ｴｶﾀﾉﾅﾁﾀﾀ･､･ﾊ･ﾟ･ｯ･ｹ､ﾎｿ･...

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

[email protected]

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

[email protected]

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


S → E → I → R → S

Susceptible

S

Removed/Recovered

R

Infective

I

Latent/Incubation

E

Infectious


S → I → R → S

Susceptible

S

Removed/Recovered

R

Infective

I

Latent/Incubation

E

Infective

I


S → E → I → R → S

Susceptible

S

Removed/Recovered

R

Infective

I

Latent/Incubation

E


S → E → I → R

Susceptible

S

Removed/Recovered

R

Infective

I

Latent/Incubation

E


S → I → R

Susceptible

S

Removed/Recovered

R

Infective

I

Latent/Incubation

E

Infective

I

Epidemic dynamics modelEpidemic dynamics model


Generic SIR model


Generic SIR model

Susceptible

S

Removed/Recovered

R

Infective

I

Latent/Incubation

E

Infective

I


Generic SIR model

dS(t)dt

= B − ΛS(t)− µSS(t)


Generic SIR model

dS(t)dt

= B − ΛS(t)− µSS(t)

Λ = Λ(S, I, R): Force of infection


Generic SIR model

dS(t)dt

= B − ΛS(t) − µSS(t)

dI(t)dt

= ΛS(t)− qI(t)− µII(t)


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)


Generic SIR model

dS(t)dt

= − ΛS(t)

dI(t)dt

= ΛS(t)− qI(t)

dR(t)dt

= qI(t)


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


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)


Kermack-McKendrick SIR model



Λ ∝ I



Λ ∝ I

dS(t)dt

= −βI(t)S(t)

dI(t)dt

= βI(t)S(t)− qI(t)

dR(t)dt

= qI(t)



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



dS(t)dt

= −βI(t)S(t)

dI(t)dt


dR(t)dt

= qI(t)



dS(t)dt

= −βI(t)S(t)

dI(t)dt




dS(t)dt

= −βI(t)S(t)

dI(t)dt


dS(t)dt

+dI(t)

dt− q

β

1S(t)

dS(t)dt

= 0



dS(t)dt

= −βI(t)S(t)

dI(t)dt


ddt

{S(t) + I(t)− q

βlog S(t)

}= 0



dS(t)dt

= −βI(t)S(t)

dI(t)dt


S(t) + I(t)− qβ

log S(t) = const.



dS(t)dt

= −βI(t)S(t)

dI(t)dt


Conserved quantity of Kermack–McKendrick SIR model

S(t) + I(t)− qβ

log S(t) = S(0) + I(0)− qβ

log S(0)




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




S(t) + I(t)− qβ

log S(t) = S(0) + I(0)− qβ

log S(0)

I

S0 q

β




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



I

S0 q

β

Final size equation for Kermack–McKendrick SIR model

S∞ − qβ

log S∞ = S(0) + I(0)− qβ

log S(0)



I

S0 q

β



S(0) >qβ

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

S(0) ≤ qβ

⇒ I(t) monotonically decreases.



β

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

of disease invasion;

β

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



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)



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


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.








The number of infectives increases. ⇒ R0 > 1;

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




dS(t)dt

= −βI(t)S(t)

dI(t)dt





dI(t)dt





dI(t)dt

= q{ β

qS(t)− 1

}I(t)




dI(t)dt

= q{ β

qS(t)− 1

}I(t)

β

qS(t) > 1 ⇒ I(t) increases;

β

qS(t) < 1 ⇒ I(t) decreases.




dI(t)dt

= q{ β

qS(t)− 1

}I(t)

Effective reproduction number Rt for Kermack–McKendrick SIR model

Rt :=β

qS(t)




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


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


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)


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


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.


Kermack-McKendrick SIRS model



Susceptible

S

Removed/Recovered

R

Infective

I

Latent/Incubation

E

Infective

I



dS(t)dt

= −βI(t)S(t) + ωR(t)

dI(t)dt


dR(t)dt

= qI(t)− ωR(t)



dS(t)dt


dI(t)dt


dR(t)dt

= qI(t)− ωR(t)

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


Kermack-McKendrick SIRS model R0 :=β

qN

dS(t)dt


dI(t)dt


dR(t)dt

= qI(t)− ωR(t)

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



qN

dS(t)dt

= −βI(t)S(t) + ω{

N − S(t)− I(t)}

dI(t)dt




qN

R0 ≤ 1 ⇒ (S(t), I(t)) →t→∞

(N, 0)

R0 > 1 ⇒ (S(t), I(t)) →t→∞

(qβ

, I∗)



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



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



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



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).


Force of infection


Force of infection

Kermack–McKendric model

Λ = βI


Force of infection


Λ = βI = γ · IN

· cN


Force of infection


Λ = βI = γ · IN

· cN

Frequency/Ratio-dependent force of infection

Λ = λIN


Force of infection


Λ = βI = γ · IN

· cN

Frequency/Ratio-dependent force of infection

Λ = λIN

= γ · IN

· C


Force of infection

Vector-borne disease transmission (mass-action type)

Λ = βHV


Force of infection


Λ = βHV = γ · VM

· cM


Force of infection


Λ = βHV

Susceptible

S

Removed/Recovered

RInfected

I

V

U

Non-carrier vector

Carrier vector


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)



dS(t)dt

= −βHV(t)S(t) + ω{

N − S(t)− I(t)}

dI(t)dt


dV(t)dt

= βMI(t){ Q

δ− V(t)

}− δV(t)



dS(t)dt

= −βHV(t)S(t) + ω{

N − S(t)− I(t)}

dI(t)dt


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.

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.

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