(mathematical epidemiology)

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1 1. 30 (mathematical epidemiology) ) * Anderson and May 1), Diekmann and Heesterbeek 3) 7) 14) NO. 538, APRIL

2 S(t), I(t), R(t) (susceptibles ) (infectives ) (recovered/removed = βs(t)i(t) = βs(t)i(t) γi(t) = γi(t) (1) β γ βi(t) (force of infection) SIR (latent period / exposed class) ( incubation period) (SEIR R S SIRS * Diekmann, Metz, Thieme ) (1) S(0) = (βs(0) γ)i(t) (2) I(t) = I(0)e (βs(0) γ)t βs(0) γ > 0 R 0 = βs(0) γ > 1 (3) R 0 < 1 R 0 (basic reproduction number) βs(0) S(0) 2 1/γ R 0 2 R 0 > 1 R 0 < 1 (threshold phenomena) 2)4)5) R 0 (1) N(t) = 2

3 S(t) + I(t) + R(t) dn(t)/ = 0 (S, I) (S, I) Ω = {(S, I) : S 0, I 0} < t < Ω S S(t) lim t S(t) = S(+ ) I(t) S < S cr = γ/β lim t I(t) = 0 S cr t (S ) < t < 2 (1) di ds = 1 + S cr S (4) (S, I) I(t) = I(0)+S(0) S(t)+S cr log S(t) S(0) (5) (5) t S( ) = S(0) + I(0) + S cr log S( ) S(0) p(t) := S(0) S(t) S(0) = 1 S(t) S(0) (6) (7) p( ) = lim t p(t) S(0) (6) 1 p( ) = e R0p( ) ζ (8) (8) ζ 0 R 0 > 1 R 0 1 p( ) < 1 ζ 0 (final size equation) 1 p( ) = e R0p( ) (9) R 0 R 0 λ (2) γ(r 0 1) R 0 = 1 + (λ/γ) 1 p( ) log(1 p( )) R 0 = p( ) (10) (1) 2 γ/β [] ζ := βi(0)/γ NO. 538, APRIL

4 3. (3) (outbreak) (endemic state) (1) b µ = b µs(t) βs(t)i(t) = βs(t)i(t) (µ + γ)i(t) = µr(t) + γi(t) (11) N(t) = S(t)+I(t)+R(t) b/µ N := b/µ = b µs(t) βs(t)i(t) = βs(t)i(t) (µ + γ)i(t) (12) Ω := {(S, I) : S 0, I 0, S + I N} (12) Ω S N [ ] βn = (γ + µ) γ + µ 1 I(t) (13) R 0 = βn γ + µ = βb µ(γ + µ) (14) ds/ = 0, di/ = 0 E 1 := (N, 0), E 2 := ( N, µ ) R 0 β (R 0 1) E 1 (disease-free steady state E 2 R 0 > 1 endemic steady state (S, I ) S N = 1 I, R 0 N = µ ) (1 1R0 µ + γ (15) (prevalence) I /N 1 1/R 0 1/(µ + γ) 1/µ R 0 R 0 1 S = 0 Ω R 0 > 1 (13) E 1 E 1 E 2 R 0 (11) (11) * R 0 4

5 4. (11) (11) = (1 v)b µs(t) βs(t)i(t) = βs(t)i(t) (µ + γ)i(t) = vb µr(t) + γi(t) (16) v S = (1 v)b/µ = (βs (µ + γ))i(t) (17) v (effective reproduction number) R v = (1 v)bβ µ(µ + γ) = (1 v)r 0 (18) (16) R v 1 R v > 1 R v 1 R v 1 v 1 1 R 0 = H (19) * R 0 H (measles) (herd immunity) [] (19) = b(1 v) µs(t) βs(t)i(t) = (µ + γ)i(t) + β(s(t) + σr(t))i(t) = bv µr(t) + γi(t) βσr(t)i(t) (20) σβ σ 0 ( (1 v)b (S, I, R ) =, 0, bv ) µ µ (21) v NO. 538, APRIL

6 R(v) = β ( (1 v)b + σbv ) µ + γ µ µ (22) [] R(v) R(0) = R 0 R(1) < 1 < R 0 v > v v R(v ) = 1 v (0, 1) R(1) > 1 R 0 > 1/σ (11) R 0 1 (16) (19) R t = βs(t) µ + γ (23) HIV/AIDS R 0 < 1 8)9) R 0 = 1 (subcritical/backward bifurcation) R 0 < 1 R 0 R 0 < 1 * (21) σ > 1 + (µ/γ) 5. 13) 11) 12) 21 6

7 1) R. M. Anderson and R. M. May (1991), Infectious Diseases of Humans: Dynamics and Control, Oxford UP, Oxford. 2) O. Diekmann, J. A. P. Heesterbeak and J. A. J. Metz (1990), On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations, J. Math. Biol. 28: ) O. Diekmann and J. A. P. Heesterbeek (2000), Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, John Wiley and Sons, Chichester. 4) K. Dietz (1993), The estimation of the basic reproduction number for infectious diseases, Statistical Methods in Medical Research 2: ) J. A. P. Heesterbeek (2002), A brief history of R 0 and a recipe for its calculation, Acta Biotheoretica 50: ) H. Inaba (2001), Kermack and McKendrick revisited: The variable susceptibility model for infectious diseases, Japan J. Indust. Appl. Math. 18(2): ) (2002),,,. 8) H. Inaba and H. Sekine (2004), A mathematical model for Chagas disease with infection-agedependent infectivity, Math. Biosci. 190: ) H. Inaba (2006), Endemic threshold results for age-duration-structured population model for HIV infection, Math. Biosci. 201: ) W. O. Kermack and A. G. McKendrick (1927), Contributions to the mathematical theory of epidemics I, Proceedings of the Royal Society 115A: (reprinted in Bulletin of Mathematical Biology 53(1/2): 33-55, 1991) 11) R. M. May and A. L. Lloyd (2001), Infection dynamics on scale-free network, Physical Review E ) M. A. Nowak and R. M. May (2000), Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford. 13) (2006),,, 54 2 : ) H. R. Thieme (2003), Mathematics in Population Biology, Princeton University Press, Princeton and Oxford. [ ( ) 2006] 1 R (4): (2004) Nokes Fine NO. 538, APRIL