ABSTRACT
In this study, a nonlinear deterministic mathematical model that evaluates two important therapeutic measures of the COVID-19 pandemic: vaccination of susceptible and treatment for infected people who are in quarantine, is formulated and rigorously analyzed. Some of the fundamental properties of the model system including existence and uniqueness, positivity, and invariant region of solutions are proved under a certain meaningful set. The model exhibits two equilibrium points: disease-free and endemic equilibrium points under certain conditions. The basic reproduction number, R0, is derived via the next-generation matrix approach, and the dynamical behavior of the model is explored in detail. The analytical analysis reveals that the disease-free equilibrium solution is locally as well as globally asymptotically stable when the associated basic reproduction number is less than unity which indicates that COVID-19 dies out in the population. Also, the endemic equilibrium point is globally asymptotically stable whenever the associated basic reproduction number exceeds a unity which implies that COVID-19 establishes itself in the population. The sensitivity analysis of the basic reproduction number is computed to identify the most dominant parameters for the spreading out as well as control of infection and should be targeted by intervention strategies. Furthermore, we extended the considered model to optimal control problem system by introducing two time-dependent variables that represent the educational campaign to susceptibles and continuous treatment for quarantined individuals. Finally, some numerical results are illustrated to supplement the analytical results of the model using MATLAB ode45. © 2023 Alemzewde Ayalew et al.
ABSTRACT
Measles is a highly contagious respiratory disease of global public health concern. A deterministic mathematical model for the transmission dynamics of measles in a population with Crowley–Martin incidence function to account for the inhibitory effect due to susceptible and infected individuals and vaccination is formulated and analyzed using standard dynamical systems methods. The basic reproduction number is computed. By constructing a suitable Lyapunov function, the disease-free equilibrium is shown to be globally asymptotically stable. Using the Center Manifold theory, the model exhibits a forward bifurcation, which implies that the endemic equilibrium is also globally asymptotically stable. To determine the optimal choice of intervention measures to mitigate the spread of the disease, an optimal control problem is formulated (by introducing a set of three time-dependent control variables representing the first and second vaccine doses, and the palliative treatment) and analyzed using Pontryagin's Maximum Principle. To account for the scarcity of measles vaccines during a major outbreak or other causes such as the COVID-19 pandemic, a Holling type-II incidence function is introduced at the model simulation stage. The control strategies have a positive population level impact on the evolution of the disease dynamics. Graphical results reveal that when the mass-action incidence function is used, the number of individuals who received first and second vaccine dose is smaller compared to the numbers when the Crowley–Martin incidence-type function is used. Inhibitory effect of susceptibles tends to have the same effect on the population level as the Crowley–Martin incidence function, while the control profiles when inhibitory effect of the infectives is considered have similar effect as when the mass-action incidence is used, or when there is limitation in the availability of measles vaccines. Missing out the second measles vaccine dose has a negative impact on the initial disease prevalence. © 2022 Elsevier B.V.
ABSTRACT
In the current era, information dissemination is more convenient, the harm of rumors is more serious than ever. At the beginning of 2020, COVID-19 is a biochemical weapon made by a laboratory, which has caused a very bad impact on the world. It is very important to control the spread of these untrue statements to reduce their impact on people's lives. In this paper, a new rumor spreading model with comprehensive interventions (background detection, public education, official debunking, legal punishment) is proposed for qualitative and quantitative analysis. The basic reproduction number with important biological significance is calculated, and the stability of equilibria is proved. Through the optimal control theory, the expression of optimal control pairs is obtained. In the following numerical simulation, the optimal control under 11 control strategies are simulated. Through the data analysis of incremental cost-effectiveness ratio and infection averted ratio of all control strategies, if we consider the control problem from different perspectives, we will get different optimal control strategies. Our results provide a flexible control strategy for the security management department.