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A mathematical model for infectious disease epidemics with behaviour change and treatment is formulated and analysed. It is indicated that behaviour modification by the population has a significant impact on the dynamics of the disease. Moreover, an optimal control theory is applied to propose the best possible combination of efforts in controlling a disease. It is shown that it may not be necessary to continuously apply treatment at a full rate to eradicate the disease, if the effort is supported by effective behaviour modification strategies.

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5In this article, a mathematical model for the transmission of COVID-19 disease is formulated 6 and analysed. It is shown that the model exhibits a backward bifurcation at R 0 = 1 when recovered 7 individuals do not develop a permanent immunity for the disease. In the absence of reinfection, it is 8 proved that the model is without backward bifurcation and the disease free equilibrium is globally 9 asymptotically stable for R 0 < 1. By using available data, the model is validated and parameter 10 values are estimated. The sensitivity of the value of R 0 to changes in any of the parameter values 11 involved in its formula is analysed. Moreover, various mitigation strategies are investigated using 12 the proposed model and it is observed that the asymptomatic infectious group of individuals may 13 play the major role in the re-emergence of the disease in the future. Therefore, it is recommended 14 that in the absence of vaccination, countries need to develop capacities to detect and isolate at least 15 30% of the asymptomatic infectious group of individuals while treating in isolation at least 50% of 16 symptomatic patients to control the disease.

A mathematical model for a transmission of TB-HIV/AIDS co-infection that incorporates prevalence dependent behaviour change in the population and treatment for the infected (and infectious) class is formulated and analyzed. The two sub-models, when each of the two diseases are considered separately are mathematically analyzed. The theory of optimal control analysis is applied to the full model with the objective of minimizing the aggregate cost of the infections and the control efforts. In the numerical simulation section, various combinations of the controls are also presented and it has been shown in this part that the optimal combination of both prevention and treatment controls will suppress the prevalence of both HIV and TB to below 3% within 10 years. Moreover, it is found that the treatment control is more effective than the preventive controls.

The novel coronavirus disease has ravaged many health systems around the world and has brought many economies to their knees. In the absence of an approved curing medicine or approved vaccine to date, the major control of the surge of infections is through use of Non-Pharmaceutical Interventions (NPIs) and imposing specific standard operating procedures (SOPs) in instances when the disease spread curbs are relaxed. It is thus essential to quantify the extent to which specific NPIs can be useful in containing the pandemic. To achieve this, we constructed a mathematical model that accounts for both person to person transmission as well as transmission through contact from pathogen-contaminated surfaces. The model assumes that there is change of behaviour resulting from the surge of the number of cases, hence a class of susceptible individuals who practise self-protection measures. Basic properties of the model including the conditions for existence and stability of steady states are explored. The model was fitted to new-cases data for South Africa and baseline parameter values were estimated. Sensitivity analysis of the model was performed to determine the most influential parameters on the disease threshold. Our results show that practising of selfprotection measures is vital in slowing the spread of the infection. In addition, it is evident from the results that minimizing contact through "physical distancing" as well as with contaminated surfaces can significantly help in containing the infection. The model was extended to account for testing and quarantining of both symptomatic and asymptomatic infected individuals. In addition, migration and potential use of a vaccine were explored. In the case of migration, the scenarios considered included aspects when there are both border control and illegal crossings as well as the case where the government is in full control with proper SOPs. Our results show that, although testing and isolating/quarantining of infected individuals is essential in

Funds from various global organizations, such as, The Global Fund, The World Bank, etc. are not directly distributed to the targeted risk groups. Especially in the so-called third-world-countries, the major part of the fund in HIV prevention programs comes from these global funding organizations. The allocations of these funds usually pass through several levels of decision making bodies that have their own specific parameters to control and specific objectives to achieve. However, these decisions are made mostly in a heuristic manner and this may lead to a non-optimal allocation of the scarce resources. In this paper, a hierarchical mathematical optimization model is proposed to solve such a problem. Combining existing epidemiological models with the kind of interventions being on practice, a 3-level hierarchical decision making model in optimally allocating such resources has been developed and analyzed. When the impact of antiretroviral therapy (ART) is included in the model, it has been shown that the objective function of the lower level decision making structure is a non-convex minimization problem in the allocation variables even if all the production functions for the intervention programs are assumed to be linear.

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