(Kermack and McKendrick) S(t),I(t), R(t) Kermack and McKendrick model SIR

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1 1 ( ) (Anderson 1991) (Kermack and McKendrick) Kermack and McKendrick (1927) S(t),I(t), R(t) Kermack and McKendrick model SIR S (t) = λs(t)i(t) I (t) = λs(t)i(t) γi(t) R (t) = γi(t) λ γ N) I (t) = λni(t) γi(t) (2) (1) R = λn γ > 1 (3) N cr = γ λ R (basic reproduction number) (Diekmann, et al. 199) R > 1 R < 1 ( ) Kermack and McKendrick model Kermack and McKendrick : disease-age (infectivity) (susceptibility) inaba@ms.u-tokyo.ac.jp 1

2 b µ Kermack and McKendrick model S (t) = b µs(t) λs(t)i(t) I (t) = λs(t)i(t) (µ + γ)i(t) (4) R (t) = µr(t) + γi(t) R(t) N(t) = S(t) + I(t) + R(t) N() = b µ N = b µ R = λn γ + µ (5) R 1 (disease-free steady state: DFSS) (S, I ) = ( b µ, ) R > 1 DFSS (endemic steady state: ESS) R > 1 SIR ( 199) (Pease 1987, Inaba 1998) (Anderson and May 1991, Iannelli 1995, Inaba 199 ) SIR (Pease 1987) s(t, τ) t τ i(t, τ) t τ r(t, τ) t τ. m µ γ(τ) τ β 1 (τ)β 2 (σ) σ τ γ, β j L + (R + ), (j = 1, 2) β 1 (τ) σ β 1 ( )β 2 (σ) β 1 ( ) := sup τ β 1 (τ). β 2 (τ) 2

3 β 1 (τ) s(t,τ) t i(t,τ) t r(t,τ) t + s(t,τ) τ = µs(t, τ) s(t, τ)β 1 ( ) β 2 (σ)i(t, σ)dσ, + i(t,τ) τ = (µ + γ(τ))i(t, τ), + r(t,τ) τ = µr(t, τ) r(t, τ)β 1 (τ) β 2 (σ)i(t, σ)dσ, (s(t, τ) + i(t, τ) + r(t, τ))dτ, i(t, ) = {β 1( )s(t, τ) + β 1 (τ)r(t, τ)}dτ β 2 (τ)i(t, τ)dτ, s(t, ) = m r(t, ) = γ(τ)i(t, τ)dτ, s(, a) = s (a), i(, a) = i (a), r(, a) = r (a), (s, i, r ). N R = Nβ 1 ( ) (6) β 2 (ζ)e µζ ζ γ(σ)dσ dζ. (7) R 1 R > 1 (Inaba 2b) s(t, τ) (Yang 2) 4 HIV mating HIV Kermack-McKendrick model S (t) = b (µ + λ(t))s(t) ( t + ) τ i(t, τ) = (µ + γ(τ))i(t, τ) (8) i(t, ) = λ(t)s(t) λ(t) = β(τ)i(t, τ)dτ C(P (t)) P (t) τ P (t) = S(t) + i(t, τ)dτ C(P ) C(x)/x R < 1, ) R > 1 (Thieme and Castillo-Chavez 1993) ( ) b R = C β(τ)e µτ τ γ(σ)dσ dτ (9) µ (Iannelli, Loro, Milner, Pugliese and Rabbiolo 1992) ( b µ 3

4 (pair formation model) R (Knolle 199, Diekmann, et al. 1991, Inaba 1997) τ l 1 (a), l 2 (a), l 3 (a) ρ, σ µ β(a) a η ρ, σ, µ, η HIV l 1(a) = (2µ + σ)l 1 (a) + ρβ(a)l 2 (a) + β(a)ηl 3 (a) l 2(a) = (σ + µ)l 1 (a) (ρ + µ)l 2 (a) + (σ + µ)l 3 (a) l 3(a) = (1 β(a))ρl 2 (a) (β(a)η + σ + 2µ)l 3 (a) l 1 () = 1, l 2 () = l 3 () = l 3 (a) = ρ(σ + µ) σ + ρ + µ (1) l 2 (a) = σ + µ σ + ρ + µ e µa (1 e (σ+ρ+µ)a ), (11) a e a s (β(ζ)η+σ+2µ)dζ (1 β(s))e µs (1 e (σ+ρ+µ)s )ds (12) R = [β(a)ρl 2 (a) + β(a)ηl 3 (a)]da (13) HIV R = S(a) := β(a) + (1 β(a)) φ(a) := a S(a)φ(a)da (14) β(τ)ηe τ a (β(ζ)η+σ+2µ)dζ dτ, (15) ρ(σ + µ) µ(σ + ρ + 2µ) e µa (1 e (σ+ρ+µ)a ) (16) S(a) a φ(a) a HIV (endemic state) Kakehashi (1998) HIV 4

5 (Nowak and May 1991, Kirschner 1996) HIV HIV Brookmeyer and Gail (1994) [1] R. M. Anderson (1991), Discussion: The Kermack-McKendrick epidemic threshold theorem, Bull. Math. Biol. 53(1/2): [2] R. M. Anderson and R. M. May (1991), Infectious Diseases of Humans: Dynamics and Control, Oxford UP, Oxford. [3] R. Brookmeyer and M. H. Gail (1994), AIDS Epidemiology: A Quantitative Approach, Oxford UP, New York Oxford. [4] O. Diekmann, J. A. P. Heesterbeak and J. A. J. Metz (199), On the definition and the computation of the basic reproduction ratio R in models for infectious diseases in heterogeneous populations, J. Math. Biol. 28: [5] O. Diekmann, K. Dietz and J. A. P. Heesterbeek (1991), The basic reproduction ratio for sexually transmitted diseases I. Theoretical considerations, Math. Biosci. 17: [6] M. Iannelli, R. Loro, F. Milner, A. Pugliese and G. Rabbiolo (1992), An AIDS model with distributed incubation and variable infectiousness: Applications to IV drug users in Latium, Italy, Eur. J. Epidemiol. 8(4): [7] (1994), HIV, 5(4),

6 [8] H. Inaba (1997), Calculating R for HIV infection via pair formation, In Advances in Mathematical Population Dynamics -Molecules, Cells and Man, O. Arino, D. Axelrod and M. Kimmel (eds.), World Scientific, Singapore: [9] H. Inaba (1998), Mathematical analysis for an evolutionary epidemic model, in Mathematical Models in Medical and Health Sciences, M. A. Horn, G. Simonett and G. F. Webb (eds.), Vanderbilt University Press, Nashville, pp [1] H. Inaba (2a), Endemic threshold and stability in an evolutionary epidemic model, to appear. [11] H. Inaba (2b), Kermack and McKendrick Revisited: The Variable Susceptibility Model for Infectious Diseases, to appear in JJIAM. [12] (199),, 7, [13] M. Kakehashi (1998), A mathematical analysis of the spread of HIV/AIDS in Japan, IMA J. Math. Appld. Med. Biol. 15: [14] 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) [15] W. O. Kermack and A. G. McKendrick (1932), Contributions to the mathematical theory of epidemics-ii. The problem of endemicity, Proceedings of the Royal Society 138A: (reprinted in Bulletin of Mathematical Biology 53(1/2): 57-87, 1991) [16] W. O. Kermack and A. G. McKendrick (1933), Contributions to the mathematical theory of epidemics-iii. Further studies of the problem of endemicity, Proceedings of the Royal Society 141A: (reprinted in Bulletin of Mathematical Biology 53(1/2): , 1991) [17] D. Kirschner (1996), Using mathematics to understand HIV immune dynamics, Notices of the AMS, 43(2): [18] H. Knolle (199), Age preference in sexual choice and the basic reproduction number of HIV/AIDS, Biom. J. 32(2): [19] J. A. J. Metz (1978), The epidemic in a closed population with all susceptibles equally vulnerable; some results for large susceptible populations and small initial infections, Acta Biotheoretica 27, 1/2: [2] M. A. Nowak and R. M. May (1991), Mathematical biology of HIV infections: Antigenic variation and diversity threshold, Math. Biosci. 16(1): [21] C. M. Pease (1987), An evolutionary epidemiological mechanism, with applications to type A influenza, Theor. Poul. Biol. 31: [22] (1992). [23] H. R. Thieme and C. Castillo-Chavez (1993), How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?, SIAM J. Appl. Math. 53(5):

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