Ebola virus disease (EVD) is a severe infection with an extremely high fatality rate spread through direct and indirect contacts. Recently, an outbreak of EVD in West Africa brought public attention to this deadly disease. We study the spread of EVD through a two-patch model. We determine the basic reproduction number, the disease-free equilibrium, two boundary equilibria and the endemic equilibrium when the disease persists in the two sub-populations for specific conditions. Further, we introduce time-dependent controls into our proposed model. We analyse the optimal control problem where the control system is a mathematical model for EVD that incorporates educational campaigns. The control functions represent educational campaigns in their respective patches, with one patch having more effective controls than the other. We aim to study how these control measures would be implemented for a certain time period, in order to reduce or eliminate EVD in the respective communities, while minimising the intervention implementation costs. Numerical simulations results are provided to illustrate the dynamics of the disease in the presence of controls.
Herpes simplex virus type 2 (HSV-2) is the most prevalent sexually transmitted disease worldwide, despite the availability of highly effective antiviral treatments. In this paper, a basic mathematical model for the spread of HSV-2 incorporating all the relevant biological details and poor treatment adherence is proposed and analysed. Equilibrium states of the model are determined and their stability has been investigated. The basic model is then extended to incorporate a time dependent intervention strategy. The aim of the control is tied to reducing the rate at which HSV-2 patients in treatment quit therapy before completion. Practically, this control can be implemented through monitoring and counselling all HSV-2 patients in treatment. The Pontryagin's maximum principle is used to characterize the optimal level of the control, and the resulting optimality system is solved numerically. Overall, the study demonstrates that though time dependent control will be effective on controlling new HSV-2 cases it may not be sustainable for certain time intervals.
Herpes simplex virus (HSV-2) triples the risk of acquiring human immunodeficiency virus (HIV) and contributes to more than 50% of HIV infections in other parts of the world. A deterministic mathematical model for the co-interaction of HIV and HSV-2 in a community, with all the relevant biological detail and poor HSV-2 treatment adherence is proposed. The threshold parameters of the model are determined and stabilities are analysed. Further, we applied optimal control theory. We proved the existence of the optimal control and characterized the controls using Pontryagin’s maximum principle. The controls represent monitoring and counselling of individuals infected with HSV-2 only and the other represent monitoring and counselling of individuals dually infected with HIV and HSV-2. Numerical results suggests that more effort should be devoted to monitoring and counselling of individuals dually infected with HIV and HSV-2 as compared to those infected with HSV-2 only. Overall, the study demonstrate that, though time dependent controls will be effective on controlling HIV cases, they may not be sustainable for certain time intervals.
HIV/AIDS has been somehow linked to prostitution for decades now. A mathematical model is presented to assess the link between prostitution and HIV transmission. The epidemic thresholds known as the reproduction numbers and equilibria for the model are determined and stabilities analyzed. Analysis of the reproduction numbers suggests that HIV/AIDS control using antiretroviral therapy is more effective in the absence of prostitution. Numerical simulations further show high levels of HIV/AIDS when percentage of prostitutes in the community is high. Results from this study suggest that effectively controlling HIV/AIDS requires strategies that address both prostitution and HIV/AIDS transmission. Addressing HIV/AIDS through condom use and antiretroviral therapy may not be enough to stem HIV/AIDS in the community as some drug/alcohol misusing prostitutes may not be able to negotiate for safe sex while they are in drunken stupor. Furthermore, prostitutes are likely to get infected by different HIV strains some of which may be resistant to the antiretroviral therapy regimen in use.
Hepatitis D virus is an infectious subviral agent that can only propagate in people infected with hepatitis B virus. In this study, we modified and further developed a recent model for early hepatitis D virus and hepatitis B virus kinetics to better reproduce hepatitis D virus and hepatitis B virus kinetics measured in infected patients during anti-hepatitis D virus treatment. The analytical solutions were provided to highlight the new features of the modified model. The improved model offered significantly better prospects for modeling hepatitis D virus and hepatitis B virus interactions.
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