Objective: Coronavirus disease 2019 (COVID-19) is a pandemic respiratory illness spreading from person-to-person caused by a novel coronavirus and poses a serious public health risk. The goal of this study was to apply a modified susceptible-exposed-infectious-recovered (SEIR) compartmental mathematical model for prediction of COVID-19 epidemic dynamics incorporating pathogen in the environment and interventions. The next generation matrix approach was used to determine the basic reproduction number $$R_0$$R 0 . The model equations are solved numerically using fourth and fifth order Runge–Kutta methods. Results: We found an $$R_0$$R 0 of 2.03, implying that the pandemic will persist in the human population in the absence of strong control measures. Results after simulating various scenarios indicate that disregarding social distancing and hygiene measures can have devastating effects on the human population. The model shows that quarantine of contacts and isolation of cases can help halt the spread on novel coronavirus.

The control of spread of HIV to reduce its effects on a population is an important role of public health. HIV testing and counselling (HTC) and eventual enrolment of infected individuals on anti retro-viral treatment (ART) as soon as possible to reduce the risk of dying is currently the main intervention against HIV. Mathematical models can be used to study the effects of HIV prevention, testing and treatment with ART on HIV patients. In this study, we employ a deterministic model to provide a quantification of HIV prevention, testing and treatment with ART as public health measurements in the fight against HIV infection. Lyapunov function has been used to derive a condition that ensures that the model system is globally asymptotically stable when R₀ is less than unity. Through sensitivity analysis, we determine the relative importance of model parameters for disease transmission. The sensitivity analysis results suggest that the effective contact rates are mechanisms fuelling HIV epidemic proliferation while ART efficacy reduces the incidence. The model is fitted to HIV surveillance data obtained from world in data website. Although the results show a high proportion of individuals with HIV in Kenya, the incidence curve is indicative of a declining HIV infection and settling at an endemic steady state. The results are suggestive of the need to promote preventive mechanism against the occurrence of new infections. Moreover, the results show that the combination of several control mechanisms would significantly reduce the spread of the disease, if we maintain the level of each control high.

HIV is one of the major causes of deaths, especially in Sub-Saharan Africa. In this paper, an in vivo deterministic model of differential equations is presented and analyzed for HIV dynamics. Optimal control theory is applied to investigate the key roles played by the various HIV treatment strategies. In particular, we establish the optimal strategies for controlling the infection using three treatment regimes as the system control variables. We have applied Pontryagin’s Maximum Principle in characterizing the optimality control, which then has been solved numerically by applying the Runge-Kutta forth-order scheme. The numerical results indicate that an optimal controlled treatment strategy would ensure significant reduction in viral load and also in HIV transmission. It is also evident from the results that protease inhibitor plays a key role in virus suppression; this is not to underscore the benefits accrued when all the three drug regimes are used in combination.

The control of HIV demands different interventions for different age groups. In the present manuscript, we formulate and analyze a mathematical compartmental models of HIV transmission within and between two age groups in Kenya. We fitted the model to data using MCMC technique and inferred the parameters. We also estimate the reproduction numbers, namely within age group transmission and between age groups transmission basic reproduction numbers. The analysis of the data revealed that there is significant difference in mean number of new HIV infections between males and females within the two age groups. More, particularly, females are highly infected with HIV as compared to their male counterparts. Calculation of the reproduction numbers within and between age groups provides insights into control that cannot be deduced simply from observations on the prevalence of infection. More specifically, the analysis showed that the per capita rate of HIV transmission was highest when there is interaction between young adults to adults and most HIV infections occurred in adult population. Furthermore, the sensitivity analysis demonstrated that the reproduction numbers depend mainly on the probabilities of infection. This results can be used to guide HIV interventions, condom distribution and antiretroviral therapy. Precisely, the results can be used to educate the young adults on practicing safe sex with their partners in order to contain the occurrence of new infections.

Malaria is one of the three most dangerous infectious diseases worldwide (along with HIV/AIDS and tuberculosis). In this paper we compare the disease dynamics of the deterministic and stochastic models in order to determine the effect of randomness in malaria transmission dynamics. Relationships between the basic reproduction number for malaria transmission dynamics between humans and mosquitoes and the extinction thresholds of corresponding continuous-time Markov chain models are derived under certain assumptions. The stochastic model is formulated using the continuous-time discrete state Galton-Watson branching process (CTDSGWbp). The reproduction number of deterministic models is an essential quantity to predict whether an epidemic will spread or die out. Thresholds for disease extinction from stochastic models contribute crucial knowledge on disease control and elimination and mitigation of infectious diseases. Analytical and numerical results show some significant differences in model predictions between the stochastic and deterministic models. In particular, we find that malaria outbreak is more likely if the disease is introduced by infected mosquitoes as opposed to infected humans. These insights demonstrate the importance of a policy or intervention focusing on controlling the infected mosquito population if the control of malaria is to be realized.