Extinction and Ergodic Stationary Distribution of COVID-19 Epidemic Model with Vaccination Effects

Abstract

Human society always wants a safe environment from pollution and infectious diseases, such as COVID-19, etc. To control COVID-19, we have started the big effort for the discovery of a vaccination of COVID-19. Several biological problems have the aspects of symmetry, and this theory has many applications in explaining the dynamics of biological models. In this research article, we developed the stochastic COVID-19 mathematical model, along with the inclusion of a vaccination term, and studied the dynamics of the disease through the theory of symmetric dynamics and ergodic stationary distribution. The basic reproduction number is evaluated using the equilibrium points of the proposed model. For well-posedness, we also test the given problem for the existence and uniqueness of a non-negative solution. The necessary conditions for eradicating the disease are also analyzed along with the stationary distribution of the proposed model. For the verification of the obtained result, simulations of the model are performed.

Computation of the Exact Forms of Waves for a Set of Differential Equations Associated with the SEIR Model of Epidemics

“…From these solutions, the suitable solutions are these which do not contain more than one relationship connecting the parameters α i of the solved equations. Such solutions include (37), (39), ( 43), ( 41), ( 56), ( 58), (60), and (62). These solutions are constructed by functions with a shape that is similar to the shape of the logistic function in Figure 2.…”

“…Epidemic models can be applied to study other processes as well, such as the spread of ideas (for overviews, see [4,54]). An important recent application of epidemic models is in research on the spread of COVID-19 [55][56][57][58][59][60][61][62][63][64][65][66][67][68]. We proceed as follows: the methodology of SEsM is described briefly in Section 2; in Section 3, we derive a sequence of nonlinear differential equations connected to the SEIR model of epidemic spread and use SEsM to obtain exact solutions of these equations; and in Section 4, the applicability of the obtained solutions for describing the evolution of epidemic waves is discussed.…”

Epidemic Waves and Exact Solutions of a Sequence of Nonlinear Differential Equations Connected to the SIR Model of Epidemics

“…Epidemic models can also be applied for the description of other processes, such as the spread of ideas (for overviews, see [ 3 , 63 ]). We also note the use of epidemic models for the study of COVID-19 spreading [ 64 , 65 , 66 , 67 , 68 , 69 , 70 , 71 , 72 , 73 , 74 , 75 , 76 , 77 , 78 ], as well as the numerical methods for obtaining solutions to such models [ 79 , 80 ].…”