Vaccination control of an epidemic model with time delay and its application to COVID-19

Abstract

This paper studies an SEIR-type epidemic model with time delay and vaccination control. The vaccination control is applied when the basic reproduction number $R_0>1$ . The vaccination strategy is expressed as a state delayed feedback which is related to the current and previous state of the epidemic model, and makes the model become a linear system in new coordinates. For the presence and absence of vaccination control, we investigate the nonnegativity and boundedness of the model, respectively. We obtain some sufficient conditions for the eigenvalues of the linear system such that the nonnegativity of the epidemic model can be guaranteed when the vaccination strategy is applied. In addition, we study the stability of disease-free equilibrium when $R_0<1$ and the persistent of disease when $R_0>1$ . Finally, we use the obtained theoretical results to simulate the vaccination strategy to control the spread of COVID-19.

Keywords: Epidemic model; State delayed feedback; Time delay; Vaccination.

Fig. 1

The flow diagram for model (1)

Fig. 2

The actual number of active infections in Italy

Fig. 3

a The time evolution of system (1) with no control when $R_0=2.0642$; b the number of active infections with no control when $R_0=2.0642$

Fig. 4

a The time evolution of system (1) with control $u_e(t)$ when $R_0=2.0642$ and b the number of active infections with control $u_e(t)$ when $R_0=2.0642$

Fig. 5

Time evolution of the vaccination function $u_e(t)$

Fig. 6

a The percentage of the total number of people who have recovered through treatment under different situations and b the percentage under vaccination situation

Fig. 7

a The time evolution of system (1) with control $u_e(t)$ when $R_0=2.0642$ and b the number of active infections with control $u_e(t)$ when $R_0=2.0642$

Fig. 8

Time evolution of the vaccination function $u_e(t)$