Asymptotic properties of an HIV/AIDS model with a time delay

A mathematical model for HIV/AIDS with explicit incubation period is presented as a system of discrete time delay differential equations and its important mathematical features are analysed. The disease-free and endemic equilibria are found and their local stability investigated. We use the Lyapunov functional approach to show the global stability of the endemic equilibrium. Qualitative analysis of the model including positivity and boundedness of solutions, and persistence are also presented. The HIV/AIDS model is numerically analysed to asses the effects of incubation period on the dynamics of HIV/AIDS and the demographic impact of the epidemic using the demographic and epidemiological parameters for Zimbabwe.     

Identification and antibacterial evaluation of bioactive compounds from Garcinia kola (Heckel) seeds

We assessed the bioactivity of G. kola seeds on Streptococcus pyogenes, Staphylococcus aureus, Plesiomonas shigelloides and Salmonella typhimurium. The crude ethyl acetate, ethanol, methanol, acetone and aqueous extracts were screened by the agar-well diffusion method and their activities were further determined by Minimum Inhibitory Concentration (MIC) and Minimum Bactericidal Concentration (MBC) assays. The extracts were fractionated by Thin Layer Chromatography (TLC). Bioautography was used to assess the activity of the possible classes of compounds present in the more active extracts. Column chromatography was used to purify the active compounds from the mixture, while GC-MS was used to identify the phytocomponents of the fractions. The inhibition zone diameters of the extracts ranged from 0-24 ± 1.1 mm, while MIC and MBC values ranged between 0.04-1.25 mg/mL and 0.081-2.5 mg/mL, respectively. The chloroform/ethyl acetate/formic acid (CEF) solvent system separated more active compounds. The MIC of the fractions ranged between 0.0006-2.5 mg/mL. CEF 3 (F3), CEF 11 (F11) and CEF 12 (F12) revealed the presence of high levels of linoleic acid, 1,2-benzenedicarboxylic acid and 2,3-dihydro-3,5-dihydroxy-6-methyl ester, respectively. The results obtained from this study justify the use of this plant in traditional medicine and provide leads which could be further exploited for the development of new and potent antimicrobials.     

HIV/AIDS Model with Delay and the Effects of Stochasticity

We present a deterministic HIV/AIDS model with delay. We then extend the model by adjoining terms capturing stochastic effects. The intensity of the fluctuations in the stochastic system is analytically evaluated using Fourier transform methods. We carry out simulations to assess differences in the dynamical behavior of the deterministic and stochastic models. Simulation results show that they are no significant differences in the behavior of the two models.     

Assessing the effects of homosexuals and bisexuals on the intrinsic dynamics of HIV/AIDS in heterosexual settings

A deterministic compartmental sex-structured HIV/AIDS model for assessing the effects of homosexuals and bisexuals on the intrinsic dynamics of the disease in heterosexual settings in which homosexuality and bisexuality issues have remained taboo is presented. The epidemic threshold and equilibria for the model are determined and stabilities are investigated. Comprehensive qualitative analysis of the model including invariance of solutions and permanence are carried out. The epidemic threshold known as the basic reproductive number suggests that heterosexuality, homosexuality, and bisexuality influence the growth of the epidemic in HIV/AIDS affected populations and the partial reproductive number (homosexuality induced or heterosexuality and bisexuality induced) with the larger value influences the overall dynamics of the epidemic in a setting. Numerical simulations of the model show that as long as one of the partial reproductive numbers is greater than unity, the disease will exist in the population. We conclude from the study that homosexuality and bisexuality enlarge the epidemic in a heterosexual setting. The theoretical study highlights the need to carry out substantial research to map homosexuals and bisexuals as it has remained unclear as to what extent this group has contributed to the epidemic in heterosexual settings especially in southern Africa, which has remained the epidemiological locus of the epidemic.     

Modelling and analysis of the effects of malnutrition in the spread of cholera

Although cholera has existed for ages, it has continued to plague many parts of the world. In this study, a deterministic model for cholera in a community is presented and rigorously analysed in order to determine the effects of malnutrition in the spread of the disease. The important mathematical features of the cholera model are thoroughly investigated. The epidemic threshold known as the basic reproductive number and equilibria for the model are determined, and stabilities are investigated. The disease-free equilibrium is shown to be globally asymptotically stable. Local stability of the endemic equilibrium is determined using centre manifold theory and conditions for its global stability are derived using a suitable Lyapunov function. Numerical simulations suggest that an increase in susceptibility to cholera due to malnutrition results in an increase in the number of cholera infected individuals in a community. The results suggest that nutritional issues should be addressed in impoverished communities affected by cholera in order to reduce the burden of the disease.