Essential Ideas of Mathematical Modeling in Population Dynamics

Transcription

1 Essential Ideas of Mathematical Modeling in Population Dynamics Toward the application for the disease transmission dynamics Hiromi SENO Research Center for Pure and Applied Mathematics Department of Computer and Mathematical Sciences Graduate School of Information Sciences Tohoku University, Japan FOR: Department of Mathematics, Universitas Indonesia, August 23 26, 2017

2 Outline of This Course Prologue Introduction to the mathematical modeling of population dynamics Modeling for malthusian and logistic growths Law of mass-action Extension of logistic equation model Time-discrete model Mathematical modeling of disease transmission dynamics Kermack McKendrick model Modeling of a discrete model for the disease transmission dynamics Mathematical nature of the discrete model Extension with the introduction of time step size Limit to zero time step size Relation to Kermack McKendrick model Basic reproduction number R 0 Key factors and idea for the more sophisticated modeling Epilogue

3 Prologue

4 Hiromi SENO Research Center for Pure and Applied Mathematics, Department of Computer and Mathematical Sciences Graduate School of Information Sciences Tohoku University mathematical expressions mathematical translations BIOLOGICAL PROBLEMS biological hypotheses & assumptions design of mathematical analyses mathematical analyses modeling hypotheses & assumptions biological researches mathematical results mathematical discussions MATHEMATICAL PROBLEMS biological translations biological discussions

5 Mathematical model for biological phenomenon Mathematical Model for Application Biology Advanced Mathematical Model Basic Mathematical Model Mathematics

6 Mathematical model for biological phenomenon Mathematical model for the systemization of knowledges for biological phenomena Mathematical model for the quantitative understanding of biological phenomena Mathematical model for the qualitative understanding of biological phenomena Mathematical model for the research on a specific biological phenomena

7 Mathematical model for biological phenomenon explanation experiment description systemization prediction model development understanding mathematical interest

8 Mathematical modeling

9 Mathematical modeling of population dynamics is the nature of the spatio-temporal variation of biological population size (i.e. density etc.).

10 Mathematical modeling of population dynamics is the nature of the spatio-temporal variation of biological population size (i.e. density etc.).

11 Mathematical modeling of population dynamics

12 Mathematical modeling of population dynamics

13 Hiromi SENO Research Center for Pure and Applied Mathematics, Department of Computer and Mathematical Sciences Graduate School of Information Sciences Tohoku University What mathematical model is reasonable from the biological viewpoint?

14 Hiromi SENO Research Center for Pure and Applied Mathematics, Department of Computer and Mathematical Sciences Graduate School of Information Sciences Tohoku University What mathematical structure is appropriate for the reasonable modeling?

15 Hiromi SENO Research Center for Pure and Applied Mathematics, Department of Computer and Mathematical Sciences Graduate School of Information Sciences Tohoku University Reasonability of modeling depends on i) purpose of theoretical research; ii) available data/knowledge/hypothesis; iii) design of mathematical analysis.

16 Introduction to the mathematical modeling of population dynamics

17 Modeling of malthusian growth

18 Modeling of malthusian growth dn(t) = rn(t) N(t) = N(0)e rt Population size at time t Intrinsic growth rate

19 Malthusian growth N t

20 Malthusian growth

21 Modeling of malthusian growth dn(t) = rn(t) N(t) = N(0)e rt

22 Modeling of malthusian growth dn(t) = rn(t) t 0 lim N(t + t) N(t) t = rn(t)

23 Modeling of malthusian growth dn(t) = rn(t) N(t + t) N(t) t rn(t)

24 Modeling of malthusian growth dn(t) = rn(t) N(t + t) N(t) N(t) t r Time-independent and density-independent constant

25 Modeling of malthusian growth { N(t + t) N(t) } N(t) t r Mean per capita increment of population size in t

26 Modeling of malthusian growth { N(t + t) N(t) } N(t) t r Mean per capita increment velocity of population size in t i.e. Mean per capita growth rate Physiological condition of individual

27 Modeling of malthusian growth dn(t) = rn(t) N(t + t) N(t) N(t) t r Time-independent and density-independent constant Mean per capita growth rate Physiological condition of individual

28 Modeling of logistic growth

29 Logistic equations [L-1] [L-2] [L-3] [L-4] dn(t) dn(t) dn(t) dn(t) = { r 0 βn(t) } N(t) { = r 0 1 N(t) } N(t) K = r 0 N(t) b{n(t)} 2 = { r 0 βn(t) } N(t) b{n(t)} 2 r 0 : intrinsic growth rate

30 Logistic equation [L-1] dn(t) = { r 0 βn(t) } N(t)

31 Logistic growth N(t) = r 0 /β 1 + { r 0 /β N(0) 1} e r 0t

32 Logistic growth N t

33 Logistic growth

34 Logistic growth

35 Logistic growth Drosophila

36 Logistic growth? Tasmanian sheep

37 Logistic equation [L-1] dn(t) = { r 0 βn(t) } N(t)

38 Modeling of logistic growth dn(t) = { r 0 N(t) }N(t) N(t + t) N(t) lim t 0 t = { r 0 N(t) }N(t)

39 Modeling of logistic growth dn(t) = { r 0 N(t) }N(t) N(t + t) N(t) t { N(t) }N(t) r 0

40 Modeling of logistic growth dn(t) = { r 0 N(t) }N(t) N(t + t) N(t) N(t) t r 0 N(t) Density-dependent Mean per capita growth rate Physiological condition of individual

41 Modeling of logistic growth N(t + t) N(t) N(t) t r 0 N(t) Density Effect

42 Modeling of logistic growth Per capita growth rate r 0 r 0 N

43 Logistic growth (!?) Per capita growth rate azuki bean beetle Callosobruchus chinensis (Linnaeus) Bruchidae Population density

44 Logistic growth (!?) Population Density azuki bean beetle Callosobruchus chinensis (Linnaeus) Bruchidae Generation

45 Logistic growth (!??) The great tit Parus major in Holland

46 Number of independent offsprings per reproductive individual Logistic growth (!?) Number of reproductive individuals

47 Logistic equations L-1 and L-2 [L-1] [L-2] dn(t) dn(t) = { r 0 βn(t) } N(t) { = r 0 1 N(t) } N(t) K

48 Logistic growth N carrying capacity t

49 Logistic equations L-1 and L-2 Carrying capacity [L-1] [L-2] dn(t) dn(t) = { r 0 βn(t) } N(t) { = r 0 1 N(t) K } N(t) [L-1] r 0 β [L-2] K Mathematical equivalence between L-1 and L-2 is not that of modeling!

50 Logistic equations L-1 and L-2 Carrying capacity [L-1] [L-2] dn(t) dn(t) = { r 0 βn(t) } N(t) { = r 0 1 N(t) K } N(t) [L-1] r 0 β [L-2] K They are significantly different from each other in the dependence of the carrying capacity on the intrinsic growth rate!

51 Logistic equation L-2 dn(t) { = r 0 1 N(t) K } N(t)

52 Logistic equations [L-1] [L-2] [L-3] [L-4] dn(t) dn(t) dn(t) dn(t) = { r 0 βn(t) } N(t) { = r 0 1 N(t) } N(t) K = r 0 N(t) b{n(t)} 2 = { r 0 βn(t) } N(t) b{n(t)} 2 r 0 : intrinsic growth rate

53 Law of mass-action

54 Law of mass-action αa + βb γc

55 Law of mass-action αa + βb γc Reaction velocity V = 1 α d[a] = 1 β d[b] = 1 γ d[c]

56 Law of mass-action in chemical reaction αa + βb γc Reaction velocity: law of kinetic mass-action V = V([A], [B], [C]) = κ [A] n A [B] n B [C] n C

57 Law of mass-action Lotka Volterra type of mass-action assumption V = V([A], [B]) = κ [A][B]

58 Logistic equation L-3 [L-3] Malthusian growth Decrease due to intra-specific reaction

59 Logistic equation L-4 [L-4] = [L-1] + [L-3] Logistic growth Decrease due to intra-specific reaction

60 Logistic equations [L-1] [L-2] [L-3] [L-4] dn(t) dn(t) dn(t) dn(t) = { r 0 βn(t) } N(t) { = r 0 1 N(t) } N(t) K = r 0 N(t) b{n(t)} 2 = { r 0 βn(t) } N(t) b{n(t)} 2 r 0 : intrinsic growth rate

61 Logistic equation with the mass-action assumption for resource consumption dn(t) dr(t) = cρr(t)n(t) = ρr(t)n(t) cf. bacteria population in culture

62 Logistic equation with the mass-action assumption for resource consumption dr/ dn/ = dr dn = ρrn cρrn = 1 c (= const.)

63 Logistic equation with the mass-action assumption for resource consumption R(t) = 1 c N(t)+C = 1 c N(t)+R(0)+ 1 c N(0)

64 Logistic equation with the mass-action assumption for resource consumption dn(t) dr(t) = cρr(t)n(t) = ρr(t)n(t) cf. bacteria population in culture

65 Logistic equation with the mass-action assumption for resource consumption dn(t) = ρr(t)n(t) = ρ { cr(0)+n(0) N(t) } N(t) Carrying capacity: cr(0)+n(0)

66 Logistic growth

67 Logistic growth

68 Logistic growth Drosophila

69 Extension of logistic equation model

70 Extended logistic equation with the generalized density effect function dn(t) = [ ] r 0 β{n(t)} α N(t) }{{} Density-dependent per capita growth rate Gilpin & Ayala (1973)

71 Extended logistic equation with the generalized density effect function dn(t) = [ r 0 β{n(t)} α] N(t)

72 Logistic growth (!?) Per capita growth rate azuki bean beetle Callosobruchus chinensis (Linnaeus) Bruchidae Population density

73 Number of independent offsprings per reproductive individual Logistic growth (!?) Number of reproductive individuals

74 Extended logistic equation with the generalized mass-action assumption dn(t) = r 0 N(t) γ{n(t)} θ }{{} Decrease due to the intraspecific reaction

75 Extended logistic equation with the generalized mass-action assumption dn(t) = r 0 N(t) γ{n(t)} θ (θ < 1)

76 Extended logistic equation dn = (r 0 βn α ) N γn }{{}}{{} θ Per capita growth rate Intraspecific reaction

77 Extended logistic equation dn(t) =(r 0 βn α )N γn θ (θ < 1) Allee effect

78 Time-discrete model of malthusian growth

79 Time-discrete model of malthusian growth dn(t) = r 0 N(t) N(t) =N(0) e rt

80 Time-discrete model of malthusian growth dn(t) = r 0 N(t) N(t) =N(0) e rt N(t) = N(0) e rt N(t + h) = N(0) e r(t+h) = e rh N(0) e rt h time step size for the discrete dynamics

81 Time-discrete model of malthusian growth dn(t) = r 0 N(t) N(t) =N(0) e rt N(t + h) =e rh N(t) h time step size for the discrete dynamics

82 Time-discrete model of malthusian growth dn(t) = r 0 N(t) N(t) =N(0) e rt N k+1 = RN k with time step size h and R = e rh.

83 Malthusian growth N t

84 Time-discrete model of logistic growth

85 Time-discrete model of logistic growth dn(t) = { r 0 βn(t) } N(t) N(t) = r 0 /β 1 + { r 0 /β N(0) 1} e r 0t

86 Time-discrete model of logistic growth dn(t) = { r 0 βn(t) } N(t) N(t) = r 0 /β 1 + { r 0 /β N(0) 1} e r 0t N(t + h) = e rh N(t) 1 + er 0 h 1 r 0 βn(t) h time step size for the discrete dynamics

87 Time-discrete model of logistic growth dn(t) = { r 0 βn(t) } N(t) N(t) = r 0 /β 1 + { r 0 /β N(0) 1} e r 0t Verhulst Beverton Holt model N k+1 = RN k 1 + bn k with time step size h, R = e rh and b = er 0 h 1 r 0 β.

88 Time-discrete model of logistic growth 4 3 N time

89 Time-discrete model of logistic growth Discretization with the Euler scheme dn(t) = { r 0 βn(t) } N(t) Ñ(t + h) Ñ(t) h = { r 0 βñ(t) } Ñ(t) h time step size for the discretization

90 Time-discrete model of logistic growth Discretization with the Euler scheme dn(t) = { r 0 βn(t) } N(t) Ñ(t + h) = { 1 + r 0 h βhñ(t) } Ñ(t) h time step size for the discretization

91 Time-discrete model of logistic growth logistic map N k+1 = r(1 N k ) N k with time step size h, r = 1 + r 0 h and N k = βhñ(kh) 1+r 0 h.

92 Time-discrete model of logistic growth logistic map ^ N k ^ N k

93 Time-discrete model of logistic growth Bifurcation diagram N^ ^ r

94 Time-discrete model of logistic growth 4 Simple Euler Scheme 3 N time

95 Time-discrete model of logistic growth The other discretization with the Euler scheme dn(t) = { r 0 βn(t) } N(t) d log N(t) = r 0 βn(t) log Ñ(t + h) log Ñ(t) h = r 0 βñ(t) h time step size for the discretization

96 Time-discrete model of logistic growth The other discretization with the Euler scheme dn(t) = { r 0 βn(t) } N(t) Ñ(t + h) =e r 0h βhñ(t) Ñ(t) h time step size for the discretization

97 Time-discrete model of logistic growth Ricker model N k+1 = R e N k N k with time step size h, R = e r 0h and N k = βhñ(kh).

98 Logistic growth (!?) Population Density azuki bean beetle Callosobruchus chinensis (Linnaeus) Bruchidae Generation

99 Mathematical modeling of disease transmission dynamics

100 Kermack McKendrick model

101 Kermack McKendrick model Susceptible S β I (1 m)q Infective I R Removed/Recovered mq

102 Kermack McKendrick model ds(t) di(t) dr(t) = βi(t)s(t)+(1 m)qi(t)+θr(t) = βi(t)s(t) qi(t) = mqi(t) θr(t) S(t) : Susceptible population size at time t; I(t) : Infective population size at time t; R(t) : Recovered immune population size at time t; N : Total population size (time-independent constant; = S(t)+I(t)+R(t))

103 Kermack McKendrick model ds(t) di(t) dr(t) = β I(t) N = β I(t) N cn S(t)+(1 m)qi(t)+θr(t) cn S(t) qi(t) = mqi(t) θr(t) ( β = βc) S(t) : Susceptible population size at time t; I(t) : Infective population size at time t; R(t) : Recovered immune population size at time t; N : Total population size (time-independent constant; = S(t)+I(t)+R(t))

104 Kermack McKendrick model Susceptible S β I N cn (1 m)q Infective I R Removed/Recovered mq

105 Kermack McKendrick SIR model (θ = 0 and m = 1) Susceptible S β I N cn Infective I R Removed/Recovered q

106 Kermack McKendrick SIR model (θ = 0 and m = 1) ds(t) di(t) dr(t) = βi(t)s(t)+(1 m)qi(t)+θr(t) = βi(t)s(t) qi(t) = mqi(t) θr(t)

107 Kermack McKendrick SIR model (θ = 0 and m = 1) S R I I time q β S

108 Kermack McKendrick SIR model (θ = 0 and m = 1) ds(t) = βi(t)s(t) di(t) = βi(t)s(t) qi(t)

109 Kermack McKendrick SIR model (θ = 0 and m = 1) 1 S(t) ds(t) = βi(t) ds(t) + di(t) = qi(t)

110 Kermack McKendrick SIR model (θ = 0 and m = 1) ds(t) + di(t) q β 1 S(t) ds(t) = 0

111 Kermack McKendrick SIR model (θ = 0 and m = 1) S(t)+I(t) q β log S(t) =const.

112 Kermack McKendrick SIR model (θ = 0 and m = 1) Conserved quantity of Kermack McKendrick SIR model S(t)+I(t) q β log S(t) =S(0)+I(0) q β log S(0)

113 Kermack McKendrick SIR model (θ = 0 and m = 1) Conserved quantity of Kermack McKendrick SIR model S(t)+I(t) q β log S(t) =S(0)+I(0) q β log S(0) I 0 q β S

114 Modeling of a discrete model for the disease transmission dynamics

115 Assumptions A population with an infectious disease of negligible fatality; Disease transmission dynamics in such a time scale that the temporal variation of total population size with birth, death and migration could be negligible; Approximation of complete mixing of individuals in terms of disease transmission process.

116 Assumptions Probability that the cumulative number of contacts of an individual with others per day, P(i); Expected cumulative number of contacts of an individual with others per day, π = i=0 ip(i); Probability that a susceptible individual is infected per contact, β; Probability that a susceptible individual with j times contacts to infectives can successfully escape the disease transmission, (1 β) j.

117 Modeling: probability about the number of contacted infectives Probability that l of j contacts are with infective at the k th day ( j l )( ) l ( Ik 1 I ) j l k N N ( ) j := l l! l!(j l)! I k N : probability that an encounter is with an infective

118 Modeling: probability about successful escape from disease transmission Probability that a susceptible can escape from the infection when he/she contacts j times with others at the k th day j l=0(1 β) l ( j l = j l=0 ( j l )( ) l ( Ik 1 I ) j l k N N ){ } Ik /N l ( (1 β) 1 I k /N } j ( { Ik /N = (1 β)+1 1 I k /N ( = 1 β I ) j k N 1 I k N 1 I k N ) j ) j

119 Discrete model for the disease transmission dynamics S k+1 = I k+1 = ( 1 β I ) j k P(j)S k +(1 m)qi k + θr k N j=0 { ( 1 1 β I ) } j k P(j)S k +(1 q)i k N j=0 R k+1 = mqi k +(1 θ)r k q : Probability of recovery with loss of infectivity; m : Probability of successful establishment of immunity; θ : Probability of loss of immunity.

120 Discrete model for the disease transmission dynamics

121 Discrete model for the disease transmission dynamics In case of Poisson distribution of contact numbers with others per day P(j) = γj e γ j! ( π = γ ) ( 1 β I ) j k P(j) =e βγi k/n j=0 N

122 Discrete model for the disease transmission dynamics S k+1 = S k e βγi k/n +(1 m)qi k + θr k I k+1 = S k (1 e βγi k/n )+(1 q)i k R k+1 = mqi k +(1 θ)r k

123 Discrete model for the disease transmission dynamics Susceptible S 1 e (1 m)q βγ I k N Infective I R mq Removed/Recovered θ=0 and m=1 SIR model

124 Discrete SIR model for the disease transmission dynamics θ=0 and m=1 Susceptible S 1 e βγ I k N Infective I R Removed/Recovered q

125 Discrete SIR model for the disease transmission dynamics θ=0 and m=1 S k+1 = S k e βγi k/n +(1 m)qi k + θr k I k+1 = S k (1 e βγi k/n )+(1 q)i k R k+1 = mqi k + (1 θ)r k

126 Discrete SIR model for the disease transmission dynamics I k S k Trajectories of our time-discrete SIR model. S(0) =0.99; I(0) =0.01; R(0) =0.0; q = 0.2; β = 0.01; γ = 50.0.

127 Mathematical nature of the discrete model

128 Discrete SIR model for the disease transmission dynamics S k+1 = S k e βγi k/n I k+1 = S k (1 e βγi k/n )+(1 q)i k

129 Discrete SIR model for the disease transmission dynamics log S k+1 S k = βγ I k N S k+1 + I k+1 = S k + I k qi k

130 Discrete SIR model for the disease transmission dynamics S k+1 + I k+1 qn βγ log S k+1 = S k + I k qn βγ log S k

131 Discrete SIR model for the disease transmission dynamics Conserved quantity of discrete SIR model S k + I k qn βγ log S k = S 0 + I 0 qn βγ log S 0

132 Extension of the discrete model with the introduction of time step size

133 Introduction of time step size h Discrete model for the disease transmission dynamics S k+1 = S k e βγi k/n +(1 m)qi k + θr k I k+1 = S k (1 e βγi k/n )+(1 q)i k R k+1 = mqi k +(1 θ)r k (S k, I k, R k ) ( S(t), I(t), R(t) ) ; (S k+1, I k+1, R k+1 ) ( S(t + h), I(t + h), R(t + h) ) ; P(j) = γj e γ j! (γh)j e γh ; γ γh; j! q qh (0 qh 1); θ θh (0 θh < 1).

134 Introduction of time step size h Discrete model wth the time step size h S(t + h) =S(t)e βγh I(t)/N +(1 m)qh I(t)+θh R(t) I(t + h) =S(t){1 e βγh I(t)/N } +(1 qh)i(t) R(t + h) =mqh I(t)+(1 θh)r(t) (S k, I k, R k ) ( S(t), I(t), R(t) ) ; (S k+1, I k+1, R k+1 ) ( S(t + h), I(t + h), R(t + h) ) ; P(j) = γj e γ j! (γh)j e γh ; γ γh; j! q qh (0 qh 1); θ θh (0 θh < 1).

135 Limit to zero time step size

136 Limit to zero time step size Discrete model with the time step size h S(t + h) =S(t)e βγh I(t)/N +(1 m)qh I(t)+θh R(t) I(t + h) =S(t){1 e βγh I(t)/N } +(1 qh)i(t) R(t + h) =mqh I(t)+(1 θh)r(t)

137 Limit to zero time step size Discrete model with the time step size h S(t + h) S(t) h = S(t) 1 e βγh I(t)/N h +(1 m)qi(t)+θr(t) I(t + h) I(t) h = S(t) 1 e βγh I(t)/N h qi(t) R(t + h) R(t) h = mqi(t) θr(t) h 0

138 Limit to zero time step size h 0 ODE model at the limit of h 0 ds(t) di(t) dr(t) = β I(t) γs(t)+(1 m)qi(t)+θr(t) N = β I(t) γs(t) qi(t) N = mqi(t) θr(t)

139 Relation to Kermack McKendrick model

140 Relation to Kermack McKendrick model ODE model at the limit of h 0 ds(t) di(t) dr(t) = β I(t) γs(t)+(1 m)qi(t)+θr(t) N = β I(t) γs(t) qi(t) N = mqi(t) θr(t)

141 Relation to Kermack McKendrick model Kermack McKendrick model ds(t) di(t) dr(t) = β I(t) N = β I(t) N cn S(t)+(1 m)qi(t)+θr(t) cn S(t) qi(t) = mqi(t) θr(t)

142 Relation to Kermack McKendrick model ODE model at the limit of h 0 ds(t) di(t) = β I(t) γs(t)+(1 m)qi(t)+θr(t) N = β I(t) γs(t) qi(t) N Kermack McKendrick model ds(t) di(t) = β I(t) N = β I(t) N cn S(t)+(1 m)qi(t)+θr(t) cn S(t) qi(t)

143 Relation to Kermack McKendrick model γ = π = cn

144 Relation to Kermack McKendrick model Correspondence of modeling to Kermack McKendrick model With Poisson distribution {P(j)} and γ = π = cn, our time-discrete SIR model coincides with Kermack McKendrick model at the limit of zero time step size. This result was expected, because of the coincidence in their modelings. Further for the SIR models,...

145 Dynamical consistency in SIR model Dynamical consistency in SIR model With Poisson distribution {P(j)} and γ = π = cn, our time-discrete SIR model has the quantitative dynamical consistency with Kermack McKendrick SIR model, independently of time step size h.

146 Dynamical consistency in SIR model Kermack McKendrick SIR model ( β = cβ) S(t)+I(t) q q log S(t) =S(0)+I(0) log S(0) cβ cβ Discrete SIR model (γ = π = cn) S k + I k q cβ log S k = S 0 + I 0 q cβ log S 0 Remark: This is independent of time step size h.

147 SIR models Relation to Kermack McKendrick model I k S k Trajectories of our time-discrete SIR model. S(0) =0.99; I(0) =0.01; R(0) =0.0; N = 1; q = 0.2; β = 0.01; γ = cn = 50.0.

148 Relation to Kermack McKendrick model Trajectories of our time-discrete model and Kermack McKendrick model. (a) m = 0.5; (b)m = 1.0. S(0) =0.999; I(0) =0.001; R(0) =0.0; q = 0.02; β = 0.1; γ = cn = 0.5; θ = 0.0; N = 1.0; h = In(b),thetrajectoryof time-discrete model is always on the solution curve of Kermack McKendrick model.

149 Basic Basic reproduction reproduction number number R R00

150 Basic reproductive number R 0 Definition of the basic reproductive number R 0 The expected number of new cases of an infection caused by an infected individual, in a population consisting of susceptible contacts only. Since the frequency of susceptibles decreases as the epidemic process goes on, the basic reproductive number defines, rigorously saying, the supremum of the expected number of new cases of an infection caused by an infected individual at the initial stage of epidemic process.

151 Basic reproductive number R 0 Definition of the basic reproductive number R 0 The expected number of new cases of an infection caused by an infected individual, in a population consisting of susceptible contacts only. R 0 > 1 R 0 < 1 Initial increase of infective frequency Monotonic disappearance of infection

152 Basic reproductive number R 0 Definition of the basic reproductive number R 0 The expected number of new cases of an infection caused by an infected individual, in a population consisting of susceptible contacts only. R 0 > 1 R 0 < 1 Invasion success of epidemic disease Invasion failure of epidemic disease

153 Discrete model for the disease transmission dynamics General formula of discrete model ( S k+1 = 1 β I ) j k P(j)S N k +(1 m)qi k + θr k j=0 { ( I k+1 = 1 1 β I ) } j k P(j)S N k +(1 q)i k j=0 R k+1 = mqi k +(1 θ)r k

154 Basic reproductive number R 0 For the general discrete model with I 0 N, R 0 = 0, ands 0 N I 1 = = { ( 1 1 β I ) } j 0 P(j)S 0 +(1 q)i 0 N j=0 j=0 { β jβi 0 P(j)+(1 q)i 0 } jp(j)+(1 q) I 0 j=0 = {β π +(1 q)}i 0

155 Basic reproductive number R 0 At the invasion stage in the discrete model Invasion success of infectious disease: I 1 > I 0 I 1 I 0 β π +(1 q) > 1

156 Basic reproductive number R 0 At the invasion stage in the discrete model Invasion success of infectious disease: I 1 > I 0 R 0 = β π q > 1

157 Kermack McKendrick model ds(t) di(t) dr(t) = βi(t)s(t)+(1 m)qi(t)+θr(t) = βi(t)s(t) qi(t) = mqi(t) θr(t) S(t) : Susceptible population size at time t; I(t) : Infective population size at time t; R(t) : Recovered immune population size at time t; N : Total population size (time-independent constant; = S(t)+I(t)+R(t))

158 Basic reproductive number R 0 Kermack McKendrick model with I(0) N, R(0) =0, ands(0) N di(t) = β I(0) cns(0) qi(0) t=0 N β I(0) cn N qi(0) N =(βcn q)i(0)

159 Basic reproductive number R 0 At the invasion stage in Kermack McKendrick model Invasion success of infectious disease: di(t) > 0 t=0 di(t) (βcn q)i(0) > 0 t=0

160 Basic reproductive number R 0 At the invasion stage in Kermack McKendrick model Invasion success of infectious disease: di(t) > 0 t=0 R 0 = βcn q > 1

161 Basic reproductive number R 0 For Kermack McKendrick model R 0 = βcn q For the discrete model R 0 = β π q

162 Consistency of R 0 The basic reproduction number R 0 is coincident between the Kermack McKendrick model and the present discrete model, with the correspondence π = cn. This result is reasonable from the coincidence in their modelings!

163 Discrete model and Kermack McKendrick model Trajectories of our time-discrete model and Kermack McKendrick model. Numerical calculation for the case that the system approaches an endemic state. S(0) =0.999; I(0) =0.001; R(0) =0.0; q = 0.02; m = 0.5; θ = 0.001; β = 0.1; γ = cn = 0.5; N = 1.0; h = 10.0.

164 Key factors and idea for the more sophisticated modeling

165 Key factors and idea for the more sophisticated modeling Spatio-temporal scales mean-field approximation, quasi-stationary state approximation, pair approximation, singular perturbation, metapopulation, renormalization Spatio-temporal heterogeneity reaction diffusion equations, lattice/cellular space, network space, non-standard mean-field approximation Human activities and community structure optimal theory, control theory, game theory, mathematical population genetics

166 Epilogue

167 Epilogue Discrete model has flexibility wider than the ODE model does with respect to the reasonable modeling, although it is in general less tractable for the mathematical analysis. Despite such potential difficulty in mathematics, it is primarily important for the theoretical research in mathematical biology that the structure of mathematical model must be reasonably determined/chosen/designed depending on the biological focus of the theoretical research with it. Such reasonable modeling necessarily requires both the biological/medical knowledge about the focused phenomenon and the mathematical knowledge/sense about the connection between the natures of phenomenon and the structures of mathematical factor applied for the modeling.