Local immunization program for susceptible-infected-recovered network epidemic model

Abstract

The immunization strategies through contact tracing on the susceptible-infected-recovered framework in social networks are modelled to evaluate the cost-effectiveness of information-based vaccination programs with particular focus on the scenario where individuals belonging to a specific set can get vaccinated due to the vaccine shortages and other economic or humanity constraints. By using the block heterogeneous mean-field approach, a series of discrete-time dynamical models is formulated and the condition for epidemic outbreaks can be established which is shown to be not only dependent on the network structure but also closely related to the immunization control parameters. Results show that increasing the immunization strength can effectively raise the epidemic threshold, which is different from the predictions obtained through the susceptible-infected-susceptible network framework, where epidemic threshold is independent of the vaccination strength. Furthermore, a significant decrease of vaccine use to control the infectious disease is observed for the local vaccination strategy, which shows the promising applications of the local immunization programs to disease control while calls for accurate local information during the process of disease outbreak.

FIG. 1.

Illustrations of the contact tracing by an infected node. The central node is an infected node with 8 susceptible neighbors, among which two are in the set Ω while the others are outside Ω. Only one, out of two Ω-nodes, is traced (and also will get vaccinated) by the central node along the contact between them (indicated by a blue and thick line).

FIG. 2.

Effect of the immunization rate on the epidemic threshold and prevalence for the random immunization case when f = 0.5: (a) final recovered size R_∞ versus the infection rate β for different values of δ; (b) contour of I_max in the (δ – β) parameter plane, where the white line indicates the theoretically predicted curve determined in Eq. (5), and the light gray region corresponds to the parameter region with zero prevalence.

FIG. 3.

Effect of the immunization rate on the epidemic threshold and prevalence for the targeted immunization case when K = 10: (a) final recovered size R_∞ versus the infection rate β for different values of δ; (b) contour of I_max in the (δ – β) parameter plane, where the white line indicates the theoretically predicted curve determined in Eq. (7), and the light gray region to the parameter region with zero prevalence.

FIG. 4.

The contour plot of the immunization efficiency, where the horizontal coordinate is the infection rate β and the vertical coordinate is the immunization strength δ. Panels (a) and (b) illustrate the random immunization case for f = 0.2 and 0.8, respectively, while panels (c) and (d) show the targeted immunization case for K = 6 and 12, respectively. The dashed lines in each panel indicate the epidemic thresholds by theoretical predictions.