Final epidemic size (Kermack-McKendrick)

Formula: for the standard SIR model the final size Z = S(0) - S(inf) satisfies the implicit relation


  Z = S(0) (1 - exp(-R0 [Z + I(0)]))

(Ma & Earn 2006, eq. 4, p.681), which in the limit I(0) -> 0, S(0) -> 1 collapses to the classical Z = 1 - exp(-R0 Z) (their eq. 5). Ma & Earn show the relation holds for arbitrary distributions of the infectious period, so R0 alone determines the final size. The root is isolated by bisection on [0, S(0)], where the residual is non-negative at 0 and strictly negative at S(0).

Usage

Finalsz(R0, s0 = 1, i0 = NULL, tol = 1e-14, max_iter = 200)

Arguments

R0: Basic reproduction number (>= 0).
s0: Initial susceptible proportion in (0, 1].
i0: Initial infectious proportion; default 1 - s0.
tol: Bracket width at which bisection stops.
max_iter: Maximum bisection steps.

Value

List with estimate, final_size, s_inf, attack_rate, R0, s0, i0, residual, iters, n, method.

References

Kermack & McKendrick (1927), Proc. R. Soc. Lond. A 115(772):700-721, doi:10.1098/rspa.1927.0118; Ma & Earn (2006), Bull. Math. Biol. 68(3):679-702, doi:10.1007/s11538-005-9047-7.

Examples

Finalsz(R0 = 5L)
#> $estimate
#> [1] 0.9930228
#> 
#> $final_size
#> [1] 0.9930228
#> 
#> $s_inf
#> [1] 0.006977154
#> 
#> $attack_rate
#> [1] 0.9930228
#> 
#> $R0
#> [1] 5
#> 
#> $s0
#> [1] 1
#> 
#> $i0
#> [1] 0
#> 
#> $residual
#> [1] 1.221245e-15
#> 
#> $iters
#> [1] 47
#> 
#> $n
#> [1] 1
#> 
#> $method
#> [1] "Final epidemic size (Kermack-McKendrick)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"