Calculation of variance in infected cases due to stochastic fluctuations

[kapawlak]

I took a simple SIR model to calculate the 1σ width for the infected using stochastic calculus in the mean field approximation for S.

The processes assumed are the following:
Infection of susceptible: (I + S → 2 I, rate r)
Death or Recovery of infected (I → ∅, rate ε)

The result is:

var(I(t)) = I(0)/[1+ coth(t/2( ε-r S(t) )] × [1+ 2 r S(t)/(ε-r S(t))] × exp[- t (ε- r S(t))]

or in terms of mean field I(t) only:

var(I(t)) = I(t)/[1 + coth( ½ Ln( I(t)/I(0) ) ] × [1- 2 t (Ln( I(t)/I(0) )- tε) /Ln( I(t)/I(0) )]

This estimation should give a reasonable reflection of variance due to stochastic fluctuations. The formula can be improved by error propagating parameter uncertainties (β, t0) if desired.

The function is closed form so it should be easy and efficient to shade in the ∓ region

Edit: Quick simulation to demonstrate what it looks like