Mathematical models to describe in vivo and in vitro immunological response to infection in humans by HIV-1 have been of major concern due to the rich variety of parameters affecting its dynamics. In this paper, HIV-1 in vivo dynamics is studied to predict and describe its evolutions in presence of ARVs using delay differential equations. The delay is used to account for the latent period of time that elapsed between HIV -CD4 + T cell binding (infection) and production of infectious virus from this host cell. The model uses four variables: healthy CD4 + T-cells (T), infected CD4 + T-cells (T * ), infectious virus (V I ) and noninfectious virus (V N ). Of importance is effect of time delay and drug efficacy on stability of disease free and endemic equilibrium points. Analytical results showed that DFE is stable for all > 0. On the other hand, there is a critical value of delay > 0, such that for all > , the EEP is stable but unstable for < . The critical value of delay is the bifurcation value where the HIV-1 in vivo dynamics undergoes a Hopf-bifurcation. This stability in both equilibria is achieved only if the drug efficacy 0 ≤ ≤ 1 is above a threshold value of . Numerical simulations show that this stability is achieved at the drug efficacy of = 0.59 and time delay of = 0.65 days. This ratifies the fact that if CD4 + T cells remain inactive for long periods of time > the HIV-1 viral materials will not be reproduced, and the immune system together with treatment will have enough time to clear the viral materials in the blood and thus the EEP is maintain.
Malaria is one of the major causes of deaths and ill health in endemic regions of sub-Saharan Africa and beyond despite efforts made to prevent and control its spread. Epidemiological models on how malaria is spread have made a substantial contribution on the understanding of disease changing aspects. Previous researchers have used Susceptible -Exposed-Infectious-Recovered (SEIR) model to explain how malaria is spread using ordinary differential equations. In this paper we develop mathematical SEIR model to define the dynamics of the spread of malaria using Delay differential equations with four control measures such as long lasting treated insecticides bed nets, intermittent preventive treatment of malaria in pregnant women (IPTP), intermittent preventive treatment of malaria in infancy (IPTI) and indoor residual spraying. The model is analyzed and reproduction number derived using next generation matrix method and its stability is checked by Jacobean matrix. Positivity of solutions and boundedness of the model is proved. We show that the disease free equilibrium is locally asymptotically stable if R 0 <1 (R 0 -reproduction number) and is unstable if R 0 >1. Numerical simulation shows that, with proper treatment and control measures put in place the disease is controlled.
Numerous models of mathematics have existed to pronounce the immunological response to contagion by human immunodeficiency virus (HIV-1). The models have been used to envisage the regression of HIV-1 in vitro and in vivo dynamics. Ordinarily the studies have been on the interface of HIV virions, CD4+T-cells and Antiretroviral (ARV). In this study, time delay, chemotherapy and role of CD8+T-cells is considered in the HIV-1 in-vivo dynamics. The delay is used to account for the latent time that elapses between exposure of a host cell to HIV-1 and the production of contagious virus from the host cell. This is the period needed to cause HIV-1 to replicate within the host cell in adequate number to become transmittable. Chemotherapy is by use of combination of Reverse transcriptase inhibitor and Protease inhibitor. CD8+T-cells is innate immune response. The model has six variables: Healthy CD4+T-cells, Sick CD4+T-cells, Infectious virus, Noninfectious virus, used CD8+T-cells and unused CD8+T-cells. Positivity and boundedness of the solutions to the model equations is proved. In addition, Reproduction number (R 0 ) is derived from Next Generation Matrix approach. The stability of disease free equilibrium is checked by use of linearization of the model equation. We show that the Disease Free Equilibrium is locally stable if and only if R 0 <1 and unstable otherwise. Of significance is the effect of CD8+ T-cells, time delay and drug efficacy on stability of Disease Free Equilibrium (DFE). From analytical results it is evident that for all τ > 0, Disease Free Equilibrium is stable when τ =0.67. This stability is only achieved if drug efficacy is administered. The results show that when drug efficacy of α 1 =0.723 and α 2 =0.723 the DFE is achieved.
The population size of every country or government is very important in planning on effective service delivery. The cost of conducting population census yearly is of great significance to the country's budget and many countries conduct population census once in a decade. This makes planning and provision of services to be based on mere approximation. Provision of free maternity services, estimation of national hospital insurance fund premium for medical care, and provision of retirement benefits, payment of allowances to the aged require accurate demographic statistics. In this study, population dynamics is described using a stochastic model, where population is put into distinct and disjoint age classes: Juvenile, sub-Adult, Adult, Resting-Adult, Senior Citizens and the Aged. These structures are assigned intra and inter group transmission rates which form the elements of transmission matrix and presented in form of a Leslie model. The model was modified to allow stochastic variation of transition parameters which is affected by demographic and environmental factors, specifically the effect of contraceptives to control population. It was found that intermittent implementation of control strategy at 50% and 70% efficacy yields a steady population growth rate of λ=1.39 and a steady population distribution of P=(23%, 10%, 23%, 18%, 23%, 20%, 6%) T .
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