On the Spread of Epidemics in a Closed Heterogeneous Population

Transcription

1 On the Spread of Epidemics in a Closed Heterogeneous Population Artem Novozhilov Applied Mathematics 1 Moscow State University of Railway Engineering (MIIT) the 3d Workshop on Mathematical Models and Numerical Methods in Biomathematics INM RAS, October 27 28, 2011 Artem Novozhilov (MIIT) October 27, / 26

2 S usceptible Īnfected R emoved Model: The population is divided into three groups: S I R The SIR ODE model (Kermack & McKendrick, 1927): Ṡ(t) = βs(t)i(t), I(t) = βs(t)i(t) γi(t), Ṙ(t) = γi(t), S(t) + I(t) + R(t) = N, S(0) = S 0, I(0) = I 0, R(0) = 0, S 0 + I 0 = N. (SIR model) Ref: Kermack & McKendrick, 1927 Artem Novozhilov (MIIT) October 27, / 26

3 Analysis of the SIR Model: 1000 I S 0 + I 0 = N (a) S(t) R(t) (b) σ S 200 I(t) t S( ) Figure: (a) The phase plane of the SIR model; (b) The integral curves Theorem Let the spread of an infectious disease in a closed population be described by the SIR model. Then epidemic occurs if and only if the basic reproductive number R 0 = Nβ/γ > 1. If epidemic occurs then the proportion of the susceptibles who escaped the infection (z = S( )/N) can be found as the only root of the equation z = exp{ R 0 (1 z)}, 0 < z < 1. Artem Novozhilov (MIIT) October 27, / 26

4 Assumptions Behind the SIR Model: A single infection triggers an autonomous process; The disease results in either complete immunity or death; Contacts are made according to the law of mass action; Individuals are infected for an exponentially distributed period of time; All individuals are equally susceptible; The population is closed; The population size is large enough to apply deterministic description. Artem Novozhilov (MIIT) October 27, / 26

5 Heterogeneous SIR Model: Here Ṡ = βsi, I = βsi γi. β = ([contact rate] P[successful contact])/n. Let us assume that P[successful contact] takes only two values: Here S(t) = S 1 (t) + S 2 (t). Ref: Gart, J.J.: Biometrics, 24: , 1968 Ṡ 1 = β 1 S 1 I, Ṡ 2 = β 2 S 2 I, I = β 1 S 1 I + β 2 S 2 I γi. Artem Novozhilov (MIIT) October 27, / 26

6 Heterogeneous SIR Model: Discrete version: Ṡ i = β i S i I, i = 1,..., n, n I = I β i S i γi, S(t) = i=1 n S i. i=1 Ref: Ball, F.: Adv Appl Prob, 17:1 22, 1985 Continuous version: s(t, ω) = β(ω)s(t, ω)i(t), t d I(t) = I(t) β(ω)s(t, ω) dω γi(t), dt Ω s(0, ω) = s 0 (ω), I(0) = I 0, S(t) = s(t, ω) dω. Ref: Novozhilov, A: Math Biosc, 215: , 2008 Ω Artem Novozhilov (MIIT) October 27, / 26

7 SI Model with Distributed Susceptibility and Infectivity: Continuous version: t s(t, ω 1) = β 1 (ω 1 )s(t, ω 1 ) β 2 (ω 2 )i(t, ω 2 ) dω 2 Ω 2 = β 1 (ω 1 )s(t, ω 1 )E t [β 2 ]I(t), t i(t, ω 2) = β 2 (ω 2 )i(t, ω 2 ) β 1 (ω 1 )s(t, ω 1 ) dω 1 Ω 1 = β 2 (ω 2 )i(t, ω 2 )E t [β 1 ]S(t), E t [β 1 ] = β(ω)p s (t, ω 1 ) dω 1 = β(ω 1 ) s(t, ω 1) dω 1, Ω 1 Ω 1 S(t) E t [β 2 ] = β(ω)p i (t, ω 2 ) dω 2 = β(ω 2 ) i(t, ω 2) dω 2, Ω 2 Ω 2 I(t) s(0, ω 1 ) = S 0 p s (0, ω 1 ), i(0, ω 2 ) = I 0 p i (0, ω 2 ). Artem Novozhilov (MIIT) October 27, / 26

8 General Theory of Heterogeneous Populations: l(t, ω) = l(t, ω)f (t, ω), l(t, ω) 0, t F (t, ω) = m 1 u i(t, G i )ϕ i (ω) + m 2 v j(t, H j )ψ j (ω), i=1 j=1 G i (t) = g i (ω)l(t, ω) dω = N(t)E t [g i ], i = 1,..., m 1, Ω H j (t) = h j (ω)p(t, ω) dω = E t [h i ], j = 1,..., m 2. Ω Ref: Karev G.P.: J Math Biol, 60: , 2010 Artem Novozhilov (MIIT) October 27, / 26

9 Equivalent ODE System: For the SIR model with distributed susceptibility Ṡ(t) = E t [β]s(t)i(t), E t [β] = β(ω)p(t, ω) dω, I(t) s(t, ω) = E t [β]s(t)i(t) γi(t), p(t, ω) = S(t), s(0, ω) = S 0 p(0, ω), I(0) = I 0, S(t) = s(t, ω) dω, Ω E t [β] = d dλ log M 0[λ], λ=q(t) q(t) = I(t), q(0) = 0, M 0 [λ] = exp{λβ(ω)}p(0, ω) dω. Ω Ω Artem Novozhilov (MIIT) October 27, / 26

10 Homogeneous Model with Nonlinear Transmission Function: Theorem The heterogeneous SIR model with distributed susceptibility can be reduced to the homogeneous SIR model with a nonlinear transmission function in the form Ṡ = h(s)i, I = h(s)i γi, S(0) = S 0, I(0) = I 0, where [ dm 1 0 h(s) = S [ξ] 1 0 dξ. ξ=s/s0] Artem Novozhilov (MIIT) October 27, / 26

11 Power Transmission Function: Power transmission function: T (S, I) = βs p I q. Ref: McCallum, H., Barlow, N. & Hone, J.: Tr Ecol Evol, 16(6):295300, Corollary The power transmission function T (S, I) = βs p I q with q = 1, p = 1 + 1/k in the homogeneous SIR model can be deduced from the mechanistic formulation of the SIR model with distributed susceptibility when the initial distribution is a gamma-distribution with the shape parameters k. Ref: Novozhilov, A.: Dyn Con Dis Impul Sys, 2009, 16: Artem Novozhilov (MIIT) October 27, / 26

12 Nonlinear Transmission Functions: Heterogeneous SI model: t s(t, ω 1) = ω 1 s(t, ω 1 ) ω 2 i(t, ω 2 ) dω 2, Ω 2 t i(t, ω 2) = ω 2 i(t, ω 2 ) ω 1 s(t, ω 1 ) dω 1, Ω 1 s(0, ω 1 ) = S 0 p s (0, ω 1 ), i(0, ω 2 ) = I 0 p i (0, ω 2 ). Corollary The power transmission function T (S, I) = βs p I q with q = 1 + 1/k 2, p = 1 + 1/k 1 in the homogeneous SIR model can be deduced from the mechanistic formulation of the SIR model with distributed susceptibility and infectivity when the initial distributions are gamma-distributions with parameters k 1, ν 1 and k 2, ν 2 respectively. Ref: Novozhilov, A.: Dyn Con Dis Impul Sys, 2009, 16: Artem Novozhilov (MIIT) October 27, / 26

13 The Final Size of an Epidemic: Theorem Let ẑ = S( )/N be the proportion of the population that escapes infection in the population provided I 0 /N 0. Then for the SIR model with distributed susceptibility, ẑ can be found as the only solution of the equation z = M 0 [ N(1 z)/γ], satisfying the condition 0 < ẑ < 1. This solution exists if and only if R 0 = E 0 [β]nγ 1 > 1. Remark: If p(0, ω) = δ(ω ω), β( ω) = const then we obtain the well-known formula z = exp{ R 0 (1 z)}. Ref: Ball, F.: Adv Appl Prob, 17:1 22, 1985 Ref: Novozhilov, A: Math Biosc, 215: , 2008 Artem Novozhilov (MIIT) October 27, / 26

14 The Final Size of an Epidemic. Examples: gamma log normal Wald beta S( ) σ 2 (0) 8 x 10 3 Figure: The size of the susceptible population that never gets infected versus the initial variance of the parameter distributions. The initial means are the same in all four cases. The parameters are S(0) = 999, I(0) = 1, γ = 20, E 0 [β] = 0.05, R 0 = 2.5 Artem Novozhilov (MIIT) October 27, / 26

15 The Final Size of an Epidemic: Theorem Let ( ) R0 M 0 (1 z) E 0 [β] ( ) R0 M 0 (1 z) E 0 [β] for any z (0, 1). Then ˆ z ẑ. Theorem For any distribution of susceptibility, the final epidemic size satisfies cv 2 (1 z ) 1 ẑ 1 z, where z is the solution to z = exp{ R 0 (1 z)} belonging to (0, 1). Ref: Catriel, G.: J. Math Biol, in press Artem Novozhilov (MIIT) October 27, / 26

16 Stochastic SIR Model: J shaped form R 0 1, π = U shaped form R 0 > 1, 0 < π < S( ) S( ) Instead of two important quantities R 0 (the basic reproductive number) and z (the final epidemic size) in the stochastic settings we have R 0, π (the probability of disease invasion) and a random variable z = S( )/N. Artem Novozhilov (MIIT) October 27, / 26

17 Stochastic vs Deterministic Epidemic Models: Deterministic model: 1. The basic reproductive number R 0 (the threshold parameter: if R 0 1 then there is no epidemic) 2. The final size of an epidemic S( ) (fixed number) Stochastic model: 1. The basic reproductive number R 0 (the threshold parameter: if R 0 > 1 then there is a possibility of a major outbreak) 2. The final size of an epidemic S( ) (random variable, which, if conditioned on the major outbreak, has an approximately normal distribution), z = S( )/N 3. The probability of a major outbreak π (π > 0 iff R 0 > 1) 4. The duration of an epidemic T (random variable) Artem Novozhilov (MIIT) October 27, / 26

18 Stochastic SIR model (Analytical Results): Let E n,m be an epidemic process with n initial susceptible and m infective individuals. Let Z n be the total number of infected individuals. Then D Z, Z n if R 0 1, P[Z < ] = 1, if R 0 > 1, P[Z < ] = 1 π. Moreover, conditional upon the epidemic becoming established n 1 Z n D τ, n, where τ is the root of 1 τ = e R 0τ, and 1 (Z n nτ) D N (0, σ 2 ), n. n Ref: Ball, F.: Adv Appl Prob, 17:1 22, 1985 Ref: Scalia Tomba, G.: J Appl Prob 23: , 1986 Artem Novozhilov (MIIT) October 27, / 26

19 Duration of an Epidemic: Probability Distribution Function Duration of the Epidemic Artem Novozhilov (MIIT) October 27, / 26

20 The Final Size of an Epidemic: Artem Novozhilov (MIIT) October 27, / 26

21 Duration of an Epidemic: 1.5 CV 0 CV 0.25 CV 0.5 Probability Distribution Function CV Duration of the Epidemic Artem Novozhilov (MIIT) October 27, / 26

22 Simulation Results: CV = 0 CV = 0.25 CV = 0.5 CV = 1 E[z] Ê[z] Var[z] Ê[T ] Var[T ] π Artem Novozhilov (MIIT) October 27, / 26

23 Conclusions: There is a general technique applicable to the study of epidemics with distributed heterogeneity Using the theory of heterogeneous populations we obtain a simple equation for the final size of an SIR epidemic It can be shown that some heterogeneous models imply simple homogeneous models The results can be applied to stochastic epidemics conditioned on the occurrence of a major outbreak The final epidemic size is very sensitive to the population heterogeneity, whereas the probability of a major outbreak together with the distribution of the duration of the epidemic are not Artem Novozhilov (MIIT) October 27, / 26

24 Bratus, A. S., Novozhilov, A. S., & Platonov, A. P.: Dynamical Systems and Models in Biology, Moscow: FizMatLit, 2010, 400 pages. Artem Novozhilov (MIIT) October 27, / 26

25 Acknowledgments: NCBI/NLM/NIH, USA for the hospitality during my stay Moscow State University of Railway Engineering, Russia for the grant for young researchers Russian Foundation for Basic Research, grant # Artem Novozhilov (MIIT) October 27, / 26

26 Thank you for your attention! Questions? site: Artem Novozhilov (MIIT) October 27, / 26