The novel Coronavirus (COVID-19) infection has become a global public health issue, and it has been a cause for morbidity and mortality of more people throughout the world. In this paper, we investigated the impacts of vaccination, other protection measures, home quarantine with treatment, and hospital quarantine with treatment strategies simultaneously using a deterministic mathematical modelling approach. No one has considered these intervention strategies simultaneously in his/her modelling approach. We examined all the qualitative properties of the model such as the positivity and boundedness of the model solutions, the disease-free and endemic equilibrium points, the effective reproduction number using next-generation matrix method, local stabilities of equilibrium points using the Routh–Hurwitz method. Using the Centre Manifold criteria, we have shown the existence of backward bifurcation whenever the COVID-19 effective reproduction number is less than unity. Moreover, we have analysed both sensitivity and numerical simulation using parameter values taken from published literature. The numerical results show that the transmission rate is the most sensitive parameter we have to control. Also vaccination, other protection measures, home quarantine with treatment, and hospital quarantine with treatment have great effects to minimize the COVID-19 transmission in the community.

In this paper, we proposed and analyzed a realistic compartmental mathematical model on the spread and control of HIV/AIDS-pneumonia coepidemic incorporating pneumonia vaccination and treatment for both infections at each infection stage in a population. The model exhibits six equilibriums: HIV/AIDS only disease-free, pneumonia only disease-free, HIV/AIDS-pneumonia coepidemic disease-free, HIV/AIDS only endemic, pneumonia only endemic, and HIV/AIDS-pneumonia coepidemic endemic equilibriums. The HIV/AIDS only submodel has a globally asymptotically stable disease-free equilibrium if R 1 < 1 . Using center manifold theory, we have verified that both the pneumonia only submodel and the HIV/AIDS-pneumonia coepidemic model undergo backward bifurcations whenever R 2 < 1 and R 3 = max R 1 , R 2 < 1 , respectively. Thus, for pneumonia infection and HIV/AIDS-pneumonia coinfection, the requirement of the basic reproduction numbers to be less than one, even though necessary, may not be sufficient to completely eliminate the disease. Our sensitivity analysis results demonstrate that the pneumonia disease transmission rate β 2 and the HIV/AIDS transmission rate β 1 play an important role to change the qualitative dynamics of HIV/AIDS and pneumonia coinfection. The pneumonia infection transmission rate β 2 gives rises to the possibility of backward bifurcation for HIV/AIDS and pneumonia coinfection if R 3 = max R 1 , R 2 < 1 , and hence, the existence of multiple endemic equilibria some of which are stable and others are unstable. Using standard data from different literatures, our results show that the complete HIV/AIDS and pneumonia coinfection model reproduction number is R 3 = max R 1 , R 2 = max 1.386 , 9.69 = 9.69 at β 1 = 2 and β 2 = 0.2 which shows that the disease spreads throughout the community. Finally, our numerical simulations show that pneumonia vaccination and treatment against disease have the effect of decreasing pneumonia and coepidemic disease expansion and reducing the progression rate of HIV infection to the AIDS stage.

HIV/AIDS and COVID-19 co-infection is a common global health and socio-economic problem. In this paper, a mathematical model for the transmission dynamics of HIV/AIDS and COVID-19 co-infection that incorporates protection and treatment for the infected (and infectious) groups is formulated and analyzed. Firstly, we proved the non-negativity and boundedness of the co-infection model solutions, analyzed the single infection models steady states, calculated the basic reproduction numbers using next generation matrix approach and then investigated the existence and local stabilities of equilibriums using Routh-Hurwiz stability criteria. Then using the Center Manifold criteria to investigate the proposed model exhibited the phenomenon of backward bifurcation whenever its effective reproduction number is less than unity. Secondly, we incorporate time dependent optimal control strategies, using Pontryagin’s Maximum Principle to derive necessary conditions for the optimal control of the disease. Finally, we carried out numerical simulations for both the deterministic model and the model incorporating optimal controls and we found the results that the model solutions are converging to the model endemic equilibrium point whenever the model effective reproduction number is greater than unity, and also from numerical simulations of the optimal control problem applying the combinations of all the possible protection and treatment strategies together is the most effective strategy to drastically minimizing the transmission of the HIV/AIDS and COVID-19 co-infection in the community under consideration of the study.

Animosity towards mathematics is a very common worldwide problem and it is usually caused by wrong information, low participation, low challenge tolerance, falling further behind, being unemployed, and avoiding the advanced math classes needed for success in many careers. In this study, we have considered and formulated the new SEATS compartmental mathematical model with optimal control theory to analyze the dynamics of university students’ animosity towards mathematics. We applied the next-generation matrix, Ruth-Hurwitz criteria, Lyapunov function, and Volterra-Lyapunov stable matrices to show local and global stability of equilibrium points of the model respectively. The study demonstrated that the animosity-free equilibrium point is both locally and globally asymptotically stable whenever the model basic reproduction number is less than unity, whereas the animosity-dominance equilibrium point is both locally and globally asymptotically stable when the model basic reproduction number is greater than unity. Finally, we applied numerical ode45 solvers using the Runge–Kutta method and we have carried out numerical simulations and shown that applying both prevention and treatment controls is the best strategy to minimize and possibly eradicate the animosity-infection in the community under consideration.

Anxiety towards mathematics is the most common problem throughout nations in the world. In this study, we have mainly formulated and analyzed a Caputo fractional order mathematical model with optimal control strategies on higher institution students’ anxiety towards mathematics. The non-negativity and boundedness of the fractional order dynamical system solutions have been analysed. Both the anxiety-free and anxiety endemic equilibrium points of the Caputo fractional order model are found, and the local stability analysis of the anxiety-free and anxiety endemic equilibrium points are examined. Conditions for Caputo fractional order model backward bifurcation are analyzed whenever the anxiety effective reproduction number is less than one. We have shown the global asymptotic stability of the endemic equilibrium point. Moreover, we have carried out the optimal control strategy analysis of the fractional order model. Eventually, we have established the analytical results through numerical simulations to investigate the memory effect of the fractional order derivative approach, the behavior of the model solutions and the effects of parameters on the students anxiety towards mathematics in the community. Protection and treatment of anxiety infectious students have fundamental roles to minimize and possibly to eradicate mathematics anxiety from the higher institutions.

In the paper, we have considered a nonlinear compartmental mathematical model that assesses the effect of treatment on the dynamics of HIV/AIDS and pneumonia coinfection in a human population at different infection stages. Our model revealed that the disease-free equilibrium points of the HIV/AIDS and pneumonia submodels are both locally and globally asymptotically stable whenever the associated basic reproduction numbers ( R H and R P ) are less than unity. Both the submodel endemic equilibrium points are locally and globally asymptotically stable whenever the associated basic reproduction numbers ( R P and R H ) are greater than unity. The full HIV/AIDS-pneumonia coinfection model has both locally and globally asymptotically stable disease-free equilibrium points whenever the basic reproduction number of the coinfection model R H P is less than unity. Using standard values of parameters collected from different kinds of literature, we found that the numerical values of the basic reproduction numbers of the HIV/AIDS-only submodel and pneumonia-only submodel are 17 and 7, respectively, and the basic reproduction number of the HIV/AIDS-pneumonia coinfection model is max 7 , 17 = 17 . Applying sensitive analysis, we identified the most influential parameters to change the behavior of the solution of the considered coinfection dynamical system are the HIV/AIDS and pneumonia transmission rates β 1 and β 2 , respectively. The coinfection model was numerically simulated to investigate the stability of the coinfection endemic equilibrium point, the impacts of transmission rates, and treatment strategies for HIV/AIDS-only, pneumonia-only, and HIV/AIDS-pneumonia coinfected individuals. Finally, we observed that numerical simulations indicate that treatment against infection at every stage lowers the rate of infection or disease prevalence.