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, R 0 , 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.

We study the triangular equilibrium points in the framework of Yukawa correction to Newtonian potential in the circular restricted three-body problem. The effects of α and λ on the mean-motion of the primaries and on the existence and stability of triangular equilibrium points are analyzed, where α ∈ − 1 , 1 is the coupling constant of Yukawa force to gravitational force, and λ ∈ 0 , ∞ is the range of Yukawa force. It is observed that as λ ⟶ ∞ , the mean-motion of the primaries n ⟶ 1 + α 1 / 2 and as λ ⟶ 0 , n ⟶ 1 . Further, it is observed that the mean-motion is unity, i.e., n = 1 for α = 0 , n > 1 if α > 0 and n < 1 when α < 0 . The triangular equilibria are not affected by α and λ and remain the same as in the classical case of restricted three-body problem. But, α and λ affect the stability of these triangular equilibria in linear sense. It is found that the triangular equilibria are stable for a critical mass parameter μ c = μ 0 + f α , λ , where μ 0 = 0.0385209 ⋯ is the value of critical mass parameter in the classical case of restricted three-body problem. It is also observed that μ c = μ 0 either for α = 0 or λ = 0.618034 , and the critical mass parameter μ c possesses maximum ( μ c max ) and minimum ( μ c min ) values in the intervals − 1 < α < 0 and 0 < α < 1 , respectively, for λ = 1 / 3 .

The existence and stability of noncollinear equilibrium points in the elliptic restricted three-body problem under the consideration of Yukawa correction to Newtonian potential are studied in this paper. The effects of various parameters (μ, ê, α, and λ) on the noncollinear equilibrium points are discussed briefly, and it is found that only ordinate of noncollinear equilibria E4,5 is affected by Yukawa correction while abscissa is affected by only mass parameter μ. The noncollinear equilibria was found linearly stable for a critical mass parameter μc. A critical point λ = ½ is also obtained for the critical mass parameter μc, and at this point, the critical mass parameter μc has maximum or minimum values according to α < 0 or α > 0, respectively.