Kazeem Oare Okosun

Impact of chemo-therapy on optimal control of malaria disease with infected immigrants

We derived and analyzed rigorously a mathematical model that describes the dynamics of malaria infection with the recruitment of infected immigrants, treatment of infectives and spray of insecticides against mosquitoes in the population. Both qualitative and quantitative analysis of the deterministic model are performed with respect to stability of the disease free and endemic equilibria. It is found that in the absence of infected immigrants disease-free equilibrium is achievable and is locally asymptotically stable. Using Pontryagins Maximum Principle, the optimal strategies for disease control are established. Finally, numerical simulations are performed to illustrate the analytical results.

A co-infection model of malaria and cholera diseases with optimal control

In this paper we formulate a mathematical model for malaria-cholera co-infection in order to investigate their synergistic relationship in the presence of treatments. We first analyze the single infection steady states, calculate the basic reproduction number and then investigate the existence and stability of equilibria. We then analyze the co-infection model, which is found to exhibit backward bifurcation. The impact of malaria and its treatment on the dynamics of cholera is further investigated. Secondly, we incorporate time dependent controls, using Pontryagins Maximum Principle to derive necessary conditions for the optimal control of the disease. We found that malaria infection may be associated with an increased risk of cholera but however, cholera infection is not associated with an increased risk for malaria. Therefore, to effectively control malaria, the malaria intervention strategies by policy makers must at the same time also include cholera control.

Stability Analysis and Optimal Control of a Vector-Borne Disease with Nonlinear Incidence

The paper considers a model for the transmission dynamics of a vector-borne disease with nonlinear incidence rate. It is proved that the global dynamics of the disease are completely determined by the basic reproduction number. In order to assess the effectiveness of disease control measures, the sensitivity analysis of the basic reproductive number R0 and the endemic proportions with respect to epidemiological and demographic parameters are provided. From the results of the sensitivity analysis, the model is modified to assess the impact of three control measures; the preventive control to minimize vector human contacts, the treatment control to the infected human, and the insecticide control to the vector. Analytically the existence of the optimal control is established by the use of an optimal control technique and numerically it is solved by an iterative method. Numerical simulations and optimal analysis of the model show that restricted and proper use of control measures might considerably decrease the number of infected humans in a viable way.

Modelling the impact of drug resistance in malaria transmission and its optimal control analysis

We derive and analyse a deterministic model for the transmission of malaria disease with drug resistance in the infectives. Firstly, we calculate the basic reproduction number, R, and investigate the existence and stability of equilibria. The system is found to exhibit backward bifurcation, with this occurrence, the classical epidemiological requirement for effective eradication of malaria, R < 1, is no longer sufficient, even though necessary. Secondly, by using optimal control theory, we derive the conditions for optimal control of the disease using Pontryagin’s Maximum Principle. Finally, numerical simulations are performed to illustrate the analytical results.

Analysis of recruitment and industrial human resources management for optimal productivity in the presence of the HIV/AIDS epidemic

The aim of this paper is to analyze the recruitment effects of susceptible and infected individuals in order to assess the productivity of an organizational labor force in the presence of HIV/AIDS with preventive and HAART treatment measures in enhancing the workforce output. We consider constant controls as well as time-dependent controls. In the constant control case, we calculate the basic reproduction number and investigate the existence and stability of equilibria. The model is found to exhibit backward and Hopf bifurcations, implying that for the disease to be eradicated, the basic reproductive number must be below a critical value of less than one. We also investigate, by calculating sensitivity indices, the sensitivity of the basic reproductive number to the model’s parameters. In the time-dependent control case, we use Pontryagin’s maximum principle to derive necessary conditions for the optimal control of the disease. Finally, numerical simulations are performed to illustrate the analytical results. The cost-effectiveness analysis results show that optimal efforts on recruitment (HIV screening of applicants, etc.) is not the most cost-effective strategy to enhance productivity in the organizational labor force. Hence, to enhance employees’ productivity, effective education programs and strict adherence to preventive measures should be promoted.

Modelling the influence of temperature and rainfall on the population dynamics of Anopheles arabiensis.

BackgroundMalaria continues to be one of the most devastating diseases in the world, killing more humans than any other infectious disease. Malaria parasites are entirely dependent on Anopheles mosquitoes for transmission. For this reason, vector population dynamics is a crucial determinant of malaria risk. Consequently, it is important to understand the biology of malaria vector mosquitoes in the study of malaria transmission. Temperature and precipitation also play a significant role in both aquatic and adult stages of the Anopheles.MethodsIn this study, a climate-based, ordinary-differential-equation model is developed to analyse how temperature and the availability of water affect mosquito population size. In the model, the influence of ambient temperature on the development and the mortality rate of Anopheles arabiensis is considered over a region in KwaZulu-Natal Province, South Africa. In particular, the model is used to examine the impact of climatic factors on the gonotrophic cycle and the dynamics of mosquito population over the study region.ResultsThe results fairly accurately quantify the seasonality of the population of An. arabiensis over the region and also demonstrate the influence of climatic factors on the vector population dynamics. The model simulates the population dynamics of both immature and adult An. arabiensis. The simulated larval density produces a curve which is similar to observed data obtained from another study.ConclusionThe model is efficiently developed to predict An. arabiensis population dynamics, and to assess the efficiency of various control strategies. In addition, the model framework is built to accommodate human population dynamics with the ability to predict malaria incidence in future.

Mathematical Analysis of a Malaria Model with Partial Immunity to Reinfection

A deterministic model with variable human population for the transmission dynamics of malaria disease, which allows transmission by the recovered humans, is first developed and rigorously analyzed. The model reveals the presence of the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with one or more stable endemic equilibria when the associated reproduction number is less than unity. This phenomenon may arise due to the reinfection of host individuals who recovered from the disease. The model in an asymptotical constant population is also investigated. This results in a model with mass action incidence. A complete global analysis of the model with mass action incidence is given, which reveals that the global dynamics of malaria disease with reinfection is completely determined by the associated reproduction number. Moreover, it is shown that the phenomenon of backward bifurcation can be removed by replacing the standard incidence function with a mass action incidence. Graphical representations are provided to study the effect of reinfection rate and to qualitatively support the analytical results on the transmission dynamics of malaria.

On a drug-resistant malaria model with susceptible individuals without access to basic amenities.

In this paper, a deterministic malaria transmission model in the presence of a drug-resistant strain is investigated. The model is studied using stability theory of differential equations, optimal control, and computer simulation. The threshold condition for disease-free equilibrium is found to be locally asymptotically stable and can only be achieved in the absence of a drug-resistant strain in the population. The existence of multiple endemic equilibria is also established. Both the Sensitivity Index (SI) of the model parameters and the Incremental Cost-Effectiveness Ratio (ICER) for all possible combinations of the disease-control measures are determined. Our results revealed among others that the most cost-effective strategy for drug-resistant malaria control is the combination of the provision of basic amenities (such as access to clean water, electricity, good roads, health care, and education) and treatment of infective individuals. Therefore, more efforts from policy-makers on the provisions of basic amenities and treatment of infectives would go a long way to combat the malaria epidemic.

Optimal control analysis of hepatitis C virus with acute and chronic stages in the presence of treatment and infected immigrants

In this paper, we consider a deterministic hepatitis C virus (HCV) model and study the impact of optimal control on the screening of immigrants and treatment of HCV on the transmission dynamics of the disease in a homogeneous population with constant immigration of susceptibles. First, we derived the condition in which disease-free equilibrium is locally asymptotically stable and established that a stable disease-free equilibrium can only be achieved in the absence of infective immigrants. Second, we investigated the impact of each control mechanism individually and the combinations of these strategies in the control of HCV. The costs associated with each of these strategies are also investigated by formulating the costs function problem as an optimal control problem, and we then use the Pontryagins Maximum Principle to solve the optimal control problems. From the numerical simulations we found that the optimal combination of treatment of acute-infected and chronic-infected individuals control strategy produced the same results as the combination of the three strategies (combination of screening of immigrants, treatment of acute-infected and chronic-infected individuals). By our model and these results, we suggest the treatment of acute-infected and chronic-infected individuals control strategy should be optimized where resources are scarce, because the implementation of the three strategies (combination of screening of immigrants, treatment of acute-infected and chronic-infected individuals) would imply additional cost.

Optimal control analysis of malaria-schistosomiasis co-infection dynamics

This paper presents a mathematical model for malaria--schistosomiasis co-infection in order to investigate their synergistic relationship in the presence of treatment. We first analyse the single infection steady states, then investigate the existence and stability of equilibria and then calculate the basic reproduction numbers. Both the single-infection models and the co-infection model exhibit backward bifurcations. We carrying out a sensitivity analysis of the co-infection model and show that schistosomiasis infection may not be associated with an increased risk of malaria. Conversely, malaria infection may be associated with an increased risk of schistosomiasis. Furthermore, we found that effective treatment and prevention of schistosomiasis infection would also assist in the effective control and eradication of malaria. Finally, we apply Pontryagins Maximum Principle to the model in order to determine optimal strategies for control of both diseases.