J. Y. T. Mugisha

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.

Mathematical Analysis of the Dynamics of Visceral Leishmaniasis in the Sudan

In this paper we develop a mathematical model to study the dynamics of visceral leishmaniasis in the Sudan. To develop this model we consider the dynamics of the disease between three different populations, human, reservoir and vector populations. The model is analyzed at equilibrium and the stability of the equilibria is analyzed. The basic reproduction number is derived, and the threshold conditions for disease elimination established. Results show that the disease can be eliminated under certain conditions. Simulations of the model show that human treatment helps in disease control, and its synergy with vector control will more likely result in the elimination of the disease.

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.

The role of backward mutations on the within-host dynamics of HIV-1

The quality of life for patients infected with human immunodeficiency virus (HIV-1) has been positively impacted by the use of antiretroviral therapy (ART). However, the benefits of ART are usually halted by the emergence of drug resistance. Drug-resistant strains arise from virus mutations, as HIV-1 reverse transcription is prone to errors, with mutations normally carrying fitness costs to the virus. When ART is interrupted, the wild-type drug-sensitive strain rapidly out-competes the resistant strain, as the former strain is fitter than the latter in the absence of ART. One mechanism for sustaining the sensitive strain during ART is given by the virus mutating from resistant to sensitive strains, which is referred to as backward mutation. This is important during periods of treatment interruptions as prior existence of the sensitive strain would lead to replacement of the resistant strain. In order to assess the role of backward mutations in the dynamics of HIV-1 within an infected host, we analyze a mathematical model of two interacting virus strains in either absence or presence of ART. We study the effect of backward mutations on the definition of the basic reproductive number, and the value and stability of equilibrium points. The analysis of the model shows that, thanks to both forward and backward mutations, sensitive and resistant strains co-exist. In addition, conditions for the dominance of a viral strain with or without ART are provided. For this model, backward mutations are shown to be necessary for the persistence of the sensitive strain during ART.

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.

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.

The dynamics of a fisheries model with feeding patterns and harvesting: Lates niloticus and Oreochromis niloticus in Lake Victoria

Abstract In this study, we develop a model based on a standard Lotka-Volterra prey–predator model that is used to predict and explain the effect of predation by the Nile perch on Nile tilapia. The effect of harvesting of both the Nile perch and Nile tilapia on the population dynamics of the fish species is investigated. The model incorporates three developmental stages (i.e. young, juveniles and mature) of each species and the effect of preference of the Nile perch for either young or juveniles Nile tilapia is pointed out. Lake Victoria being a vast lake, the carrying capacity is not a vital parameter considered in the model. Both qualitative and numerical analyses of the model are carried out. The results of this study show that predation by the Nile perch is one of the main causes of fish stock depletion in Lake Victoria. The findings further show that uncontrolled human exploitation (over fishing) of the stock leads to loss of fish biodiversity in Lake Victoria. The basic predation numbers R 0 , show that increasing the harvesting rate of the mature Nile perch leads to stable stationary states. However, increasing the predation rates of the Nile perch leads to unstable ecosystem.