This paper proposes and analyses a mathematical model for the transmission dynamics of malaria with fourtime dependent control measures in Kenya: insecticide treated bed nets (ITNs), treatment, indoor residual spray (IRS), and intermittent preventive treatment of malaria in pregnancy (IPTp). We first considered constant control parameters and calculate the basic reproduction number and investigate existence and stability of equilibria as well as stability analysis. We proved that if
Malaria is a leading cause of mortality and morbidity among the underfive group and the pregnant women in SubSaharan Africa [
Malaria transmission is highly variable across Kenya because of the different transmission intensities driven by climate and temperature. Kenya has four malaria epidemiological zones: the endemic areas, the seasonal malaria transmission, the malaria epidemic prone areas, and the low risk malaria areas [
Mathematical modelling has become an important tool in understanding the dynamics of disease transmission and in decision making processes regarding intervention programs for disease control. Mathematical models provide a framework for understanding the transmission dynamics for malaria and can be used for the optimal allocation of different interventions against malaria [
IPTp is one of the WHO recommended prevention therapies for the pregnant women. IPTp has been shown to be effective in reducing maternal and infant mortality that are related to malaria for the most atrisk group for malaria [
In this paper, a model for malaria transmission dynamics with four control strategies is formulated and analyzed. We then formulate an optimal control problem and derive expressions for the optimal control for the malaria transmission model with four control variables and then use the optimal control theory to study the effectiveness of all possible combinations of four malaria preventive measures among the pregnant women and children under five years of age.
The model is formulated by considering the human and mosquito subgroups. The considered model consists of population of susceptible
Figure
Malaria model with interventions.
The susceptible humans (pregnant women and children under the age of five) (
Susceptible mosquitoes (
The state variables of the model are represented and described in Table
State variables of the malaria model.
Variable  Description 


Number of susceptible individuals (pregnant and under 5) at time 



Number of exposed individuals (pregnant and under 5) at time 



Number of infectious humans (pregnant and under 5) at time 



Number of recovered humans (pregnant and under 5) at time 



Number of susceptible mosquitoes at time 



Number of exposed mosquitoes at time 



Number of infectious mosquitoes at time 



Total number of individuals (pregnant and under 5) at time 



Total number of pregnant individuals at time 



Total mosquito population at time 
Description of parameter variables of the malaria model.
Parameter  Description 


Mosquito contact rate with human 



Mosquito biting rate 



Probability of human getting infected (the probability of transmission of infection from an infectious mosquito to a susceptible human provided there is a bite) 



Probability of a mosquito getting infected (the probability of transmission of infection from an infectious human to a susceptible mosquito provided there is a bite) 



Per capita natural death rate of humans 



Per capita natural death rate of mosquitoes 



Per capita rate of loss of immunity by recovered individuals 



Humans progression rate from exposed to infected 



Mosquitoes progression rate from exposed to infected 



Recruitment rate of human by birth and by getting pregnant 



Recruitment of mosquitoes by birth 



Per capita disease induced death rate for humans (pregnant and under 5) 



Proportion of spontaneous individual recovery 



Force of infection for susceptible humans (pregnant and under 5) to exposed individuals 



Force of infection for susceptible pregnant humans to exposed individuals 



Force of infection for susceptible mosquitoes to exposed mosquitoes 
Prevention and control variables in the model.
Variable  Description 


Preventive measure using insecticidetreated bed nets (ITNs) 



The control effort on treatment of infectious individuals 



Preventing measure using indoor residual spray (IRS) 



Preventive measure using intermittent preventive treatment for pregnant women (IPTp) 



Rate constant due to use of indoor residual spray 



Rate constant due to use of treatment effort 



Rate constant due to use of insecticidetreated bed nets 
The following assumptions have been used in the formulation of the model:
Population for human and mosquito being constant (no immigrants).
No recovery for infected mosquitoes.
Mosquitoes not dying due to disease infection.
All parameters in the model being nonnegative.
The total population size for the human is
We will assume that the control parameters are constant so as to determine the basic reproduction number, steady states, and their stability as well as the bifurcation analysis.
We describe the basic properties and analysis of the formulated malaria model with control strategies through mathematical analysis of the formulated model.
All the state variables and parameters for model (
The feasible solutions set for model (
System (
The matrices
The diseasefree equilibrium point for system (
The Jacobian matrix
Next, we study the global behavior of the diseasefree equilibrium for system (
The DFE,
We consider the following Lyapunov function:
On the boundary when
Therefore the largest compact invariant
Next, we investigate the endemic equilibrium and its stability of system (
First we determine the existence of the endemic equilibrium points.
The endemic equilibrium (
Substituting and solving for
It follows that
there is a unique endemic equilibrium if
there is a unique endemic equilibrium if
there are two endemic equilibria if
there are no endemic equilibria otherwise.
Thus the results of this section can be summarized in the following theorem.
If
Item (iii) indicates the possibility of backward bifurcation in model (
To apply the theory, we introduce dimensionless state variables into system (
The system of (
Therefore system (
A right eigenvector associated with the eigenvalue zero is
Therefore the following result is established.
Model (
Finally, we will investigate the global stability of the endemic equilibrium in the feasible region.
The global behavior of the endemic equilibrium of model (
If
Following Li and Muldowney [
What remains is to find conditions for which the Bendixson criterion given by (
Choose now matrix
Therefore
Sensitivity analysis is to assess the relative impact of each of the parameters of the basic reproductive number. The normalized forward sensitivity index of the reproduction number with respect to these parameters given in Table
Sensitivity indices (SI) of
Parameter  Sensitivity indices  

Endemic  Seasonal  Epidemic  Low risk  

−0.0402531  −0.0402531  −0.0402531  −0.0402531 

−1.07211  −1.07211  −1.07211  −1.07211 

0.00038817  0.00038817  0.00038817  0.00038817 

0.22445  0.22445  0.22445  0.22445 

0.5  0.5  0.5  0.5 

0.5  0.5  0.5  0.5 

1  1  1  1 

−0.4987  −0.4987  −0.4987  −0.4987 

0.5  0.5  0.5  0.5 

−0.01818  −0.02048  −0.01639  −0.02563 

−0.322497  −0.322497  −0.322497  −0.322497 

−0.13695  −0.10336  −0.11321  −0.03508 

1  1  1  1 
Following Chitnis et al. [
The sensitivity index of the model parameters is given by
Parameter values for the full malaria model.
Parameter  Estimated value  Source  

Endemic  Epidemic  Seasonal  Low risk  





KNBS (2009 Census estimates) [ 


Estimated  


Estimated  


Chitnis et al. [  


Estimated  


Estimated  


Blayneh et al. [  


Estimated  





KNBS (estimates based on 2009 Census) [ 


Niger and Gumel [  


Chiyaka et al. [  


Assumed  


KNBS and ICF Macro [  


Assumed  


Assumed  


Blayneh et al. [  





Estimated 





Estimated 





Estimated 





KNBS (2009 Census estimate) [ 





KNBS (2009 Census estimate) [ 





Estimated 
We consider the case of timedependent control variables. The malaria dynamics model is extended and an optimal control problem is formulated. We formulate an optimal control model for malaria disease in order to determine optimal malaria control strategies (ITNs, IRS, IPTp, and treatment) with minimal implementation cost.
For the optimal control problem of the given system, we consider the control variables
Our aim with the given objective function is to minimize total number of mosquito population
We seek an optimal control
The existence of an optimal control can be proved by using the theorem given in Fleming and Rishel [
the set of controls and corresponding state variables is nonempty.
the control set
right hand side of each equation of the state system in (
there exist constants
The state and the control variables of system (
This concludes existence of an optimal control since the state variables are bounded. Hence we have the following theorem.
Given the objective functional
Lagrangian for a problem discusses how the techniques come and Hamiltonian helps in solving for the adjoint variable. In order to find an optimal solution, first we find the Lagrangian and Hamiltonian for the optimal control problem (
In order to find the necessary conditions for this optimal control, we apply Pontryagin’s Maximum Principle [
If
The state equation is
Given the optimal controls
To determine the adjoint equations and the transversality conditions we use the Hamiltonian
In order to minimize Hamiltonian,
Solving
The optimality system is solved using the forwardbackward fourthorder RungeKutta scheme in
The parameters in model (
In addition the effect of the different intervention strategies is estimated as
We simulated malaria model with intervention strategies to find the dynamics of human and mosquito populations. It is observed that the control strategy leads to decrease in the number of infected human (
Simulations showing the dynamics of human population with intervention strategies for the endemic setting.
The figure shows steady decrease in susceptible human population at the initial period as the exposure of humans to disease increases. Thereafter graph of susceptible humans increases as the exposed and infected human population decreases due to positive effect of the intervention strategies being implemented.
In this section, we investigate numerically the effect of the several optimal control strategies on the spread of malaria. We compare the numerical results from the simulation using one control and various combinations of two, three, and four control strategies. This was done by comparing when there were not any intervention strategies and when there were intervention strategies. There are 15 different control strategies for each of the four different epidemiological zones in Kenya that are explored. We use the case of endemic zone with the case of one control variable, two control variables, three control variables, and all the four control variables being in use for the illustration purpose.
The results in Figures
Simulations of the model showing the effect of treatment on the spread of malaria for the endemic setting.
Simulations of the model showing the effect of ITNs and treatment on the spread of malaria for the endemic setting.
Simulations of the model showing the effect of ITNs, treatment, and IRS on the spread of malaria for the endemic setting.
Simulations of the model showing the effect of ITNs, treatment, IRS, and IPTp on the spread of malaria for the endemic setting.
There are several combinations for the different settings as described below.
Results of only one intervention strategy for the endemic epidemiological zones is as follows.
Results of combining 2 intervention strategies for the endemic epidemiological zones are as follows.
Results of combining three intervention strategies for the endemic epidemiological zones are as follows.
Results of combining the four intervention strategies for the endemic epidemiological zones are as follows.
The same procedure is repeated for other combination for strategies and for other epidemiological zones (epidemic, seasonal, and low). Based on the simulation findings for the highest number of infections being inverted at a lower cost, the combined use of treatment and IRS reduces the infected human and mosquito population faster at a lower cost for the endemic settings (105 infections at
In this paper, we formulated a mathematical model for the transmission dynamics of malaria with four timedependent control measures in Kenya. We first consider control parameters to be constant and perform mathematical analysis of the model. The analysis showed that there exists a domain where the model is epidemiologically and mathematically well posed. Stability analysis of the model showed that the diseasefree equilibrium is globally asymptotically stable if
We then consider the case of timedependent control variable from where we formulated an optimal control problem and derived expressions for the optimal control for the malaria model with four control variables with an aim of minimizing the number of malaria infections in humans (derive optimal prevention and treatment strategies) while keeping the cost low. In the optimal control problem, use of one control at a time and the different combination of interventions can be explored to investigate and compare the effects of the control strategies on malaria eradication for different transmission settings. The analysis of the model showed that the state and optimal control can be calculated using the optimality system. The optimality system is comprised of the state system, the adjoint/costate system, initial conditions at
The results of the optimal control problem will be able to show which intervention or combination of the different intervention strategies has the highest impact on the control of the disease especially for different transmission settings. Our results shows that an effective IRS use and treatment will be beneficial to the community for the control of malaria disease (infected human and mosquito population) faster at a lower cost for the endemic settings. This is slightly different from the findings of Agusto et al. [
The findings show that, for the epidemic prone areas, the optimal control strategy for reducing the infected human and mosquito population was the use of treatment and IRS. This is slightly different from Agusto et al. [
The results show that for seasonal areas much impact will be felt when treatment is used which is different from Mwamtobe et al. [
The results show that, for the low risk areas, just the use of ITNs and treatment will be sufficient to reduce infected human and mosquito population. This is comparable to Silva and Torres [
These findings support the WHO concerns on the capability of only one intervention strategy in reducing malaria transmission. The findings are however applicable to the designing of intervention strategies for malaria especially when costs aspects are of concern. This modelling approach also addresses effectiveness of the recommended intervention for atrisk group of malaria (pregnant women) by the WHO. The modelling approach has also been implemented in the
To the best of our knowledge, this is the first ever optimal control modelling and simulation of malaria intervention strategies in free
In this paper, we derived and analyzed a deterministic model for the transmission of malaria disease which incorporated the use of insecticidetreated bed nets (ITNs), treatment, indoor residual spray (IRS), and intermittent preventive treatment for pregnant women (IPTp) and performed optimal control analysis of the model. We first consider constant control parameters from where we investigate existence and stability of equilibria as well as stability analysis. We proved that if
We then consider the timedependent control case from where we derived and analyzed the necessary conditions for the optimal control of the disease. There were 15 different control strategies for each of the four different epidemiological zones in Kenya that were explored using control plots (numerical simulations) which compared when there were not any intervention strategies and when there were intervention strategies. Using the optimal control approach we found that the combined use of treatment and IRS would reduce the highest number of infected human and mosquito population faster at a lower cost for the endemic settings. For the epidemic prone settings the use of treatment and IRS has more impact in reducing the infected human and mosquito population. For seasonal areas much impact will be felt when treatment is used. For the low risk areas, just the use of ITNs and treatment will be sufficient to reduce infected human and mosquito population. This was deduced from the intervention which takes shorter time to start reducing the number of infected mosquitoes and humans.
We conclude that, according to our model, the optimal control strategy for malaria control in endemic areas was the combined use of treatment and IRS; in epidemic prone areas it was the use of treatment and IRS; in seasonal areas it was the use of treatment; and in low risk areas was the use of ITNs and treatment. Control programs that follow these strategies can effectively reduce the spread of malaria disease in different malaria transmission settings in Kenya.
The geometric approach to global stability is as described by Li and Muldowney for proving Theorem
The general method considered is the one developed by Li & Muldowney [
Assume that the following hypotheses hold
Function
Point
Let
As a consequence, the following theorem holds [
Assume that conditions
Recommendation: in order to reduce malaria transmission in Kenya, the National Malaria Control Programme should tailormake different intervention strategies for different epidemiological zones since some strategies work better than others in a resource limited setting. In addition the designed interventions that are preventive in nature should be implemented across all the different settings. There will be a need for the National Control Programme to create awareness on the malaria preventive measures.
No competing interests exist.
All the authors have significant contribution to this paper and the final form of this paper is approved by all authors.
This work was (partially) funded by an African Doctoral Dissertation Research Fellowship (ADDRF) award offered by African Population and Health Research Centre (APHRC) in partnership with the International Development Research Centre (IDRC). Phase 4 of the ADDRF program is funded by IDRC (Grant no. 107508001). This work was also supported by National Commission for Science and Technology (NACOSTI/ RCD/ST&I 6th Call PhD/225).