Heterogeneous SEIR Model

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Gabriela Gomes et al.

Individual variation in susceptibility or exposure to SARS-CoV-2 lowers the herd immunity threshold

medRxiv, 21 May 2020

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7239079/

https://www.medrxiv.org/content/10.1101/2020.04.27.20081893v3.full.pdf

https://www.medrxiv.org/content/10.1101/2020.04.27.20081893v3.supplementary-material

Abstract:

As severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) spreads, the susceptible subpopulation is depleted causing the incidence of new cases to decline. Variation in individual susceptibility or exposure to infection exacerbates this effect. Individuals that are more susceptible or more exposed tend to be infected earlier, depleting the susceptible subpopulation of those who are at higher risk of infection. This selective depletion of susceptibles intensifies the deceleration in incidence. Eventually, susceptible numbers become low enough to prevent epidemic growth or, in other words, the herd immunity threshold (HIT) is reached.

Although estimates vary, simple calculations suggest that herd immunity to SARS- CoV-2 requires 60-70% of the population to be immune.

By fitting epidemiological models that allow for heterogeneity to SARS-CoV-2 outbreaks across the globe, we show that variation in susceptibility x (or exposure x) to infection reduces these estimates.

We integrate continuous distributions q(x) of susceptibility or connectivity in otherwise basic epidemic models (SEIR) for COVID-19 and show that as the coefficient of variation (CV) increases from 0 to 4, the herd immunity threshold declines from over 60% to less than 10%.

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................................................................................................................................................................... (eq. 1, "SEIR model")

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Notation

S(x) = number of individuals with susceptibility x
E(x) = number of individuals ("exposeds") who originally

had susceptibility x,
became exposed & infected but
are neither symptomatic nor infectious

I(x) = number of individuals who originally had susceptibility x and became infectious

x = susceptibility

<x> = mean susceptibility factor at epidemic onset (<x> = 1?).

is the average force of infection upon susceptible individuals in a population of size N,
corresponds to β I/N in homogeneous SIR model above.

ρ is a factor measuring the infectivity of individuals in compartment E in relation to those in I ............................................................. (ρ = 0.5)
= basic reproduction number .............................................................................................................................. (eq. 2)
1/δ = latency period ............................................................................................................................................................................... (4 days)
1/γ = average recovery time ................................................................................................................................................................. (4 days)
q(x): prior [in contradistinction to posterior] to the epidemic, susceptibility x is described by a probability density function q(x) with

mean q(x) = 1 and
CV = < (x- 1)² > explored as a parameter.

q(x) = probability density function of the gamma distribution ("Gamma Distribution")

q(x).png

with (note: γ and β used here are different from the parameters used in eq. 2)

γ being the shape parameter,
μ the location parameter,
β the scale parameter,
the gamma function

the coefficient of variation

Fig. J1: q[x] = probability density function of the gamma distribution. Abscissa = x, ordinate = q(x), μ = 1, β = 1,

γ =

0.111 (CV = 3)
1 (CV = 1)
2.111 (CV = 0.7)

Figure 1: The effect of variation in susceptibility to infection on the size of epidemics. Suppressed wave and subsequent dynamics. Blue bars are confirmed new cases and overlaid red bars represent deaths.

basic reproduction number R_o

[which drops from an initial

R₀ = 2.69 (susceptibility diagrams, upper part of Fig. 1) and
R₀ = 2.82 (connectivity diagrams, lower part of Fig. 1)

in March due to a lockdown and is assumed to increase to the pre-lockdown value along with the gradual relaxation of the lockdown conditions] and

effective reproduction number R_eff = R_o S(t)/N = R_o s(t), which initially drops and rises in parallel with R₀, but eventually falls together with the decreasing fraction s(t)(?)]

are displayed on bottom panels. Blue shades represent socialdistancing (intensity reflected in R_o trends and shade density).

Susceptibility factors x were implemented as gammadistributions.
Consensus parameter values (Materials and Methods):

δ = 1/4 per day;
gamma = 1/4 per day; and
ρ = 0.5.
Fraction of infected individuals identified as positive (reporting fraction): p = 0.1.
R₀ and social distancing parameters were estimated by Bayesian inference as described in Supplementary Materials.

Curves represent median model predictions from 10⁴posterior samples. Orange shades represent 95% credible intervals.

Figure 3: Herd immunity threshold with variation in susceptibility and exposure to infection. Curves generated with the model (Equation 1) with gamma distributed susceptibility (black) or connectivity (gray) assuming R_o = 3: (solid) herd immunity threshold (HIT); (dashed) final size of uncontrolled epidemic. Vertical lines indicate coefficients of individual variation for several infectious diseases according to literature: (solid green) susceptibility or exposure to malaria [Amazon 1.79 (6), Africa 2.05 (7)]; (solid blue) susceptibility or exposure to tuberculosis [Portugal 2.37, Brazil 3.33 (8)]; (dotted red) infectiousness for SARS-CoV-1 [Singapore 2.62, Beijing 2.64 (9)]; (dotted black) infectiousness for SARS-CoV-2 [3.22 (10)]

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