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感染症伝染ダイナミクスの数理モデル初歩A First Step into Mathematical Model on The Epidemic Dynamics of Infectious Disease

瀬野裕美Hiromi SENO

東北大学大学院情報科学研究科Graduate School of Information Sciences, Tohoku University, Sendai, Japan

seno@math.is.tohoku.ac.jp

FOR: 明治大学 MIMS 現象数理学拠点リモートセミナー14:30–16:00, 6 August, 2020

OutlineOutline

Prologue: Epidemic dynamics of infectious disease

Epidemic dynamics modelGeneric SIR modelKermack–McKendrick SIR modelBasic reprodution number R0

Kermack–McKendrick SIRS modelForce of infection

Epilogue: Various factors on the epidemic dynamics

Prologue:

Epidemic dynamics of infectious disease

Prologue:

Epidemic dynamics of infectious disease

Epidemic dynamics of infectious diseaseEpidemic dynamics of infectious disease

S → E → I → R → S

Susceptible

S

Removed/Recovered

R

Infective

I

Latent/Incubation

E

Epidemic dynamics of infectious diseaseEpidemic dynamics of infectious disease

S → E → I → R → S

Susceptible

S

Removed/Recovered

R

Infective

I

Latent/Incubation

E

Infectious

Epidemic dynamics of infectious diseaseEpidemic dynamics of infectious disease

S → I → R → S

Susceptible

S

Removed/Recovered

R

Infective

I

Latent/Incubation

E

Infective

I

Epidemic dynamics of infectious diseaseEpidemic dynamics of infectious disease

S → E → I → R → S

Susceptible

S

Removed/Recovered

R

Infective

I

Latent/Incubation

E

Epidemic dynamics of infectious diseaseEpidemic dynamics of infectious disease

S → E → I → R

Susceptible

S

Removed/Recovered

R

Infective

I

Latent/Incubation

E

Epidemic dynamics of infectious diseaseEpidemic dynamics of infectious disease

S → I → R

Susceptible

S

Removed/Recovered

R

Infective

I

Latent/Incubation

E

Infective

I

Epidemic dynamics modelEpidemic dynamics model

Epidemic dynamics modelEpidemic dynamics model

Generic SIR model

Epidemic dynamics modelEpidemic dynamics model

Generic SIR model

Susceptible

S

Removed/Recovered

R

Infective

I

Latent/Incubation

E

Infective

I

Epidemic dynamics modelEpidemic dynamics model

Generic SIR model

dS(t)dt

= B − ΛS(t)− µSS(t)

Epidemic dynamics modelEpidemic dynamics model

Generic SIR model

dS(t)dt

= B − ΛS(t)− µSS(t)

Λ = Λ(S, I, R): Force of infection

Epidemic dynamics modelEpidemic dynamics model

Generic SIR model

dS(t)dt

= B − ΛS(t) − µSS(t)

dI(t)dt

= ΛS(t)− qI(t)− µII(t)

Epidemic dynamics modelEpidemic dynamics model

Generic SIR model

dS(t)dt

= B − ΛS(t) − µSS(t)

dI(t)dt

= ΛS(t)− qI(t)− µII(t)

dR(t)dt

= qI(t)− µRR(t)

Epidemic dynamics modelEpidemic dynamics model

Generic SIR model

dS(t)dt

= − ΛS(t)

dI(t)dt

= ΛS(t)− qI(t)

dR(t)dt

= qI(t)

Epidemic dynamics modelEpidemic dynamics model

Generic SIR model

dS(t)dt

= − ΛS(t)

dI(t)dt

= ΛS(t)− qI(t)

dR(t)dt

= qI(t)

ddt

{S(t) + I(t) + R(t)

}= 0

Epidemic dynamics modelEpidemic dynamics model

Generic SIR model

dS(t)dt

= − ΛS(t)

dI(t)dt

= ΛS(t)− qI(t)

dR(t)dt

= qI(t)

S(t) + I(t) + R(t) = N(time-independent constant total population size)

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

Λ ∝ I

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

Λ ∝ I

dS(t)dt

= −βI(t)S(t)

dI(t)dt

= βI(t)S(t)− qI(t)

dR(t)dt

= qI(t)

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

I

RS

time

10 20 30 40 50

0.2

0.4

0.6

0.8

1.0

I

S0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

q

β

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

dS(t)dt

= −βI(t)S(t)

dI(t)dt

= βI(t)S(t)− qI(t)

dR(t)dt

= qI(t)

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

dS(t)dt

= −βI(t)S(t)

dI(t)dt

= βI(t)S(t)− qI(t)

dR(t)dt

= qI(t)

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

dS(t)dt

= −βI(t)S(t)

dI(t)dt

= βI(t)S(t)− qI(t)

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

dS(t)dt

= −βI(t)S(t)

dI(t)dt

= βI(t)S(t)− qI(t)

dS(t)dt

+dI(t)

dt− q

β

1S(t)

dS(t)dt

= 0

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

dS(t)dt

= −βI(t)S(t)

dI(t)dt

= βI(t)S(t)− qI(t)

ddt

{S(t) + I(t)− q

βlog S(t)

}= 0

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

dS(t)dt

= −βI(t)S(t)

dI(t)dt

= βI(t)S(t)− qI(t)

S(t) + I(t)− qβ

log S(t) = const.

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

dS(t)dt

= −βI(t)S(t)

dI(t)dt

= βI(t)S(t)− qI(t)

Conserved quantity of Kermack–McKendrick SIR model

S(t) + I(t)− qβ

log S(t) = S(0) + I(0)− qβ

log S(0)

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

Conserved quantity of Kermack–McKendrick SIR model

S(t) + I(t)− qβ

log S(t) = S(0) + I(0)− qβ

log S(0)

I

RS

time

10 20 30 40 50

0.2

0.4

0.6

0.8

1.0

I

S0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

q

β

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

Conserved quantity of Kermack–McKendrick SIR model

S(t) + I(t)− qβ

log S(t) = S(0) + I(0)− qβ

log S(0)

I

S0 q

β

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

Conserved quantity of Kermack–McKendrick SIR model

S(t) + I(t)− qβ

log S(t) = S(0) + I(0)− qβ

log S(0)

I

RS

time

10 20 30 40 50

0.2

0.4

0.6

0.8

1.0

I

S0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

q

β

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

I

S0 q

β

Final size equation for Kermack–McKendrick SIR model

S∞ − qβ

log S∞ = S(0) + I(0)− qβ

log S(0)

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

I

S0 q

β

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

S(0) >qβ

⇒ I(t) increases in the early periodof disease invasion;

S(0) ≤ qβ

⇒ I(t) monotonically decreases.

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

β

qS(0) > 1 ⇒ I(t) increases in the early period

of disease invasion;

β

qS(0) ≤ 1 ⇒ I(t) monotonically decreases.

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

β

qS(0) > 1 ⇒ I(t) increases in the early period

of disease invasion;

β

qS(0) ≤ 1 ⇒ I(t) monotonically decreases.

Basic reproduction number for Kermack–McKendrick SIR model

R0 :=β

qS(0)

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIR model

R0 > 1 ⇒ I(t) increases in the early periodof disease invasion;

R0 ≤ 1 ⇒ I(t) monotonically decreases.

Basic reproduction number for Kermack–McKendrick SIR model

R0 :=β

qS(0) ⪅ R0 :=

β

qN

Epidemic dynamics modelEpidemic dynamics model

Basic reproduction number R0

Epidemic dynamics modelEpidemic dynamics model

Basic reproduction number R0

In the biological context, the basic reproduction number

R0 is defined as the expected number of new cases of

an infection caused by an infective individual, in a pop-

ulation::::::::::::::::::::::::::::::::::::::::::consisting of susceptible contacts only.

Epidemic dynamics modelEpidemic dynamics model

Basic reproduction number R0

In the biological context, the basic reproduction number

R0 is defined as the expected number of new cases of

an infection caused by an infective individual, in a pop-

ulation::::::::::::::::::::::::::::::::::::::::::consisting of susceptible contacts only.

※ The definition is independent of the epidemic situation in the population!

Epidemic dynamics modelEpidemic dynamics model

Basic reproduction number R0

In the biological context, the basic reproduction number

R0 is defined as the expected number of new cases of

an infection caused by an infective individual, in a pop-

ulation::::::::::::::::::::::::::::::::::::::::::consisting of susceptible contacts only.

※ The definition is independent of the epidemic situation in the population!

R0 > 1 ⇒ The number of infectives (likely) increases;

R0 < 1 ⇒ The number of infectives decreases.

Epidemic dynamics modelEpidemic dynamics model

Basic reproduction number R0

In the biological context, the basic reproduction number

R0 is defined as the expected number of new cases of

an infection caused by an infective individual, in a pop-

ulation::::::::::::::::::::::::::::::::::::::::::consisting of susceptible contacts only.

※ The definition is independent of the epidemic situation in the population!

The number of infectives increases. ⇒ R0 > 1;

The number of infectives decreases. ⇒ R0 < 1 (likely)

Epidemic dynamics modelEpidemic dynamics model

Basic reproduction number R0

Kermack-McKendrick SIR model

dS(t)dt

= −βI(t)S(t)

dI(t)dt

= βI(t)S(t)− qI(t)

Epidemic dynamics modelEpidemic dynamics model

Basic reproduction number R0

Kermack-McKendrick SIR model

dI(t)dt

= βI(t)S(t)− qI(t)

Epidemic dynamics modelEpidemic dynamics model

Basic reproduction number R0

Kermack-McKendrick SIR model

dI(t)dt

= q{ β

qS(t)− 1

}I(t)

Epidemic dynamics modelEpidemic dynamics model

Basic reproduction number R0

Kermack-McKendrick SIR model

dI(t)dt

= q{ β

qS(t)− 1

}I(t)

β

qS(t) > 1 ⇒ I(t) increases;

β

qS(t) < 1 ⇒ I(t) decreases.

Epidemic dynamics modelEpidemic dynamics model

Basic reproduction number R0

Kermack-McKendrick SIR model

dI(t)dt

= q{ β

qS(t)− 1

}I(t)

Effective reproduction number Rt for Kermack–McKendrick SIR model

Rt :=β

qS(t)

Epidemic dynamics modelEpidemic dynamics model

Basic reproduction number R0

Kermack-McKendrick SIR model

dI(t)dt

= q{ β

qS(t)− 1

}I(t)

Effective reproduction number Rt for Kermack–McKendrick SIR model

Rt :=β

qS(t) ≤ R0 :=

β

qN

Epidemic dynamics modelEpidemic dynamics model

William Ogilvy Kermack Anderson Gray McKendrick(26 April 1898 – 20 July 1970) (8 September 1876 – 30 May 1943)

References: Davidson, J. N. (1971) ”William Ogilvy Kermack. 1898-1970”. Biographical Memoirsof Fellows of the Royal Society, 17: 399–429. doi:10.1098/rsbm.1971.0015;http://www-history.mcs.st-and.ac.uk/Biographies/McKendrick.html

Epidemic dynamics modelEpidemic dynamics model

Kermack, W, McKendrick, A (1991) Contributions to the mathematicaltheory of epidemics – I. Bulletin of Mathematical Biology, 53(1-2): 33–55.doi:10.1007/BF02464423.Reprinted fromthe Proceedings of the Royal Society, Vol. 115A, pp. 700–721 (1927)

Kermack, W, McKendrick, A (1991) Contributions to the mathematicaltheory of epidemics – II. The problem of endemicity. Bulletin ofMathematical Biology, 53(1-2): 57–87. doi:10.1007/BF02464424.Reprinted fromthe Proceedings of the Royal Society, Vol. 138A, pp. 55–83 (1932)

Kermack, W, McKendrick, A (1991) Contributions to the mathematicaltheory of epidemics – III. Further studies of the problem of endemicity.Bulletin of Mathematical Biology, 53(1-2): 89–118.doi:10.1007/BF02464425.Reprinted fromProceedings of the Royal Society, Vol. 141A, pp. 94–122 (1933)

Epidemic dynamics modelEpidemic dynamics model

Epidemic dynamics modelEpidemic dynamics model

I

RS

time

10 20 30 40 50

0.2

0.4

0.6

0.8

1.0

I

S0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

q

β

Epidemic dynamics modelEpidemic dynamics model

H. Inaba, Age-Structured Population Dynamics in Demography andEpidemiology, Springer, Singapore, 2017.

齋藤保久・佐藤一憲・瀬野裕美, 数理生物学講義【展開編】 ー数理モデル解析の講究ー, 共立出版, 東京, 2017.

日本数理生物学会 (編), 『数』の数理生物学 (瀬野裕美 責任編集), シリーズ数理生物学要論, 第１巻, 共立出版, 東京, 2008.

稲葉寿, 感染症の数理モデル, 培風館, 東京, 2008.

稲葉寿, 数理人口学, 東京大学出版会, 東京, 2002.

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIRS model

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIRS model

Susceptible

S

Removed/Recovered

R

Infective

I

Latent/Incubation

E

Infective

I

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIRS model

dS(t)dt

= −βI(t)S(t) + ωR(t)

dI(t)dt

= βI(t)S(t)− qI(t)

dR(t)dt

= qI(t)− ωR(t)

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIRS model

dS(t)dt

= −βI(t)S(t) + ωR(t)

dI(t)dt

= βI(t)S(t)− qI(t)

dR(t)dt

= qI(t)− ωR(t)

S(t) + I(t) + R(t) = N

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIRS model R0 :=β

qN

dS(t)dt

= −βI(t)S(t) + ωR(t)

dI(t)dt

= βI(t)S(t)− qI(t)

dR(t)dt

= qI(t)− ωR(t)

S(t) + I(t) + R(t) = N

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIRS model R0 :=β

qN

dS(t)dt

= −βI(t)S(t) + ω{

N − S(t)− I(t)}

dI(t)dt

= βI(t)S(t)− qI(t)

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIRS model R0 :=β

qN

R0 ≤ 1 ⇒ (S(t), I(t)) →t→∞

(N, 0)

R0 > 1 ⇒ (S(t), I(t)) →t→∞

(qβ

, I∗)

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIRS model R0 :=β

qN

R0 ≤ 1 ⇒ (S(t), I(t)) →t→∞

(N, 0)

R0 > 1 ⇒ (S(t), I(t)) →t→∞

(qβ

, I∗)

I∗ =1 − 1/R0

1 + q/ωN

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIRS model R0 :=β

qN

R0 ≤ 1 ⇒ (S(t), I(t)) →t→∞

(N, 0) 【感染症不在平衡状態】disease-free equilibrium state

R0 > 1 ⇒ (S(t), I(t)) →t→∞

(qβ

, I∗)

I∗ =1 − 1/R0

1 + q/ωN

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIRS model R0 :=β

qN

R0 ≤ 1 ⇒ (S(t), I(t)) →t→∞

(N, 0) 【感染症不在平衡状態】disease-free equilibrium state

R0 > 1 ⇒ (S(t), I(t)) →t→∞

(qβ

, I∗) 【感染症定着平衡状態】endemic equilibrium state

I∗ =1 − 1/R0

1 + q/ωN

Epidemic dynamics modelEpidemic dynamics model

Kermack-McKendrick SIRS model

I

I

R

R

S

S

time

time

I

S

I

S

R0 = 2.5;

q = 17 ;

ω = 0.1, 0.5;

(S(0), I(0), R(0))= (0.99N, 0.01N, 0.0).

Epidemic dynamics modelEpidemic dynamics model

Force of infection

Epidemic dynamics modelEpidemic dynamics model

Force of infection

Kermack–McKendric model

Λ = βI

Epidemic dynamics modelEpidemic dynamics model

Force of infection

Kermack–McKendric model

Λ = βI = γ · IN

· cN

Epidemic dynamics modelEpidemic dynamics model

Force of infection

Kermack–McKendric model

Λ = βI = γ · IN

· cN

Frequency/Ratio-dependent force of infection

Λ = λIN

Epidemic dynamics modelEpidemic dynamics model

Force of infection

Kermack–McKendric model

Λ = βI = γ · IN

· cN

Frequency/Ratio-dependent force of infection

Λ = λIN

= γ · IN

· C

Epidemic dynamics modelEpidemic dynamics model

Force of infection

Vector-borne disease transmission (mass-action type)

Λ = βHV

Epidemic dynamics modelEpidemic dynamics model

Force of infection

Vector-borne disease transmission (mass-action type)

Λ = βHV = γ · VM

· cM

Epidemic dynamics modelEpidemic dynamics model

Force of infection

Vector-borne disease transmission (mass-action type)

Λ = βHV

Susceptible

S

Removed/Recovered

RInfected

I

V

U

Non-carrier vector

Carrier vector

Epidemic dynamics modelEpidemic dynamics model

A model for the vector-borne disease transmission

dS(t)dt

= −βHV(t)S(t) + ωR(t)

dI(t)dt

= βHV(t)S(t)− qI(t)

dR(t)dt

= qI(t)− ωR(t)

dU(t)dt

= Q − βMI(t)U(t)− δU(t)

dV(t)dt

= βMI(t)U(t)− δV(t)

Epidemic dynamics modelEpidemic dynamics model

A model for the vector-borne disease transmission

dS(t)dt

= −βHV(t)S(t) + ω{

N − S(t)− I(t)}

dI(t)dt

= βHV(t)S(t)− qI(t)

dV(t)dt

= βMI(t){ Q

δ− V(t)

}− δV(t)

Epidemic dynamics modelEpidemic dynamics model

A model for the vector-borne disease transmission

dS(t)dt

= −βHV(t)S(t) + ω{

N − S(t)− I(t)}

dI(t)dt

= βHV(t)S(t)− qI(t)

dV(t)dt

= βMI(t){ Q

δ− V(t)

}− δV(t)

Basic reproduction number

R0 :=(

βHN · 1δ

)︸ ︷︷ ︸human infection bya carrier vector

×(

βMQδ

· 1q

)︸ ︷︷ ︸production of carriervectors by an infective

Epilogue:

Various factors on the epidemic dynamics

Epilogue:

Various factors on the epidemic dynamics

Various factors on the epidemic dynamicsVarious factors on the epidemic dynamics

Route of disease transmission;

Condition of public health;

Condition of medical treatment;

Cultural/social custom in the daily life;

Social response under the cultural/political/economicbackground.

Various factors on the epidemic dynamicsVarious factors on the epidemic dynamics

Route of disease transmission;

Condition of public health;

Condition of medical treatment;

Cultural/social custom in the daily life;

Social response under the cultural/political/economicbackground.

Various factors on the epidemic dynamicsVarious factors on the epidemic dynamics

Route of disease transmission;

Condition of public health;

Condition of medical treatment;

Cultural/social custom in the daily life;

Social response under the cultural/political/economicbackground.

Various factors on the epidemic dynamicsVarious factors on the epidemic dynamics

Route of disease transmission;

Condition of public health;

Condition of medical treatment;

Cultural/social custom in the daily life;

Social response under the cultural/political/economicbackground.

Various factors on the epidemic dynamicsVarious factors on the epidemic dynamics

Route of disease transmission;

Condition of public health;

Condition of medical treatment;

Cultural/social custom in the daily life;

Social response under the cultural/political/economicbackground.

Various factors on the epidemic dynamicsVarious factors on the epidemic dynamics

Route of disease transmission;

Condition of public health;

Condition of medical treatment;

Cultural/social custom in the daily life;

Social response under the cultural/political/economicbackground.

H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology,Springer, Singapore, 2017.

齋藤保久・佐藤一憲・瀬野裕美, 数理生物学講義【展開編】 ー数理モデル解析の講究ー, 共立出版, 東京, 2017.

瀬野裕美, 数理生物学講義【基礎編】 ー数理モデル解析の初歩ー, 共立出版, 東京, 2016.

L.J.S. Allen, 生物数学入門 ― 差分方程式・微分方程式の基礎からのアプローチ ―, 共立出版, 東京, 2011. (竹内・佐藤・守田・宮崎 監訳)

関村利朗・山村則男（編）, 理論生物学の基礎, 海游舎, 東京, 2012.

日本数理生物学会 (編), 『数』の数理生物学 (瀬野裕美 責任編集), シリーズ数理生物学要論,第１巻, 共立出版, 東京, 2008.

稲葉寿, 感染症の数理モデル, 培風館, 東京, 2008.

瀬野裕美, 数理生物学 ― 個体群動態の数理モデリング入門 ―, 共立出版, 東京, 2007.

稲葉寿, 数理人口学, 東京大学出版会, 東京, 2002.