EPIDEMIC MODELS I REPRODUCTION NUMBERS AND FINAL SIZE RELATIONS FRED BRAUER

Transcription

1 EPIDEMIC MODELS I REPRODUCTION NUMBERS AND FINAL SIZE RELATIONS FRED BRAUER 1

2 THE SIR MODEL Start with simple SIR epidemic model with initial conditions S = βsi I = (βs α)i, S() = S, I() = I, S + I = N Flow chart 2

3 Integration gives α and [(S(t) + I(t)] dt = S + I S = N S ln S S = β I(t)dt = β α [N S ] [ = R 1 S ] N 3

4 Generalize to SEIR model S = βsi S() = S E = βsi κe E() = E I = κe αi I() = I R = αi R() = Flow chart Basic reproduction number is R = βn α Final size relation is ln S S = R [ 1 S ] N 4

5 Generalize to SEIR model with infectivity in exposed stage S = βs(i + εe) S() = S E = βs(i + εe) κe E() = E I = κe αi I() = I R = αi R() = Basic reproduction number is Final size relation is ln S S = R R = βn α + εβn κ [ 1 S ] N εβ κ I 5

6 A SIMPLE TREATMENT MODEL Now add treatment at a rate γ to the basic model Assume treatment moves infectives to a class T with infectivity decreased by a factor δ and with a recovery rate η treatment continues so long as an individual remains infective Treatment is beneficial, Flow chart η > δα 6

7 Model is S = βs(i + δt), S() = S I = βs(i + δt) (α + γ)i, I() = I T = γi ηt, T() = Integration of the first equation, the sum of the first two equations, and the third equation gives ln S [ = R(γ) 1 S ] S N The quantity R(γ) = βn [ 1 + δγ ] α + γ η again represents the mean number of secondary infections caused by a single infective introduced into a fully susceptible population and is a decreasing function of γ if η > δα 7

8 THE AGE OF INFECTION MODEL Let S(t) denote the number of susceptibles at time t and ϕ(t) the total infectivity at time t, and ϕ (t) the total infectivity at time t of those individuals who were already infected at time t = Let A(τ) be the total infectivity of members of the population with infection age τ Age of infection epidemic model is S = βsϕ ϕ(t) = ϕ (t) + = ϕ (t) + t t Basic reproduction number is R = βn and final size relation is ln S S = R βs(t τ)ϕ(t τ)a(τ)dτ [ S (t τ)]a(τ)dτ A(τ)dτ, ( 1 S N ) 8

9 EXAMPLE: THE STAGED PROGRESSION EPIDEMIC Consider an epidemic with progression from S (susceptible) through k infected stages I 1,I 2,,I k with the distribution of stay in stage i given by P i, meaning that the fraction of infectives who enter stage i and are still in stage i a time τ after entering the stage is P i (τ), with P i () = 1, P(t)dt <, and P i non-negative and monotone non-decreasing Assume that in stage i the relative infectivity is ε i Then S (t) = βs(t)ϕ(t) and the infectivity ϕ(t) is k ϕ(t) = ε i I i (t) i=1 The basic reproduction number is k R = βn ε i i=1 P i (t)dt, 9

10 General final size relation is ln S [ = R S 1 S ] β N Initial term satisfies [(N S )A(t) ϕ (t)]dt [(N S )A(t) ϕ (t)]dt If all initial infectives have infection-age zero at t =, ϕ (t) = [N S ]A(t), and Then final size relation is [ϕ (t) (N S )A(t)]dt = ln S S = R ( 1 S ) N If initial infectives are outside the population under study, I =, and final size relation is ln S ( = R S 1 S ) S 1

11 INITIAL EXPONENTIAL GROWTH RATE For the simple SIR model, if t is small, S N, and the equation for I is approximately I = (βn α)i = (R 1)αI, and solutions grow exponentially with growth rate (R 1)α The exponential initial growth rate r can be measured (?), and then we have an estimate R = 1 + r α More complicated models are approximated for small t by linear systems, whose solutions have an exponential growth rate given by the largest eigenvalue of the coefficient matrix Thus for the SEIR model, the initial exponential growth rate r < α(r 1) is the (unique if R > 1) positive eigenvalue of [ ] κ βn κ α 11

12 For the age of infection model, an epidemic means that the disease-free equilibrium, with S = N and all infected variables zero is unstable An epidemic means that the equilibrium S = N,ϕ = is unstable To find equilibria, we need to use the limit equation S = βsϕ ϕ(t) = to find equilibria βs(t τ)ϕ(t τ)a(τ)dτ The linearization at the equilibrium S = N,ϕ = is u (t) = βnv(t) v(t) = βn v(t τ)a(τdτ The characteristic equation is the condition on λ that the linearization have a solution u = u e λt,v = v e λt, and this is just βn e λτ A(τ)dτ = 1 The initial exponential growth rate is the solution λ of this equation 12

13 EPILOGUE The deeper knowledge Faust sought Could not from the Devil be bought But now we are told By theorists bold That all you need is R naught - RMMay (Lord May of Oxford) 13