In this study, we have formulated a mathematical model based on a system of ordinary differential equations to study the dynamics of typhoid fever disease incorporating protection against infection. The existence of the steady states of the model are determined and the basic reproduction number is computed using the next generation matrix approach. Stability analysis of the model is carried out to determine the conditions that favour the spread of the disease in a given population. Numerical simulation of the model carried showed that an increase in protection leads to low disease prevalence in a population.

In this paper, a deterministic co-infection model incorporating protection from infection for both pneumonia infection and HIV/AIDS is considered. The model is shown to be positively invariant as well as bounded. Specifically we consider the case of maximum protection against pneumonia and the case of maximum protection against HIV/AIDS. In both cases, the endemic states are shown to exist provided the reproduction number for each case is greater than unity. Furthermore, by use of a suitable Lyapunov function, the endemic states have been shown to be globally asymptotically stable. Numerical sim-2070 Joyce K. Nthiiri et al. ulations indicate that enhanced protection against a disease lowers the rate of infection or disease prevalence.

We develop a two-patch migration model of the classical Lotka-Voltera predator-prey system with a time lag in the migration between patches. We show that when the migration rate is less than the prey growth rate the species in at least one patch survives. When the migration rate is greater than the prey growth rate, the species in both Adu A.M. Wasike et al. patches do not survive. Furthermore, when the migration rate is equal to the prey growth rate, the population will oscillate.

The model describing the interaction between the predator and prey species is referred to as a predator-prey model. The migration of these species from one patch to another may not be instantaneous. This may be due to barriers such as a swollen river or a busy infrastructure through the natural habitat. Recent predator-prey models have either incorporated a logistic growth for the prey population or a time delay in migration of the two species. Predator-prey models with logistic growth that integrate time delays in density-dependent migration of both species have been given little attention. A Rosenzweig-MacAurther model with density-dependent migration and time delay in the migration of both species is developed and analyzed in this study. The Analysis of the model when the prey migration rate is greater than or equal to the prey growth rate, the two species will coexist, otherwise, at least one species will become extinct. A longer delay slows down the rate at which the predator and prey population increase or decrease, thus aecting the population density of these species. The prey migration due to the predator density does not greatly affect the prey density and existence compared to the other factors that cause the prey to migrate. These factors include human activities in the natural habitats like logging and natural causes like bad climatic conditions, limited food resources and overpopulation of the prey species in a patch among others.

In this paper a within host mathematical model for Human Immunodeficiency Virus (HIV) transmission incorporating treatment is formulated. The model takes into account the efficacy of combined antiretroviral treatment on viral growth and T cell population in the human blood. The existence of an infection free and positive endemic equilibrium is established. The basic reproduction number R0 is derived using the method of next generation matrix. We perform local and global stability analysis of the equilibria points and show that if R0<1, then the infection free equilibrium is globally asymptotically stable and theoretically the virus is cleared and the disease dies out and if R0>1, then the endemic equilibrium is globally asymptotically stable implying that the virus persists within the host. Numerical simulations are carried out to investigate the effect of treatment on the within host infection dynamics.