Monkeypox (MPX), similar to both smallpox and cowpox, is caused by the monkeypox virus (MPXV). It occurs mostly in remote Central and West African communities, close to tropical rain forests. It is caused by the monkeypox virus in the Poxviridae family, which belongs to the genus Orthopoxvirus. We develop and analyse a deterministic mathematical model for the monkeypox virus. Both local and global asymptotic stability conditions for disease-free and endemic equilibria are determined. It is shown that the model undergo backward bifurcation, where the locally stable disease-free equilibrium coexists with an endemic equilibrium. Furthermore, we determine conditions under which the disease-free equilibrium of the model is globally asymptotically stable. Finally, numerical simulations to demonstrate our findings and brief discussions are provided. The findings indicate that isolation of infected individuals in the human population helps to reduce disease transmission.

The pandemic of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) took the world by surprise. Following the first outbreak of COVID-19 in December 2019, several models have been developed to study and understand its transmission dynamics. Although the spread of COVID-19 is being slowed down by vaccination and other interventions, there is still a need to have a clear understanding of the evolution of the pandemic across countries, states and communities. To this end, there is a need to have a clearer picture of the initial spread of the disease in different regions. In this project, we used a simple SEIR model and a Bayesian inference framework to estimate the basic reproduction number of COVID-19 across Africa. Our estimates vary between 1.98 (Sudan) and 9.66 (Mauritius), with a median of 3.67 (90% CrI: 3.31–4.12). The estimates provided in this paper will help to inform COVID-19 modeling in the respective countries/regions.

In this paper, we develop a new disease-resistant mathematical model with a fraction of the susceptible class under imperfect vaccine and treatment of both the symptomatic and quarantine classes. With standard incidence when the associated reproduction threshold is less than unity, the model exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium. It is then proved that this phenomenon vanishes either when the vaccine is assumed to be 100% potent and perfect or the Standard Incidence is replaced with a Mass Action Incidence in the model development. Furthermore, the model has a unique endemic and disease-free equilibria. Using a suitable Lyapunov function, the endemic equilibrium and disease free equilibrium are proved to be globally-asymptotically stable depending on whether the control reproduction number is less or greater than unity. Some numerical simulations are presented to validate the analytic results.

The novel coronal virus has spread across more than 213 countries within the space of six months causing devastating public health hazard and monumental economic loss. In the absence of clinically approved pharmaceutical intervention, attentions are shifted to non-pharmaceutical controls to mitigate the burden of the novel pandemic. In this regard, a ten mutually exclusive compartmental mathematical model is developed to investigate possible effects of both pharmaceutical and non-pharmaceutical controls incorporating both private and government’s quarantine and treatments. Several reproduction numbers were calculated and used to determine the impact of both control measures as well as projected benefits of social distancing, treatments and vaccination. We investigate and compare the possible impact of social distancing incorporating different levels of vaccination, with vaccination programme incorporating different levels of treatment. Using the officially published South African COVID-19 data, the numerical simulation shows that the community reproduction threshold will be 30 when there is no social distancing but will drastically reduced to 5 (about 83% reduction) when social distancing is enforced. Furthermore, when there is vaccination with perfect efficacy, the community reproduction threshold will be 4 which increases to 12 (about 67% increment) with-out vaccination. We also established that the implementation of both interventions is enough to curtail the spread of COVID-19 pandemic in South Africa which is in confirmation with the recommendation of the world health organization.

We develop a virus-resistant HIV-1 mathematical model with behavioural change in HIV-1 resistant non-progressors. The model has both disease-free and endemic equilibrium points that are proved to be locally asymptotically stable depending on the value of the associated reproduction numbers. In both models, a non-linear Goh{Volterra Lyapunov function was used to prove that the endemic equilibrium point is globally asymptotically stable for special case while the method of Castillo-Chavez was used to prove the global asymptotic stability of the disease-free equilibrium point. In both the analytic and numerical results, this study shows that in the context of resistance to HIV/AIDS, total abstinence can also play an important role in protection against this notorious infectious disease.