Complex Dynamics of An Epidemic Model With Saturated Media Coverage and Recovery

Abstract

During the outbreak of emerging infectious diseases, media coverage and medical resource play important roles in affecting the disease transmission. To investigate the effects of the saturation of media coverage and limited medical resources, we proposed a mathematical model with extra compartment of media coverage and two nonlinear functions. We theoretically obtained that saturated recovery significantly contributes the occurrence of backward bifurcation and rich dynamics. Then it is reasonable to only considering nonlinear recovery, we theoretically showed that backward bifurcation can occur and multiple equilibria may coexist under certain conditions in this case. And numerical simulations reveals the rich dynamic behaviors, including forward-backward bifurcation, Hopf bifurcation, Saddle-Node bifurcation, Homoclinic bifurcation and unstable limit cycle. Comparing the system with linear recovery, where the threshold dynamic are almost completely characterized by a threshold condition called the basic reproduction number, we concluded that only saturated media impact hardly induces the complicated dynamics, while the nonlinear recovery function, associated with limitation of medical resources, may induce the coexistence of the disease-free equilibrium (DFE) and a endemic state or multiple endemic states, which means that the limitation of medical resources causes much difficulties in eliminating the infectious diseases.

FOMO (fate of online media only) in infectious disease modeling: a review of compartmental models

“…For ease of comparison, the following notation is adopted— for information transmission rate terms, S for susceptibles, E for exposed Individuals, I for infected/infectious Individuals, A for asymptomatics, R for recovered individuals and M for the media compartment Reference Basic Model Effect of online media Transmission term [ 28 ] SEIR Creation of an aware class Susceptibles interact with the media at a rate to become aware, where represents the daily normalized number of tweets about the infectious disease at any given time [ 34 ] SEIR Modification of the transmission term Media-induced quarantine The transmission rate between susceptible and infected individuals is reduced by a factor due to awareness driven by media reports where represents the weight of the media effect Terms / represent the quarantine rates due to media reports [ 29 ] SIR Creation of aware classes Unaware individuals interact with the media at a rate to become aware [ 35 ] SEI Modification of the transmission term The transmission rate between susceptible and infected individuals is reduced by a factor due to awareness driven by media reports, where α is a measure of the media effect [ 36 ] SEIR Creation of aware classes Modification of transmission terms Susceptibles interact with the media at a rate to become aware Asymptomatics interact with the media at a rate to go into quarantine The transmission rate between susceptible and infected individuals is reduced by where ρ is a half saturation constant [ 37 ] …”

“…However, the diffusion of information among individuals is faster than the spread of a disease and that of demographic processes [ 42 ]. By considering the time variation between information dissemination, epidemiological and demographic processes, Li et al [ 35 ] constructed a multi-scale model describing the disease/information transmission with aware/unaware classes and a media term. Since the spread of awareness due to information dissemination is faster than population growth, the dynamical behavior of the system was analyzed using the theory of the slow-fast system [ 51 , 55 ].…”

“…This means that there may be a limit to the amount of information presented, explored and absorbed by the public [ 76 ] and other forms should be considered. Li and Xiao [ 35 ] considered a saturated function of newly observed infectious individuals (Table 2 ) in which the media reporting rate initially rises with newly observed infections. This gradually decelerates until a plateau is reached where the media reporting rate remains constant regardless of newly observed infections similar to the Holling type 2 functional response.…”