This paper concentrates on establishing the existence of multiple weak solutions for a specific type of elliptic equations that involve a Hardy potential and have mixed boundary conditions. The main goal of the study is to establish an existence result of at least three different weak solutions thanks to variational techniques, Hardy inequality, and a particular theorem called the Bonanno–Marano type three critical points theorem.

The 2022–2023 global monkeypox (Mpox) virus outbreak, which affected more than 100 nations, has highlighted the importance of effective public health measures and predictive modeling techniques. In this work, we formulated a mathematical model (MD) of Mpox to assess the impact of early therapeutic education on seasonal variation. To guarantee epidemiological viability, we demonstrated the boundedness and positivity of the model. The next-generation matrix method was employed to determine the basic reproduction number and evaluate the stability of the disease-free equilibrium at the local and global levels. We assessed the existence of an arbitrary equilibrium. The center manifold theorem was employed to assess the backward bifurcation, which shows that imperfect vaccination causes the backward bifurcation. An evaluation of the existence of an endemic equilibrium point was also conducted. The global stability of an endemic equilibrium point was established through the application of a nonlinear Lyapunov function of the Goh-Voltterra type. Data fitting was done to show the model's fitness and estimate the data's parameters. Sensitivity analysis was ascertained. Seasonal variation was assessed. In a numerical simulation, we assessed five controls, which indicated that early therapeutic education is the most effective way to control the re-emergence of Mpox disease in the human (HUM) population.

Lassa fever is a zoonotic viral disease that is contagious. In this research, a nonlinear deterministic mathematical model of human-to-human transmission of Lassa fever is formulated, which concentrates on the impact of early tracing of contacts and proper burial. The goal of this research was to assess the impact of tracing contacts and the proper burial of deceased individuals. We performed the test of the existence of equilibrium on the model. We computed control and basic reproduction numbers, ${\mathcal{R}}_c$ and ${\mathcal{R}}_0$, using the next-generation matrix approach, and we were able to prove the global stability of the disease-free equilibrium using the comparison method. We ascertained that the equilibrium is globally asymptotically stable if ${\mathcal{R}}_c<1$. We also show the existence of an endemic equilibrium point, where a unique endemic equilibrium has been found. Regarding the global stability of the endemic equilibrium point, we are able to prove the stability of the endemic equilibrium point globally using the Goh-Volterra type of Lyapunov function, which shows that the endemic equilibrium point is globally asymptotically stable if ${\mathcal{R}}_c>1$, with the condition that if the disease-induced death rates, the hospitalization rate of quarantined and untraced individuals, and the reversion rate of quarantined and traced individuals are all equals to zero. Numerical simulation suggests that, if the traced contact rate can be made high, it can help in controlling Lassa fever disease in society, which will also lead to a decline in infectious individuals in society. If the proper burial rate can be made almost perfect, it can help in controlling Lassa fever disease in society, which will lead to a decline in the number of infectious individuals in society. The optimal control plot shows that the campaign to increase personal hygiene and public knowledge of the actions that increase the risk of sickness is more effective in controlling Lassa fever.