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. 2021 Aug;18(181):20210186.
doi: 10.1098/rsif.2021.0186. Epub 2021 Aug 4.

Triple contagion: a two-fears epidemic model

Joshua M Epstein  1 , Erez Hatna  1 , Jennifer Crodelle  2
Affiliations

Triple contagion: a two-fears epidemic model

Joshua M Epstein et al. J R Soc Interface. 2021 Aug.

Abstract

We present a differential equations model in which contagious disease transmission is affected by contagious fear of the disease and contagious fear of the control, in this case vaccine. The three contagions are coupled. The two fears evolve and interact in ways that shape distancing behaviour, vaccine uptake, and their relaxation. These behavioural dynamics in turn can amplify or suppress disease transmission, which feeds back to affect behaviour. The model reveals several coupled contagion mechanisms for multiple epidemic waves. Methodologically, the paper advances infectious disease modelling by including human behavioural adaptation, drawing on the neuroscience of fear learning, extinction and transmission.

Keywords: behaviour; dynamical systems; epidemic modelling; neuroscience.

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Figures

Figure 1.
Figure 1.
Flow diagram of equations (2.1)–(2.8).
Figure 2.
Figure 2.
Plots for Scenario 1 (contagious disease only). (a) The proportions of susceptibles without fear (S), disease-fearful susceptibles (Sfd) and vaccine-fearful susceptibles (Sfv). (b) Vaccination rate (v). (c) The proportion of infectives (I). (d) The proportion of recovered (Rnat) and vaccinated (Rvac) individuals. Note that about 80% of the population become infected with the disease.
Figure 3.
Figure 3.
Plots for Scenario 2 (contagious disease + fear of the disease). (a) The proportions of susceptibles without fear (S), disease-fearful susceptibles (Sfd) and vaccine-fearful susceptibles (Sfv). (b) Vaccination rate (v). (c) The proportion of infectives (I). (d) The proportion of recovered (Rnat) and vaccinated (Rvac) individuals. Note that about 66% of the population become infected with the disease.
Figure 4.
Figure 4.
Plots for Scenario 3 (contagious disease + fear of the disease + vaccinations). (a) The proportions of susceptibles without fear (S), disease-fearful susceptibles (Sfd) and vaccine-fearful susceptibles (Sfv). (b) Vaccination rate (v). (c) The proportion of infectives (I). (d) The proportion of recovered (Rnat) and vaccinated (Rvac) individuals. Note that about 38% of the population become infected with the disease.
Figure 5.
Figure 5.
Plots for Scenario 4 (contagious disease + fear of the disease + vaccinations + fear of the vaccinations. (a) The proportions of susceptibles without fear (S), disease-fearful susceptibles (Sfd), and vaccine-fearful susceptibles (Sfv). (b) Vaccination rate (v). (c) The proportion of infectives (I). (d) The proportion of recovered (Rnat) and vaccinated (Rvac) individuals. Note that about 46% of the population become infected with the disease.
Figure 6.
Figure 6.
The effect of changing the relative risk, p, on disease spread. (a) The proportion of infectives (I) versus time. (b) The total fraction of the population that contracts the disease. All parameters other than p are as in Scenario 2 (table 3, column S2).
Figure 7.
Figure 7.
The effect of the effective contact rate of fear loss αf on disease spread. (a) The proportion of infectives (I) versus time. (b) The total fraction of the population that contacts the disease. All parameters other than αf remain as in table 3, column S2.
Figure 8.
Figure 8.
The effect of changing the fears contact rates (βfd, βfv) on the fraction of the population (a) vaccinated and (b) infected with the disease.
Figure 9.
Figure 9.
The effect of changing the fears’ contact rates (βfd, βfv). (a) The number of peaks in the infectives curve as a function of the two contact rates. (b) The infectives curve for a case of one peak. (c) The infectives curve for a case of no peak (no outbreak). (d) Five infectives curves with two peaks. (e) The total number of infected persons for each of the five cases.
Figure 10.
Figure 10.
The effect of changing the fraction of adverse effects from vaccinations (σ) on (a) the proportion of infectives, (b) the vaccination rate and (c) the proportion of susceptibles that fear the vaccine.
Figure 11.
Figure 11.
The effect of changing the relative risk, p, on disease spread. (a) The proportion of infectives (I) versus time. (b) The total fraction of the population that contracts the disease. All parameters other than p are as in Scenario 2 (table 3, column S2).
Figure 12.
Figure 12.
Example simulations and Rn values for two sample values of p. The reproduction number (Rn) value for the disease plotted over time for (a) p = 0.11 and (b) p = 0.12. Simulations of the proportion of the population in each compartment for (c) p = 0.11 and (d) p = 0.12.
Figure 13.
Figure 13.
Classic SIR model (Scenario 1). (a) The reproduction number (Rn) value for the no-fear case plotted over time. (b) The proportion of susceptible (S) and recovered (Rnat) individuals. (c) The proportion of infectives (I).
Figure 14.
Figure 14.
Fear epidemic. (a) The reproduction number (Rn) value for the fear of the disease plotted over time. (b) The proportion of susceptible (S), disease-fearful (Sfd) and vaccinated (Rvac) individuals. (c) The proportion of infectives (I).
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References

    1. Kermack WO, McKendrick A. 1927A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A 115, 700-721. (10.1098/rspa.1927.0118) - DOI
    1. Brauer F. 2005The Kermack–Mckendrick epidemic model revisited. Math. Biosci. 198, 119-131. (10.1016/j.mbs.2005.07.006) - DOI - PubMed
    1. Epstein JM, Parker J, Cummings D, Hammond R. 2008Coupled contagion dynamics of fear and disease: mathematical and computational explorations. PLoS ONE 3, e3955. (10.1371/journal.pone.0003955) - DOI - PMC - PubMed
    1. Bootsma MCJ, Ferguson NM. 2007The effect of public health measures on the 1918 influenza pandemic in U.S. cities. Proc. Natl Acad. Sci. USA 104, 7588-7593. (10.1073/pnas.0611071104) - DOI - PMC - PubMed
    1. Centers for Disease Control and Prevention, CDC covid data tracker. Trends in number of Covid-19 cases and deaths in the US reported to CDC, by state/territory. January 2021.

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