Timely pandemic countermeasures reduce both health damage and economic loss:
Generality of the exact solution
27 February 2023
@ OIST
ISCO2023@OISTJ. Phys. Soc. Jpn., in press (2023)

Mathematics

Physics

Mathematical Models of Infectious Diseases

Thermodynamics?

Medicine: Public healthTransmission route

Optimal pandemic control
Airborne transmission
Tsuyoshi HONDOU

Economics

Graduate School of Science
Tohoku University(Medical economics)

Pandemics
by SARS-CoV-2

aerosol nature

Question mark over policy“Countermeasures at the lastminute” (2020〜)

Daily cases in Tokyo
東京の新規感染者（6/21〜8/13）⼈数7000

6000

5000

4000

3000

2000

“To save economics, no serious measure
until Stage 4 (explosively increasing)”

00

It’s still Japan’s policy today, while this question may be intuitively obvious.

1020

304050

6月20日を起点とする経過日数
Daysafter June 20, 2021

Is it economical?

Daily cases in Tokyo

LOG

Stage 1: infection is very rare
Stage 2: infection is increasing slowly neglect
Stage 3: infection is increasing fastStage 4: infection is explosively increasing

Explosively?
“急拡⼤“？

1000

Exponential increase
Days after June 20, 2021

60

Pioneering & suggestive
Control engineer

Originally mathematician!

Abstract
The British government has been debating how to escape fromthe lockdown without provoking a resurgence of the COVID-19
disease. There is a growing recognition of the damage thelockdown has caused to economic and social life. This paper
presents a simple cost‒benefit analysis inspired by optimal
control theory and incorporating the SIR model of disease
propagation. It also reports simulations informed by thetheoretical discussion. The optimal path for government
intervention is computed under a variety of conditions. Theseinclude a cap on the permitted level of infection to avoid overloadof the health system, and the introduction of a test and tracesystem. We quantify the benefits of early intervention to control
the disease. We also examine how the governmentʼs valuation oflife influences the optimal path. A 10-week lockdown is only
optimal if the value of life for COVID-19 victims exceeds £10m.
The study is based on a standard but simple epidemiologicalmodel, and should therefore be regarded as presenting amethodological framework rather than giving policy prescriptions.

Optimal pandemic control
Timing and strength of countermeasure
Numerical approach

Economists

many papers

Lacks predictability:
Socio-economic parameters: divers

For predictable knowledge

Exact solution

Is indispensable
Not only for COVID
But also for future pandemics

Inspiration from Rowthorn & Maciejowski
Effective reproduction number Rt • Keeping the infected population• SIR models• U.K. socio-economic condition• How to regulate after lockdown• Cost-benefit analysis

constant: or equivalently Rt〜1
• Moderate intervention beadopted soon after lockdown

Analogy to “quasi-static process”
in thermodynamics.
orisentropic process

lockdown
Weeks after lockdownTypical isentropic process

Carnot’s cycle

Reversible process
ΔStotal=0
Irrevessible process
ΔStotal>0

Figure from Wikipedia

S: entropyath A2
P
Near Crisis

I (T)

I (T1)PathA
1

Log(Infected population)

To answer the previous question

Grasoshepprtn
AtaPI0

BhT1

Which protocol is better,
Path A (grasshopper) andPath B (Ant)?
”Last minute“ or “early andmoderate”

T20

1. Compare two processes
analytically
2. Then, generalize it

Wave period: assumed to be Finite
As spring (fall) comes, it naturally relaxes.

T=T1+ T2tairborne natureSimple assumptions for theoryØDynamics:exponential expansion

Rt: Effective reproduction number

infected population I(t)Infected population
Δ(t)>0: Increase
Δ(t)<0: Decrease

Derived from mathematical models of infection diseases
under small & moderate infection rate (except near herd immunity) see next slideØCost-benefit analysis: conventional in COVID economics
Total Cost: Intervention cost C(Δ) + Medical cost M(I(t))
1.
2.

Intervention cost (C(Δ)) : economic damage by countermeasure later slide
Medical cost (M): increasing function of infected population(I)Validity of assumption 1: Exponential dynamics
Except near (temporal)
herd immunity,
exponential dynamics isvalid
Numerical simulation
just for demonstration

S(t)≒S(0) unless “I(t) is order of 1”
S(t): susceptible persons at time tRef.
••

Vynnycky and White: An Introduction ofInfectious Disease Modeling (Oxford Univ.
Press, 2010)
Hondou, J.Phys.Soc.Jpn 2021

Cost Benefit Analysis
Intervention cost C(Rt): Convex downward• Total cost ＝Intervention cost（C)＋Medical cost（M)• Assume: Countermeasures: in order of cost-effectiveness⇩• Intervention cost of unit timeC(RN) = 0 if no countermeasure(RN: natural effective reproduction number)function of the effective reproduction number
C(Rt)C(Rt): convex downward“diminishing utility” in economics

Just example

countermeasure

2A
hta
P

I (T)

I (T1)*+ δ1-δ 2*
Δ=ΔR
R =ΔPathA
1ΔLog(Infected population)

Now: compare 2 processes

Bht Δ*aP R=ΔI0(Δ*+δ1)T1+(Δ*−δ2)T2=Δ*T

T20

T1

T=T1+ T2t

(ΔR= Rt − 1)2Path AI (T)

I (T1)
1PathA1

Δ *- δ 2

Δ*+δ

(ΔR= Rt − 1)

Log(Infected population)

Intervention cost, C(Δ)
for paths A & path B

I0thaPB

Δ*T20

T1

T=T1+ T2t

With the equality

Since C(Δ) is convex downward
C(Δ）*C(Δ-δ₂）*C(Δ）*C(Δ+δ₁）
０*０ Δ-δ₂C(Δ*−δ2) > C(Δ*)+Kδ2
Convexdownward

C(Δ*+δ1) > C(Δ*)−Kδ1
Note for mathematicians*Δ* Δ+δ₁

Δ

Differentiability of C(Δ）is notnecessary. Continuity is sufficient.
See https://arxiv.org/abs/2209.12805
for general proof.

Inequality holds for intervention costs

In “last minute” policy, intervention cost is larger than inmoderate & early measure.

⬇

Medical cost: Mpath A (t) ≧ Mpath B(t)Total Cost:

2
Path AI (T)

I (T1)PathA
1

I(t)：path A ≧ path B

Log(Infected population)

Medical & Total Cost

I0B

T20Total cost: Path A ≧ Path Bfor any δ1, δ2

thaPT1T=T1+ T2t

Inequalities are also valid for curves
Valid for any integrable Rt(t) thatLog(Infected population)

Ipath A(t) ≧Ipath B(t) and Ipath A(T) =Ipath B(T)

I(t)

rett
beAs we can decompose any path into pairsof small straight segments, A1i and A2i
corresponding to a segment Bi of path B,
the inequalities hold for a curved line.

0Tt

Conclusion & Discussion 1/2
Early and moderate intervention is better than last minute policy for either cost C, M and thus total cost.

Theorem (inequality for pandemic measure)

Even for curved line
The same inequalities hold for Intervention cost C
or Medical cost M alone.

for Ipath A(t) ≧ Ipath B(t)• This includes the previous result (J. Phys. Soc. Jpn. 2021) as a special
solution to the cyclic condition: I(0) = I(T)• Excess infected population in Path A compared to path B(t)⇒ irreversible costs “economic irreversibility”
Similar to thermodynamics (entropy production)

Conclusion & Discussion 2/2• Robustness comes from simple & natural assumptions:• Medical cost：(can be non-linear)• Exponential dynamics for Infected population as to Δt (=Rt-1）• Intervention cost C: convex downward as to Δ=Rt-1
Differentiability: not necessary, continuity is sufficient• Early and moderate measure: better for both Life and Economy• Dogma: Life vs Economy⇐ often unproductive• Applicable to future pandemic, irrespective of herd immunity• Core knowledge for complex real-world systems: Role of exact solutions

(a) lozenge
I (T)

(b)

alci
ed tsr M osocfh ictbo nomrtte Ecoe
B danI (T)path CI00

T1hC

P

1I0
T2 T=T1+ T2Total cost: Path C < path A（benefit for early measure）

tal0
Bottom right:cdie

Mnee stw steb cono ic
it ti ompe onm EcoC dan

pathatB

212
path A

pathA

Log(Infected population)

More general perspectiveTt

This theorem includes the previous result (2021) as a
special solution to the cyclic condition: I(0) = I(T)