Winston Garira

A mathematical analysis of the effects of control strategies on the transmission dynamics of malaria.

We formulate a deterministic model with two latent periods in the non-constant host and vector populations, in order to theoretically assess the potential impact of personal protection, treatment and possible vaccination strategies on the transmission dynamics of malaria. The thresholds and equilibria for the model are determined. The model is analysed qualitatively to determine criteria for control of a malaria epidemic and is used to compute the threshold vaccination and treatment rates necessary for community-wide control of malaria. In addition to having a disease-free equilibrium, which is locally asymptotically stable when the basic reproductive number is less than unity, the model exhibits the phenomenon of backward bifurcation where a stable disease-free equilibrium coexists with a stable endemic equilibrium for a certain range of associated reproductive number less than one. From the analysis we deduce that personal protection has a positive impact on disease control but to eradicate the disease in the absence of any other control measures the efficacy and compliance should be very high. Our results show that vaccination and personal protection can suppress the transmission rates of the parasite from human to vector and vice-versa. If the treated populations are infectious then certain conditions should be satisfied for treatment to reduce the spread of malaria in a community. Among the interesting dynamical behaviours of the model, numerical simulations show a backward bifurcation which gives a challenge to the designing of effective control measures.

Modeling HIV/AIDS and Tuberculosis Coinfection

An HIV/AIDS and TB coinfection model which considers antiretroviral therapy for the AIDS cases and treatment of all forms of TB, i.e., latent and active forms of TB, is presented. We begin by presenting an HIV/AIDS-TB coinfection model and analyze the TB and HIV/AIDS submodels separately without any intervention strategy. The TB-only model is shown to exhibit backward bifurcation when its corresponding reproduction number is less than unity. On the other hand, the HIV/AIDS-only model has a globally asymptotically stable disease-free equilibrium when its corresponding reproduction number is less than unity. We proceed to analyze the full HIV-TB coinfection model and extend the model to incorporate antiretroviral therapy for the AIDS cases and treatment of active and latent forms of TB. The thresholds and equilibria quantities for the models are determined and stabilities analyzed. From the study we conclude that treatment of AIDS cases results in a significant reductions of numbers of individuals progressing to active TB. Further, treatment of latent and active forms of TB results in delayed onset of the AIDS stage of HIV infection.

Effects of treatment and drug resistance on the transmission dynamics of malaria in endemic areas

We present a mathematical model for malaria treatment and spread of drug resistance in an endemic population. The model considers treated humans that remain infectious for some time and partially immune humans who are also infectious to mosquitoes although their infectiousness is always less than their non immune counterparts. The model is formulated by considering delays in the latent periods in both mosquito and human populations and in the period within which partial immunity is lost. Qualitative analysis of the model including positivity and boundedness of solutions is performed. Analysis of the reproductive numbers shows that if the treated humans become immediately uninfectious to mosquitoes then treatment will always reduce the number of sensitive infections. If however treated humans are infectious then for treatment to effectively reduce the number of sensitive infections, the ratio of the infectious period of the treated humans to the infectious period of the untreated humans multiplied by the ratio of the transmission rate from a treated human to the transmission rate of an untreated human should be less than one. Our results show that the spread of drug resistance with treatment as a control strategy depends on the ratio of the infectious periods of treated and untreated humans and on the transmission rates from infectious humans with resistant and sensitive infections. Numerical analysis is performed to assess the effects of treatment on the spread of resistance and infection. The study provides insight into the possible intervention strategies to be employed in malaria endemic populations with resistant parasites by identifying important parameters.

Tuberculosis transmission model with chemoprophylaxis and treatment.

A tuberculosis model which incorporates treatment of infectives and chemoprophylaxis is presented. The model assumes that latently infected individuals develop active disease as a result of endogenous re-activation, exogenous re-infection and disease relapse, though a small fraction is assumed to develop active disease soon after infection. We start by formulating and analyzing a TB model without any intervention strategy that we extend to incorporate chemoprophylaxis and treatment of infectives. The epidemic thresholds known as reproduction numbers and equilibria for the models are determined, and stabilities analyzed. The reproduction numbers for the models are compared to assess the possible community benefits achieved by treatment of infectives, chemoprophylaxis and a holistic approach of these intervention strategies. The study shows that treatment of infectives is more effective in the first years of implementation (≈ 10 years) as treatment results in clearing active TB immediately and there after chemoprophylaxis will do better in controlling the number of infectives due to reduced progression to active TB.

Mathematical Analysis of a Two Strain HIV/AIDS Model with Antiretroviral Treatment

A two strain HIV/AIDS model with treatment which allows AIDS patients with sensitive HIV-strain to undergo amelioration is presented as a system of non-linear ordinary differential equations. The disease-free equilibrium is shown to be globally asymptotically stable when the associated epidemic threshold known as the basic reproduction number for the model is less than unity. The centre manifold theory is used to show that the sensitive HIV-strain only and resistant HIV-strain only endemic equilibria are locally asymptotically stable when the associated reproduction numbers are greater than unity. Qualitative analysis of the model including positivity, boundedness and persistence of solutions are presented. The model is numerically analysed to assess the effects of treatment with amelioration on the dynamics of a two strain HIV/AIDS model. Numerical simulations of the model show that the two strains co-exist whenever the reproduction numbers exceed unity. Further, treatment with amelioration may result in an increase in the total number of infective individuals (asymptomatic) but results in a decrease in the number of AIDS patients. Further, analysis of the reproduction numbers show that antiretroviral resistance increases with increase in antiretroviral use.

Modelling Immune Response and Drug Therapy in Human Malaria Infection

A new intra-host model of malaria that describes the dynamics of the blood stages of the parasite and its interaction with red blood cells and immune effectors is proposed. Local and global stability of the disease free equilibrium are investigated. Conditions for existence and uniqueness of the endemic equilibrium are derived. An intra-host basic reproductive number is identified. We deduce that drugs based on inhibiting parasite production are more effective than those based on inhibiting merozoite invasion of erythrocytes. We extend the model to incorporate, in addition to immune response, drug therapy, following treatment with antimalarial drugs. Using stability analysis of the model, it is shown that infection can be eradicated within the host if the drug efficacy level exceeds a certain threshold value. It will persist if the efficacy is below this threshold. Numerical simulations are done to verify the analytic results and illustrate possible behaviour of the models.

Modelling the effects of pre-exposure and post-exposure vaccines in tuberculosis control

Epidemic control strategies alter the spread of the disease in the host population. In this paper, we describe and discuss mathematical models that can be used to explore the potential of pre-exposure and post-exposure vaccines currently under development in the control of tuberculosis. A model with bacille Calmette-Guerin (BCG) vaccination for the susceptibles and treatment for the infectives is first presented. The epidemic thresholds known as the basic reproduction numbers and equilibria for the models are determined and stabilities are investigated. The reproduction numbers for the models are compared to assess the impact of the vaccines currently under development. The centre manifold theory is used to show the existence of backward bifurcation when the associated reproduction number is less than unity and that the unique endemic equilibrium is locally asymptotically stable when the associated reproduction number is greater than unity. From the study we conclude that the pre-exposure vaccine currently under development coupled with chemoprophylaxis for the latently infected and treatment of infectives is more effective when compared to the post-exposure vaccine currently under development for the latently infected coupled with treatment of the infectives.

Modelling circumcision and condom use as HIV/AIDS preventive control strategies

We present a sex-structured model for heterosexual transmission of HIV/AIDS with explicit incubation period for modelling the effect of male circumcision as a preventive strategy for HIV/AIDS. The model is formulated using integro-differential equations, which are shown to be equivalent to delay differential equations with delay due to incubation period. The threshold and equilibria for the model are determined and stabilities are examined. We extend the model to incorporate the effects of condom use as another preventive strategy for controlling HIV/AIDS. Basic reproductive numbers for these models are computed and compared to assess the effectiveness of male circumcision and condom use in a community. The models are numerically analysed to assess the effects of the two preventive strategies on the transmission dynamics of HIV/AIDS. We conclude from the study that in the continuing absence of a preventive vaccine or cure for HIV/AIDS, male circumcision is a potential effective preventive strategy of HIV/AIDS to help communities slow the development of the HIV/AIDS epidemic and that it is even more effective if implemented jointly with condom use. The study provides insights into the possible community benefits that male circumcision and condom use as preventive strategies provide in slowing or curtailing the HIV/AIDS epidemic.

MATHEMATICAL ANALYSIS OF THE TRANSMISSION DYNAMICS OF SCHISTOSOMIASIS IN THE HUMAN-SNAIL HOSTS

The spread and persistence of schistosomiasis are some of the more complex host parasite processes to model mathematically because of the different larval forms assumed by the parasite and the requirement of two hosts during the life cycle. We construct a deterministic mathematical model to study the transmission dynamics of schistosomiasis where the miracidia and cercariae dynamics are incorporated. The model is analyzed to gain insights into the qualitative features of the equilibrium which allows the determination of the basic reproductive number. Conditions for existence of the endemic equilibrium are discussed and its local stability is determined using the Center Manifold Theory. Analytical and numerical techniques are employed to assess the conditions of containment and persistence of schistosomiasis. Our results show that control strategies that target the transmission of the disease from the snail to man will be more effective in the control of the disease than those that block the transmission from man to snail.

MATHEMATICAL MODELING OF CHEMOTHERAPY OF HUMAN TB INFECTION

This work assesses the impact of the first line drug regimen on active disease control under the stipulated time of tuberculosis (TB) treatment. In an effort to understand why a robust immune response mechanism sometimes fails to completely control TB infection, we first developed a model that captures the human immune response mechanisms to Mycobacterium tuberculosis (Mtb) infection. We then extended the model to include drug therapy. The drug therapy model is used to assess the potency of the recommended six-month TB drug chemotherapy in infected individuals. The efficacy of each drug was explored and observations show that low drug efficacy values result in extension of treatment period. The numerical results confirm typical clinical disease progression patterns noticed in individuals under TB therapy. The drug model simulations and analysis show that administration of the recommended first line three-drug regimen normally cures the TB infection. Using the model, we established that only Isoniazid monotherapy drug treatment, and any combination therapy of two drugs including Isoniazid are potent enough to resolve the TB infection.