The heterogeneous mixing model of COVID-19 with interventions

Abstract

The emergence of mutant strains of COVID-19 reduces the effectiveness of vaccines in preventing infection, but remains effective in preventing severe illness and death. This paper established a heterogeneous mixing model of age groups with pharmaceutical and non-pharmaceutical interventions by analyzing the transmission mechanism of breakthrough infection caused by the heterogeneity of protection period under the action of vaccine-preventable infection with the original strain. The control reproduction number R_c of the system is analyzed, and the existence and stability of equilibrium are given by the comparison principle. Numerical simulation was conducted to evaluate the vaccination program and intervention measures in the customized scenario, demonstrating that the group-3 coverage rate p₃ plays a key role in R_c. It is proposed that accelerating the rate of admission and testing is conducive to epidemic control by further fitting data of COVID-19 transmission in real scenarios. The findings provide a general modeling idea for the emergence of new vaccines to prevent infection by mutant strains, as well as a solid theoretical foundation for mainland China to formulate future vaccination strategies for new vaccines. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

Keywords: COVID-19; Control reproduction number; Heterogeneous mixing; Prevent infection; Vaccination strategies.

Fig. 1

Illustration of the basic modeling structure. The light green plane and light blue plane are the maps of COVID-19 transmission in age group-$k$ and $k+1$, respectively. The rose lines represent age group-$k$ becomes age group-$(k+1)$ due to aging.

Fig. 2

Illustration of contact pattern. The left panel is contact pattern of 16 groups, and the right panel is 4 groups. The degree of color is proportional to the number of contacts.

Fig. 3

The 2D contour map of $R_c$ with $\beta =\frac{1}{3}$. Panels (a)–(c) show the effect of simultaneous inoculation of “group-3 and group-4”, “group-3 and group-2”, and “group-4 and group-2” on $R_c$ respectively. The colorbar represents the value of $R_c$.

Fig. 4

Panels (a)–(c) represent the $\Delta R_c$, magnitudes of gradient $|\nabla R_c|$, negative gradient direction represented by streamline of “group-2 and group-3” strategy respectively.

Fig. 5

Left panel is the 3D curved surface of $R_c$, the upper part of light pink surface is $R_c<1$. Right panel is slice map correspond with left panel, X–Y plane is coverage rate combination of group-2 and 3, $Z$-axis represents the change of $\beta$, the colorbar is the value of $R_c$.

Fig. 6

The fitting results and PRCCs for Shanghai. Panel (a) is the fitting results of the reported local hospitalized cases in Shanghai from April 11 to May 16, 2022. Panel (b) is the value of partial rank correlation coefficient between control reproduction number and parameters $p$, ${\psi }_I$, $\theta$, $\beta$.