Livingstone S. Luboobi

A mathematical model for the dynamics of malaria in a human host and mosquito vector with temporary immunity

Abstract In the paper, we propose a model that tracks the dynamics of malaria in the human host and mosquito vector. Our model incorporates some infected humans that recover from infection and immune humans after loss of immunity to the disease to join the susceptible class again. All the new borne are susceptible to the infection and there is no vertical transmission. The stability of the system is analyzed for the existence of the disease-free and endemic equilibria points. We established that the disease-free equilibrium point is globally asymptotically stable when the reproduction number, R 0 ⩽ 1 and the disease always dies out. For R 0 > 1 the disease-free equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable. Thus, due to new births and immunity loss to malaria, the susceptible class will always be refilled and the disease becomes more endemic.

On oscillatory pattern of malaria dynamics in a population with temporary immunity

We use a model to study the dynamics of malaria in the human and mosquito population to explain the stability patterns of malaria. The model results show that the disease-free equilibrium is globally asymptotically stable and occurs whenever the basic reproduction number, R0 is less than unity. We also note that when R0>1, the disease-free equilibrium is unstable and the endemic equilibrium is stable. Numerical simulations show that recoveries and temporary immunity keep the populations at oscillation patterns and eventually converge to a steady state.

A mathematical model for the dynamics of HIV/AIDS with gradual behaviour change

An HIV/AIDS model that incorporates gradual behaviour change is formulated with a variable force of infection for the adult population. The variability is modelled using a general function of time since introduction of the initial infective and exemplified for three specific functions. Expressions for the time taken for the reproductive number to reduce to unity and expressions for the time taken to attain a stationary steady state are deduced and discussed. Model projections for urban, peri-urban and rural Uganda are compared with corresponding antenatal clinic sites prevalence trends. The analysis shows that the dramatic decline in HIV prevalence in Uganda in the early 1990s was only possible through drastic declines in the force of infection. Since prevalence was high and reductions in frequency of sexual acts was minimal, the huge reduction could be attributed to reductions in probability of transmission per sexual act probably due to increased selective condom use among high risk sexual partnerships since overall condom use was low.

Modeling the impact of climate change on the dynamics of Rift Valley Fever

A deterministic SEIR model of rift valley fever (RVF) with climate change parameters was considered to compute the basic reproduction number ℛ 0 and investigate the impact of temperature and precipitation on ℛ 0. To study the effect of model parameters to ℛ 0, sensitivity and elasticity analysis of ℛ 0 were performed. When temperature and precipitation effects are not considered, ℛ 0 is more sensitive to the expected number of infected Aedes spp. due to one infected livestock and more elastic to the expected number of infected livestock due to one infected Aedes spp. When climatic data are used, ℛ 0 is found to be more sensitive and elastic to the expected number of infected eggs laid by Aedes spp. via transovarial transmission, followed by the expected number of infected livestock due to one infected Aedes spp. and the expected number of infected Aedes spp. due to one infected livestock for both regions Arusha and Dodoma. These results call for attention to parameters regarding incubation period, the adequate contact rate of Aedes spp. and livestock, the infective periods of livestock and Aedes spp., and the vertical transmission in Aedes species.

Mathematical modeling of liver enzyme elevation in HIV mono-infection

HIV-infected individuals are increasingly becoming susceptible to liver disease and, hence, liver-related mortality is on a rise. The presence of CD4+ in the liver and the presence of C-X-C chemokine receptor type 4 (CXCR4) on human hepatocytes provide a conducive environment for HIV invasion. In this study, a mathematical model is used to analyse the dynamics of HIV in the liver with the aim of investigating the existence of liver enzyme elevation in HIV mono-infected individuals. In the presence of HIV-specific cytotoxic T-lymphocytes, the model depicts a unique endemic equilibrium with a transcritical bifurcation when the basic reproductive number is unity. Results of the study show that the level of liver enzyme alanine aminotransferase (ALT) increases with increase in the rate of hepatocytes production. Numerical simulations reveal significant elevation of alanine aminotransferase with increase in viral load. The findings presuppose that while liver damage in HIV infection has mostly been associated with HIV/HBV coinfection and use of antiretroviral therapy (ART), it is possible to have liver damage solely with HIV infection.

Mathematical Model for HIV/AIDS with Complacency in a Population with Declining Prevalence

An HIV/AIDS model incorporating complacency for the adult population is formulated. Complacency is assumed a function of number of AIDS cases in a community with an inverse relation. A method to find the equilibrium state of the model is given by proving a stated theorem. An example to illustrate use of the theorem is also given. Model analysis and simulations show that complacency resulting from dependence of HIV transmission on number of AIDS cases in a community leads to damped periodic oscillations in the number of infectives with oscillations more marked at lower rates of progression to AIDS. The implications of these results to public health with respect to monitoring the HIV/AIDS epidemic and widespread use of antiretroviral (ARV) drugs is discussed.

Stochastic Model for In-Host HIV Dynamics with Therapeutic Intervention

Conference paper presented at “The 2nd EAUMP Conference” on 22nd – 25th August 2012. Arusha - Tanzania

Threshold and stability results for a malaria model in a population with protective intervention among high‐risk groups

Abstract We develop a mathematical model for the dynamics of malaria with a varying population for which new individuals are recruited through immigration and births. In the model, we assume that non‐immune travellers move to endemic regions with sprays, smear themselves with jelly that is repellent to mosquitoes on arrival in malarious regions, others take long term antimalarials, and pregnant women and infants receive full treatment doses at intervals even when they are not sick from malaria (commonly referred to as intermittent preventive therapy). We introduce more features that describe the dynamics of the disease for the control strategies that protect the above vulnerable groups. The model analysis is done and equilibrium points are analyzed to establish their local and global stability. The threshold of the disease, the control reproduction number, is established for which the disease can be eliminated.

The dynamics, causes and possible prevention of Hepatitis E outbreaks.

Rapidly spreading infectious diseases are a serious risk to public health. The dynamics and the factors causing outbreaks of these diseases can be better understood using mathematical models, which are fit to data. Here we investigate the dynamics of a Hepatitis E outbreak in the Kitgum region of northern Uganda during 2007 to 2009. First, we use the data to determine that is approximately 2.25 for the outbreak. Secondly, we use a model to estimate that the critical level of latrine and bore hole coverages needed to eradicate the epidemic is at least and respectively. Lastly, we further investigate the relationship between the co-infection factor for malaria and Hepatitis E on the value of for Hepatitis E. Taken together, these results provide us with a better understanding of the dynamics and possible causes of Hepatitis E outbreaks.

Using contaminated tools fuels outbreaks of Banana Xanthomonas wilt: An optimal control study within plantations using Runge–Kutta fourth-order algorithms

Optimal control theory is applied to a system of ordinary differential equations modeling banana Xanthomonas wilt within plantations. The objective is to reduce the proportion of infected plants by use of controls representing two types of preventive methods: vector and contaminated tool prevention. The optimal controls are characterized in terms of the optimality system, which is solved analytically and numerically for several scenarios.