the three interventions strategies. Based on these simulation results of the optimal control problem, the intervention strategies are then ranked according to their increasing order of total number of infection averted. We have that strategy A averts the least number of the disease infections, followed by strategy B, and strategy C which averted the most number of infections as shown in Table 2. The ICER value for each strategy is computed using the equation (14) as follows:

$$\text{ICER(A)}=\frac{972.9036-0}{880.3-0}\approx 1.1,\text{ICER(B)}=\frac{934.3206-972.9036}{2868.0042-880.3}\approx -0.0194,\text{ICER(C)}=\frac{1219.6-934.3206}{2940.8231-2868.0042}\approx 3.9.$$

From ICER values of the strategies, we observed that ICER (C) is greater than ICER (A). It follows that, strategy C is more costly and less effective than strategy A. Therefore, it is better to remove strategy C from the list of alternative interventions and then the strategy A will be compared with the strategy B, by re-calculating ICER for each as follows.

$$\text{ICER(A)}=\frac{972.9036-0}{880.3-0}\approx 1.1,\text{ICER(B)}=\frac{934.3206-972.9036}{2868.0042-880.3}\approx -0.0194.$$

The result of comparison between two strategies, indicates that strategy B is strongly dominated by the strategy A. Thus, the strategy A is more costly and less effective than the strategy B. Consequently, the strategy B (treatment control) is most cost-effective of all the strategies for control of melioidosis infection under consideration for this particular study. This result agrees with the result of the ACER method obtained earlier. Furthermore, figures for the total number of infections averted, the total cost, ACER values and ICER values of the strategies are given below in Fig. 8, Fig. 9, Fig. 10 and Fig. 11, respectively.

Figure 8.

Total number of infected averted for control strategies.

Figure 9.

Total cost produced by control strategies.

Figure 10.

ACER values for all strategies.

Figure 11.

ICER values for all strategies.

5. Conclusion

In this work, we formulated and analyzed an optimal control problem that demonstrates the effectiveness of different control functions for eliminating the epidemic, as well as the overall cost-effectiveness of the introduced controls. The formulated optimal control model is a continuation of the basic model presented in [14]. The model is extended by incorporating two time-dependent control variables, namely personal prevention $u_1(t)$ and treatment control $u_2(t)$. The effect of these control variables on the basic reproduction of the non-autonomous system is illustrated graphically in Fig. (1), and the results are discussed. The standard results for the existence and characterization of optimal controls are established to obtain the optimality system for the optimal control problem. The optimality system is simulated by using the forward-backward sweep method in the MATLAB program to assess the impact of the three alternative optimal strategies by comparing them without controls on the transmission dynamics of the melioidosis epidemic: Strategy A-implementation of personal prevention control only, Strategy B-implementation of treatment control only, and Strategy C-implementation of the combination of both personal prevention and treatment controls. The numerical results show that the Strategy C will effectively reduce the number of infected individuals (both asymptomatic and symptomatic) in the population as demonstrated in Figs. 4. Additionally, the Strategy B has also a significant impact in reducing the infected classes as depicted in Fig. 3. These results are confirmed quantitatively in terms of the total number of infections averted. It is observed that the Strategy C averted the highest number of infectious individuals, followed by the Strategy B, and the Strategy A, which averted the least number of infectious individuals, as demonstrated in Table 2 and Fig. (8). However, the cost-effectiveness analysis carried out reveals that the Strategy B is the most cost-effective intervention strategy of all the three intervention strategies as confirmed quantitatively by ACER and ICER in Table 2, and in Figure 10, Figure 11. This shows that the treatment control strategy should be recommended to diminish the spread of the disease when available resources are limited. Therefore, using all the control strategies as suggested in [44] is not recommended regarding the costs of the optimal control strategies.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Additional information

No additional information is available for this paper.

CRediT authorship contribution statement

Habtamu Ayalew Engida: Writing – review & editing, Writing – original draft, Supervision, Software, Methodology, Investigation, Formal analysis, Conceptualization. Duncan Kioi Gathungu: Visualization, Supervision, Resources, Methodology. Melkamu Molla Ferede: Visualization, Supervision, Resources. Malede Atnaw Belay: Visualization, Supervision, Resources. Patiene Chouop Kawe: Visualization, Supervision, Resources. Bilali Mataru: Visualization, Supervision, Resources.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data included in this article/supplementary material/referenced in the article.