Modelling the Dynamics of Endemic Malaria Disease with Imperfect Quarantine and Optimal Control

: Malaria is an infectious disease caused by Plasmodium parasite and it is transmitted among humans through bites of female Anopheles mosquitoes. In this paper, a new deterministic mathematical model for the endemic malaria disease transmission that incorporates imperfect quarantine and optimal control is proposed. Impact of various intervention strategies in the community with varying population at time t are analyzed using mathematical techniques. Further, the model is analyzed using stability theory of differential equations and the basic reproduction number is obtained from the largest eigenvalue of the next-generation matrix. Conditions for local and global stability of disease free, local stability of endemic equilibria and bifurcations are determined in terms of the basic reproduction number. The Center manifold theory is used to analyze the bifurcation of the model. It is shown that the model exhibit both a backward and a forward bifurcation. Reducing the biting rate of the quarantined people is advice able to minimize the spread of endemic malaria disease. The optimal control is designed by applying Pontryagins’s Maximum Principle (PMP) with four control strategies namely, insecticide treated nets, screening, treatment and indoor residual spray. The best strategy to control endemic malaria disease is the combination that incorporated all four control strategies.


Introduction
Malaria is the dangerous one among infectious disease. It is caused by plasmodium parasites that are transmitted among humans through the bites of female Anopheles mosquitoes. And it is also the largest burden disease for these people living in poor countries, especially, in Sub Saharan Africa, causing high mortality and morbidity [1]. In 2018 (World Health Organization 2019 report), nearly 228 million malaria cases occur worldwide, out of which 405,000 million die every year [2,3]. above 40% of the world's population in more than 80 countries and regions are still under the risky of contracting malaria.
About 80% of malaria death are concentrated in 15 countries most of them in Africa [4,5]. IN recent, reduction in the number of malaria related cases are due to the global efforts of the current malaria interventions, such as decreasing mosquito breeding sites, sleeping under insecticide-treated nets (ITN), indoor residual spraying (IRS) with insecticides, are used for reducing malaria vectors and their bites, timely treatment with artemisinin-based combination therapies (ACTs) and chemoprevention for most vulnerable such as intermittent preventive treatment for pregnant women (IPTp) recommended by WHO. As global effort increases, it is necessary to know how these interventions can be implemented alongside one another.
Quarantine is also one of the public health control strategy of infectious diseases. The strategy focuses on isolation of infectious individuals from contacting with susceptible individuals or healthy populations. This control measure is effective to control and eliminate newly emerging infectious diseases caused by unidentified infectious agents.
Optimal control applications are important to approximate the efficacy of various policies and control measures. It is also important to cost estimation analysis of the examined control strategies. The theory of optimal control has been Imperfect Quarantine and Optimal Control more successfully used in decision making in various applications after the development of Pontryagins's maximum principle (1962).
Mathematical models of the dynamics of malaria transmission are useful in providing a better insight into the behavior of the disease. These models have played a great role in influencing the decision making processes regarding intervention strategies for controlling and eliminating the spread of malaria. The study of malaria using mathematical modeling began in 1911 with Ronald Ross [6,7]. Others have studied the transmission of malaria using SIR model for humans and SI for the mosquitoes. These are: Alemu Geleta Wedajo, Boka Kumsa Bole, Purnachandra Rao Koya [8,9], Tuwiine, Mugisha and Luboobi [10] developed a compartment model for the spread of malaria with susceptible-infectedrecovered-susceptible (SIRS) pattern for human and susceptible-infected (SI) pattern for mosquitoes. Yang, Wei, and Li have proposed SIR for the human and SI for the vector compartment model and define the reproduction number, R 0 and show the existence and stability of the disease-free equilibrium and an endemic equilibrium [11].
Feng and Thieme formulated a perfect quarantine model where a proportion of infected people stay at home and do not infect anybody and showed that the model can give rise to sustained oscillations [12]. Hethcote et al. analyzed six types of SIQS and SIQR models to explore which one can produce periodic solutions [13]. Gumel et al used models to examine the effectiveness of quarantine and isolation on the control of SARS outbreaks [14]. Pandey et al developed a compartmental model for Ebola transmission to assess the effectiveness of non-pharmaceutical interventions for curtailing the epidemic in Liberia [15].
Erdem et al. studied the impact of imperfect quarantine on the dynamics of an SIR-type model [16]. X. Jin et al, mathematical analysis of the Ross-Macdonald malaria model with quarantine using SIQ-SI model type [17]. K. O. Okosun et al. (2013) derived and analyzed a malaria disease transmission mathematical model that includes insecticide treated net, treatment and indoor residual spray and applied optimal control strategy to study a possible treatment of infective humans that blocks transmission to mosquitoes in controlling the spread of malaria [18]. Suresh (1978) formulated and analyzed an optimal control problem with a simple epidemic model to examine effect of a quarantine program [19].
The purpose of the study of endemic malaria disease model system (1) with imperfect quarantine strategy is to reduce the number of susceptible mosquitoes bites from or contacts with malaria infectious humans and explore the effect of the strategy in the malaria control and elimination.

Model Formulation
The ordinary differential equations that describe the interactions between the human and mosquito population is formulated and described by Otieno et al. [20]. In this paper, a deterministic compartmental model is formulated and analyzed. The model is formulated based on the assumptions of [17] by incorporating imperfect quarantine that is, we classify the infectious human as Exposed quarantined individuals with no disease clinical symptoms for the time being, but sharing common environment or home with these may have continuous opportunities of bite from malaria parasite carrier denoted by , Infected quarantined with disease clinical symptoms denoted by and Hospitalized (infected isolated individuals these are already getting treatment) and denoted by .
The is the force of infection from mosquito to human where, is the rate of probability of human getting infected, is the mosquito contact rate with human and is mosquito biting rate and ' is the rate of reduction of mosquito bites for quarantined human compartments. Note that, ' = 1 corresponds to perfect quarantine, ' = 0 corresponds to no quarantine, and 0 < ' < 1 corresponds to imperfect quarantine. The Infected non-quarantined individuals move to hospitalized (isolated class) with respective rate * or recovery class by getting partial immunity at a rate +. They also die because of natural and disease induced death rates at and , respectively. The Exposed quarantined individuals either die from natural causes at a rate of or move to Infected quarantined class after developing disease symptoms at a rate -. Infected quarantined individuals are move to hospitalized (infected isolated) with respective rate * . or recovery class by getting partial immunity at a rate + / . They also die because of natural and disease induced death rates at and , respectively. Hospitalized (infected isolated human class) move to recovery class by getting partial immunity at a rate + . or die because of natural and disease induced death rates because of natural and disease induced death rates at and , respectively. These infectious individuals progress to partially immune group (recovered class), either partially immune group losses immunity and becomes again susceptible at a rate 0 or die from natural death at a rate .

Susceptible mosquitoes
are recruited at the rate Λ . They either die due to natural death at a rate of or move to Infected class by acquiring malaria through contact with infectious humans with respective rate where is the Probability of a mosquito getting infected. Infected mosquitoes are die because of natural and disease induced death rates and , respectively.
From the law of conservation, the total number of bites by mosquitoes equal to the total number of bites on humans (i.e., = implies = .   The forces of infection on humans and mosquitoes respectively denoted and given by

Existence and Positivity of Solutions
In this sub section, the malaria model governed by the system of equation (1) is epidemiologically and mathematically well posed will be shown. Its feasible region is also denoted and given by

Theorem 1
The solution R , , , , , , , S of the system of equation (1)     Similarly, the solutions of the model variables representing mosquito populations 9 , = are confined in the feasible This shows that the feasible region of the model system (1) is bounded and is given by    , then without loss of generality we have an inequality 3 34 ≥ − + , + * + + and its general solution is given by ≥ VWX f 5 − + , + * + + g ≥ 0. Therefore ≥ 0 for > 0.

Existence of Disease Free Equilibrium Points
The disease-free equilibrium point of the model is its steady state solutions without infection or disease. Consider the disease free-equilibrium points denoted and given by:

Reproduction Number
The basic reproduction number denoted by 5 is the average number of secondary infectious infected by an infective individual during his or her whole course of disease in case that all of the population are susceptible [21]. It helps to explore whether an infection will expand through the population or go away from the population. In order to determine the stability of system (1) the threshold condition for the establishment of the disease is necessary to be obtained. Here the reproduction number is calculated using the next generation matrix method that is developed by van den Driessche and Watmough [22]. The local asymptotic stability occur if 5 < 1 and instability occur if 5 > 1.

Global Stability of the Disease-Free Equilibrium Point
To establish the global stability of disease free-equilibrium two conditions are considered. Castillo-Chavez et-al [23].

Bifurcation Analysis
The sub-threshold occurrence of multiple endemic equi libria stated in Theorem 4, is the result of forward or back ward at 5 = 1. Now, we study the Centre manifold near the criticality by using the approach developed in [24,25,26]. Based on Center Manifold theory (Gumel and Song, 2008; Castillo-Chavez and Song, 2004) and general Centre manifold theory [27], we carry out a bifurcation analysis of model system (1) at 5 = 1. Not that, the normal form representing the dynamics of the system on the Centre manifold is given by ÂÃ = ®Â / +¯ÄÂ, where, for j =1, 2…, n (10) Here, the symbol Ä denotes a bifurcation parameter to be chosen, Ç Ê s denote the right hand side of system (1), W denotes the state vector, W 5 the disease-free equilibrium E 0, Å AE denotes the differential operator with respect to W , Å Ò denotes the differential operator with respect to Ä, and È and ] denote the right and left eigenvectors, respectively, corresponding to the null eigenvalue of the Jacobian matrix of system (1), evaluated at W 5 for Ä = 0.
To apply the above result, the following simplification and change of variables are made on system (1). Let = W . , = W / , = W ' , = W ' , = W Ž , = W @ , = W Ó , and = W`, so, = W . + W / + W ' + W ' + W Ž + W @ and = W Ó, + W `. Mor over, by using the vector notation W = W ., W /, W ', W ', W Ž, W @, W Ó, W `, p , the system (1) can be written in the form We choose the rate of transmission of infection from an infectious mosquito to a susceptible human, , as the bifurcation parameter. We observe that 5 = 1 is equivalent to: So that the disease free-equilibrium 5 is locally asymptotically stable when < * and unstable when > * . Hence, = * is a bifurcation value. The Jacobian matrix of system (1) evaluated at 5 for = * is given by Where, det Ö 5 − @ = 0, Since the first and seventh columns contain only diagonal terms they give two negative eigenvalues i.e., . = − , / = − , then deleting rows and columns of the first and fifth of Ö 5 we have: In the same way, the fifth column of Ö . 5 contains only diagonal term which also forms a negative eigenvalue i.e., ' = −•. The remaining five eigenvalues are obtained from the sub-matrix The eigen values of the matrix Ö / 5 are the roots of the characteristic equation Thus, (16) implies that the Jacobian Ö 5 , * of the linearized system has a simple zero eigenvalue and the other eigenvalues have negative real part. Therefore the diseasefree equilibrium E 0 is a nonhyperbolic equilibrium. To compute the coefficients (10) and (11) There fore; for È`> 0, ]`> 0 we have, •q 5 + q / + • / f• ' • ' q ' + -• ' q Ž + * . q @ g + • . • ' • ' q ' + *q @ 1 − Theorem 5: If Ψ > 0 and Ψ > Γ , then ® > 0 and Ψ < 0 ensures that ® < 0. If ® > 0 and ¯ > 0, then the model system (1) undergo a backward bifurcation at 5 = 1 , otherwise it will exhibit a forward bifurcation. Hence the endemic equilibrium * * is locally asymptotically stable.

Analysis of the Model with Optimal Control
In this section, on model system (1) we also incorporate four time dependent control measures namely, (i) the use of insecticide treated bed net (ITN) u . t = u . as preventive measure i.e., to reduce the number of bites from mosquitoes as they physically provide a barrier between the infectious mosquitoes and the susceptible humans, and also to reduce the population of the mosquitoes by killing them after they land on the treated net. (ii) the effort of screening of quarantined individuals u / t = u / , which helps them to identify whether or not they are with disease symptom, (iii) treatment with drugs u ' t = u ' , treating individuals who developed symptoms of the disease, and (iv) the use of Indoor Residual Spray (IRS), u ' t = u ' as preventive measure i.e., insecticide spray on the breeding site of mosquitoes reduces the number of mosquito populations by killing these rest indoors after feeding.

Numerical Simulations Results
In this section, numerical simulations are performed to illustrate the effects of malaria control measures by applying different control strategies. We apply the parameter values listed in Tables 2 and 3 to obtain numerical results for the optimal system by using a forward-backward iterative method [31].

Controlling Endemic Malaria Disease Using Imperfect Quarantine Strategy
In this strategy, we simulated the model system (1) by incorporating imperfect quarantine to reduce the number of susceptible mosquitos' bites from or contacts with malaria infectious humans.  during the implementation of the strategy. From the figure, it is clearly seen that the graphs were exponentially decreased and smaller in number at the end of implementation of intervention time above in the case with imperfect quarantine 0 < ' < 1 compered to in case without quarantine ' = 0. From this we conclude that imperfect quarantine strategy plays a great role in reducing the numbers of susceptible mosquitoes bites from or contacts with malaria infectious humans and hence eliminate the spread and transmission of the disease through the human populations.
To examine the impact of the combination of each control and elimination of malaria disease, we used the following strategy: (i) Implementing ITN § . and screening § / as intervention (ii) Implementing ITN and IRS as intervention (iii) Implementing screening and treatment as intervention (iv) Implementing ITN, screening and treatment as intervention (v) Implementing ITN, treatment and IRS as intervention (vi) Implementing screening, treatment and IRS as intervention (vii) Implementing ITN, screening, and IRS as intervention (viii) Implementing ITN, screening, treatment effort and IRS as intervention

Controlling with Insecticide Treated Net ITN and Screening
In this case, we simulated the model by incorporating optimized Insecticide Treated Net and screening as disease control strategy.  In Figue 4 (a) and (b) above, there is a small number difference between the states with control § . ≠ 0, § / ≠ 0. § ' = § ' = 0 represented by blue color and without controls § . = § / = § / = § ' = 0 represented by red color. It is clearly seen from the figure that, both the number of infected humans and hospitalized (infected isolated) humans are exponentially decreased with time but their numbers cannot be zero at final time of implementation of the strategy. From this we can conclude that using only the combination of insecticide treated net ITN and screening, it is possible to reduce the number of malaria infectious individuals even without treating asymptomatic individuals.

Controlling with Insecticide Treated Net ITN and Indoor Residual Spray IRS
This case, we simulated the model by incorporating optimized insecticide treated net and Indoor Residual spray IRS as disease control strategy to optimize the objective function J

Control with Screening and Treatment
In this case, we simulated the model by incorporating optimized screening and treatment as disease control strategy to optimize the objective function J.  humans and recovered humans during the implementation of the strategy with control and without control represented by blue and red color respectively. From the figure, it is clearly seen that the numbers of hospitalized (infected isolated) humans are decreased more with time incase with control than without control. Similarly, the number of recovered humans are large incase with control while their numbers are small incase without control at the final time of implementation of the strategy.

Control with Preventive Insecticide Treated Net ITN, Screening and Treatment
In this case, we simulated the model by incorporating optimized insecticide treated net ITN, screening and treatment as disease control strategy to optimize the objective function J. , hospitalized (infected isolated) and infected mosquitoes during the implementation of the strategy with control and without control represented by blue and red color respectively. From the figure, it is clearly seen that the numbers of infected non-quarantined humans and hospitalized (infected isolated) are smaller at the end of implementation of intervention time above in the case with control than without control. Similarly, the number of infected mosquitoes are large incase without control while their numbers are small incase with control at final time of implementation of the strategy. The reason is that applying optimized the combination of insecticide treated net ITN, screening and treatment only control intervention decreases more the burden of the disease than a combination of two controls intervention but it cannot be eradicate the disease in the community.

Control with Combination of Screening, Treatment and Indoor Residual Spray IRS
In this case, we simulated the model by incorporating optimized indoor residual spray IRS, screening and treatment as disease control strategy to optimize the objective function J. Figure 8 (a), (b) and (c), represents the numbers of infected non-quarantined , hospitalized (infected isolated) and infected mosquitoes . From the figure, it is clearly seen that the numbers of infected nonquarantined humans and hospitalized (infected isolated) are decreased more with time incase with control than without control but their number cannot be zero at the final time of the implementation of the strategy. Similarly, the number of infected mosquitoes are large incase without control while their numbers are small incase with control at final time of implementation of the strategy. The reason is that applying optimized the combination of indoor residual spray IRS, screening and treatment only control intervention decreases more the burden of the disease than a combination of two controls intervention but it cannot be eradicate the disease in the community.

Control with Insecticide Treated Net ITN, Treatment and Indoor Residual Spray IRS
In this strategy, we applied a combination of treatment, Insecticide Treated Net ITN and Indoor Residual Spray IRS to the endemic malaria disease model system (1) as control strategy. In Figure 9 (a), (b) and (c) a small number difference is seen between states with control § . ≠ 0, § ' ≠ 0. § ' ≠ 0, § / = 0 and without controls § . = § / = § ' = § ' = 0 . It is clearly seen from the figure that, the number of infected non-quarantined humans, hospitalized (infected isolated) humans and infected mosquitoes are decreased more with time incase with control than without control I v * with u 2 , u 3 and ,u 4 are not equal to zero but u 1 = 0 but their number cannot be zero at the final time of the implementation of the strategy. The reason is that applying optimized the combination of ITN, treatment, and IRS only control intervention, decreases more the burden of the disease than a combination of the two controls intervention but it cannot be eradicate the disease in the community.

Control with Insecticide Treated Net ITN, Screening and Indoor Residual Spray IRS
In this strategy, we applied a combination of screening, insecticide treated net ITN and indoor residual spray IRS to the endemic malaria disease model system (1) as control strategy.
In Figure 10 (a), (b) a small number difference is seen between states with control § . ≠ 0, § / ≠ 0. § ' ≠ 0, § ' = 0 represented by blue color and without controls § . = § / = § ' = § ' = 0 represented by red color. It is clearly seen from the figure that, the number of infected non-quarantined humans and infected mosquitoes are decreased more with time incase with control than without control but their number cannot be zero at the final time of the implementation of the strategy. The reason is that applying optimized the combination of ITN, screening, and IRS only control intervention, decreases more the burden of the disease than a combination of the two controls intervention but it cannot be eradicate the disease in the community.  I v * with all u 1 ,u 2 , u 3 and u 4 are not equl to zero

Control with Screening, Treatment with Drugs, Insecticide Treated Net and Indoor Residual Spray
In this strategy, we applied a combination of screening, treatment, Insecticide Treated Net ITN and Indoor Residual Spray IRS to the endemic malaria disease model system (1) as control strategy.
In Figure 11 (a), (b) and (c) above, a small number difference is seen between states with control § . ≠ 0, § / ≠ 0. § ' ≠ 0, § ' ≠ 0 and without controls § . = § / = § ' = § ' = 0 . It is clearly seen from the figure that, the number of infected non-quarantined humans, hospitalized (infected isolated) humans and infected mosquitoes are exponentially more decreased with time incase with control than without control but their number cannot be zero at the final time of the implementation of the strategy. The reason is that applying optimized the combination of ITN, screening, treatment and IRS only control intervention, decreases furthermore the burden of the disease than a combination of the three controls intervention but it cannot be eradicate the disease in the community.

Discussion and Conclusion
In this paper, we formulated and analyzed a deterministic model that incorporates both imperfect quarantine and optimal control strategy to investigate their roles in case of endemic malaria disease control and elimination. We analyzed the dynamical behavior of the model in term of the basic reproduction number 5 and also obtained a sufficient condition for both local and global asymptotic stability of the disease-free equilibrium 5 and local asymptotic stability of endemic equilibrium * * . The model system (1) exhibit both backward and forward bifurcations at 5 = 1 .
The impact of imperfect quarantine strategy on endemic malaria persistence clearly seen on Figure 3 (figure showing Susceptible mosquitoes with and with no control parameter ' ). From this we conclude that in order to minimize the burden of malaria disease from the community, reducing the biting rate of the quarantined people is advice able than to quarantine more infected people at earlier infection stage.
The optimal control includes the use of insecticide treated nets, screening of infectious humans, treatment of infective humans and indoor residual spray to reduce the number of malaria transmitter vectors by means of spraying on the place where they choose for rest and breed. We perform and analyzed the necessary conditions for the optimal control of the disease model system (1). From this we conclude that, (i) a combination of insecticide treated net and indoor residual spray is the best alternative combination of controls to reduce the numbers of infected nonquarantined humans and mosquitoes, when combinations of bi-controls are considered. (ii) Both combinations of insecticide treated net-indoor residual spray-screening and insecticide treated netindoor residual spray treatment are the best alternative combinations of controls to reduce the numbers of infected non-quarantined humans, isolated humans and mosquitoes, when combinations of tri-controls are considered. (iii)Furthermore, the best combination is the one that incorporated all four control strategies.