Discussion Paper Series
2020-01
Infectious Disease, Dynamic and Static Externalities
in Two-Location Model:
Theoretical and Simulation Analyses
Yonemoto, Kiyoshi
January 5, 2021

Discussion Papers can be downloaded:
http://www1.tcue.ac.jp/home1/c-gakkai/dp/dp20-01

Infectious Disease, Dynamic and StaticExternalities in Two-Location Model:
Theoretical and Simulation Analyses

Kiyoshi Yonemoto

Keywords
COVID-19, infectious disease, agglomeration economy, optimal control

Address for correspondence: Kiyoshi Yonemoto,Department of Regional Policy, Takasaki City University of Economics,
1300 Kaminamie-machi, Takasaki, Gunma 370-0801 Japan.
E-mail: yonemoto@tcue.ac.jp
Fax: 81-27-343-4830
1

1. Introduction
This study theoretically characterizes the optimal control path of a two-location
economy in which production/consumption has a positive externality while infectious
disease, such as COVID-19, has a negative (dynamic) externality.
The model is based on Bobashev et al. (2011), Bethune and Korinek (2020), Alvarez
et al. (2020), and Eichenbaum et al. (2020), Akamatsu, et al. (2020) and Olivares et al.
(2020), which follow the seminal works of Kremack and McKendrick (1927) andAnderson and May (1991), but taking into account the fact that COVID-19 has beenmostly transmitted between regions not by migrants but by travelers or commuters, it
assumes the residents do not migrate. Instead, it assumes the disease is transmitted
through the economic activities of the residents over the regions as well as within eachregion.

2. The Model
The basic model is described as follows. Consider an economy with two locations: 𝑖 =1, 2 (e.g., the center and the suburb in a city.) The population of region 𝑖 is denoted by𝑁𝑖 and assumed to be constant over time. The number of susceptible, infected, andrecovered residents in location 𝑖 in period 𝑡 are denoted by 𝑆𝑖𝑡 , 𝐼𝑖𝑡 , and 𝑅𝑖𝑡 ,2respectively. Their changes are affected by the domestic activity levels of the residents
of type 𝑍 = 𝑆, 𝐼, 𝑅, those of the residents who visit the other location (and back home,)
and those of the residents who are visiting from the other region, represented by

𝛽0𝑖𝑖𝑡 ,

𝛽1𝑖𝑗𝑡 , and 𝛽2𝑗𝑖𝑡 :̇ = −(𝛽0𝑖𝑖𝑡 𝐼𝑖𝑡 𝑆𝑖𝑡 + 𝛽1𝑖𝑗𝑡 𝐼𝑗𝑡 𝑆𝑖𝑡 + 𝛽2𝑗𝑖𝑡 𝐼𝑗𝑡 𝑆𝑖𝑡 )𝑆𝑖𝑡𝐼𝑖𝑡̇ = 𝛽0𝑖𝑖𝑡 𝐼𝑖𝑡 𝑆𝑖𝑡 + 𝛽1𝑖𝑗𝑡 𝐼𝑗𝑡 𝑆𝑖𝑡 + 𝛽2𝑗𝑖𝑡 𝐼𝑗𝑡 𝑆𝑖𝑡 − 𝛾𝑖 𝐼𝑖𝑡 ,(1)𝑅̇𝑖𝑡 = 𝛾𝑖 𝐼𝑖𝑡
where 𝛾𝑖 is the rate of recovery. Also, assume that each activity level is a linear functionof the consumption of the corresponding good.
𝑆 𝐼
𝑆 𝐼
𝑆 𝐼
𝛽0𝑖𝑖𝑡 = 𝛽0 𝑥𝑖𝑖𝑡
𝑥𝑖𝑖𝑡 , 𝛽1𝑖𝑗𝑡 = 𝛽1 𝑥𝑖𝑗𝑡
𝑥𝑗𝑗𝑡 , 𝛽2𝑗𝑖𝑡 = 𝛽2 𝑥𝑖𝑖𝑡
𝑥𝑗𝑖𝑡 ,(2)
𝑍𝑥∙∙𝑡
≤ 1,

𝑍 = 𝑆, 𝐼, 𝑅.

The utility is characterized by:𝑍𝑢𝑖𝑡={𝑍𝑍𝑢(𝑥𝑖𝑖𝑡, 𝑥𝑖𝑗𝑡; 𝑋𝑖𝑡 , 𝑋𝑗𝑡 ) for 𝑍 = 𝑆, 𝑅,𝑍𝑍𝑢(𝑥𝑖𝑖𝑡, 𝑥𝑖𝑗𝑡; 𝑋𝑖𝑡 , 𝑋𝑗𝑡 ) − 𝑐 for 𝑍 = 𝐼,

(3)

where production/consumption externality 𝑋𝑖𝑡 is characterized by,𝑍𝑍𝑋𝑖𝑡 ≡ ∑(𝜙𝑐 + 𝜙0 𝑍𝑖𝑡 𝑥𝑖𝑖𝑡+ 𝜙2 𝑍𝑗𝑡 𝑥𝑗𝑖𝑡) , 𝑍 = 𝑆, 𝐼, 𝑅.

(4)𝑍

Specify 𝑢 as follows:𝜎𝜎𝑍
𝑍
𝑍
𝑍𝑢(𝑥𝑖𝑖𝑡, 𝑥𝑖𝑗𝑡; 𝑋𝑖𝑡 , 𝑋𝑗𝑡 ) = 𝑋𝑖𝑡 (𝑥𝑖𝑖𝑡) + 𝜃𝑋𝑗𝑡 (𝑥𝑖𝑗𝑡) ,For the command optimum, the maximand is,3

(5)

𝑇𝑆𝑆𝐼𝐼𝑉 = ∫ 𝑒 −𝜌𝑡 ∑{𝑆𝑖𝑡 𝑢(𝑥𝑖𝑖𝑡, 𝑥𝑖𝑗𝑡; 𝑋𝑖𝑡 , 𝑋𝑗𝑡 ) + 𝐼𝑖𝑡 [𝑢(𝑥𝑖𝑖𝑡, 𝑥𝑖𝑗𝑡; 𝑋𝑖𝑡 , 𝑋𝑗𝑡 ) − 𝑐]0𝑖(6)𝑅
𝑅+𝑅𝑖𝑡 𝑢(𝑥𝑖𝑖𝑡, 𝑥𝑖𝑗𝑡; 𝑋𝑖𝑡 , 𝑋𝑗𝑡 )}𝑑𝑡 − Φ(𝐼1𝑇 , 𝐼2𝑇 ),𝑗 ≠ 𝑖.

The Hamiltonian is:𝑆𝑆
𝐻 = 𝑒 −𝜌𝑡 {(𝑁1 − 𝐼1𝑡 − 𝑅1𝑡 )𝑢(𝑥11𝑡, 𝑥12𝑡; 𝑋1𝑡 , 𝑋2𝑡 )𝐼
𝐼𝑅
𝑅+𝐼1𝑡 [𝑢(𝑥11𝑡, 𝑥12𝑡; 𝑋1𝑡 , 𝑋2𝑡 ) − 𝑐] + 𝑅1𝑡 𝑢(𝑥11𝑡, 𝑥12𝑡; 𝑋1𝑡 , 𝑋2𝑡 )𝑆
𝑆+(𝑁2 − 𝐼2𝑡 − 𝑅2𝑡 )𝑢(𝑥22𝑡, 𝑥21𝑡; 𝑋2𝑡 , 𝑋1𝑡 )𝐼𝐼𝑅
𝑅+𝐼2𝑡 [𝑢(𝑥22𝑡, 𝑥21𝑡; 𝑋2𝑡 , 𝑋1𝑡 ) − 𝑐] + 𝑅2𝑡 𝑢(𝑥22𝑡, 𝑥21𝑡; 𝑋2𝑡 , 𝑋1𝑡 )}(7)

𝐼 {(𝑁+λ1𝑡
1 − 𝐼1𝑡 − 𝑅1𝑡 )[𝛽0 11𝑡 𝐼1𝑡 + (𝛽1 12𝑡 + 𝛽2 21𝑡 )𝐼2𝑡 ] − 𝛾1 𝐼1𝑡 }+λ𝐼2𝑡 {(𝑁2 − 𝐼2𝑡 − 𝑅2𝑡 )[𝛽0 22𝑡 𝐼2𝑡 + (𝛽1 21𝑡 + 𝛽2 12𝑡 )𝐼1𝑡 ] − 𝛾2 𝐼2𝑡 }𝑅+λ1𝑡
𝛾1 𝐼1𝑡 + λ𝑅2𝑡 𝛾2 𝐼2𝑡 .

3. Characterizing the Externalities
3.1 Optimality Conditions
For type S, the optimality conditions are:𝜕𝐻
𝑁𝑖 − 𝐼𝑖𝑡 − 𝑅𝑖𝑡𝜎𝑆 𝜎−𝜌𝑡𝜙0(𝑁𝑖 − 𝐼𝑖𝑡 − 𝑅𝑖𝑡 ) (+ 𝑆 ) 𝑋𝑖𝑡 (𝑥𝑖𝑖𝑡)
𝑆 =𝑒𝑋𝑖𝑡
𝜕𝑥𝑖𝑖𝑡𝑥𝑖𝑖𝑡
𝐼𝐼+λ𝐼𝑖𝑡 (𝛽0 𝑥𝑖𝑖𝑡
𝐼𝑖𝑡 + 𝛽2 𝑥𝑗𝑖𝑡𝐼𝑗𝑡 )(𝑁𝑖 − 𝐼𝑖𝑡 − 𝑅𝑖𝑡 )𝐼𝐼+λ𝑗𝑡𝛽1 𝑥𝑗𝑖𝑡𝐼𝑖𝑡 (𝑁𝑗 − 𝐼𝑗𝑡 − 𝑅𝑗𝑡 ) = 0, 𝑖 = 1, 2,4(8)𝜕𝐻𝑁𝑖 − 𝐼𝑖𝑡 − 𝑅𝑖𝑡𝜎
𝑆 𝜎−𝜌𝑡[(𝑁𝑖 − 𝐼𝑖𝑡 − 𝑅𝑖𝑡 )𝜃 (𝜙2+ 𝜙0 𝑆 ) 𝑋𝑗𝑡 (𝑥𝑖𝑗𝑡)
𝑆 =𝑒𝑋𝑗𝑡
𝜕𝑥𝑖𝑗𝑡
𝑥𝑖𝑗𝑡
𝜎

𝑆+ 𝜙2 (𝑁𝑗 − 𝐼𝑗𝑡 − 𝑅𝑗𝑡 )(𝑁𝑖 − 𝐼𝑖𝑡 − 𝑅𝑖𝑡 )(𝑥𝑗𝑗𝑡) ]𝐼𝐼
𝐼+λ𝐼𝑖𝑡 𝛽1 𝑥𝑗𝑗𝑡
𝐼𝑗𝑡 (𝑁𝑖 − 𝐼𝑖𝑡 − 𝑅𝑖𝑡 ) + λ𝑗𝑡
𝛽2 𝑥𝑗𝑗𝑡
𝐼𝑖𝑡 (𝑁𝑗 − 𝐼𝑗𝑡 − 𝑅𝑗𝑡 ) = 0,𝑖 = 1, 2, 𝑖 ≠ 𝑗.
Also, the ones for the co-state variables are,𝑑λ𝐼𝑖𝑡
𝜕𝐻=−𝑑𝑡𝜕𝐼𝑖𝑡
𝑆 𝐼 (𝑁𝑆 𝐼𝑆 𝐼= −λ𝐼𝑖𝑡 [𝛽0 𝑥𝑖𝑖𝑡
𝑥𝑖𝑖𝑡 𝑖 − 2𝐼𝑖𝑡 − 𝑅𝑖𝑡 ) − (𝛽1 𝑥𝑖𝑗𝑡
𝑥𝑗𝑗𝑡 + 𝛽2 𝑥𝑖𝑖𝑡𝑥𝑗𝑖𝑡 )𝐼𝑗𝑡 − 𝛾𝑖 ]

(9)
𝐼𝑆 𝐼
𝑆 𝐼−λ𝑗𝑡(𝛽1 𝑥𝑗𝑖𝑡
𝑥𝑖𝑖𝑡 + 𝛽2 𝑥𝑗𝑗𝑡
𝑥𝑖𝑗𝑡 )(𝑁𝑗 − 𝐼𝑗𝑡 − 𝑅𝑗𝑡 ) + λ𝑅𝑖𝑡 𝛾𝑖 + 𝑒 −𝜌𝑡 𝑐,𝑑λ𝑅𝑖𝑡𝜕𝐻=−= 0.
𝑑𝑡
𝜕𝑅𝑖𝑡
Transversality conditions are:
λ𝐼𝑖𝑇 =𝜕Φ(𝐼1𝑇 , 𝐼2𝑇 ), λ𝑅𝑖𝑇 = 0,
𝜕𝐼𝑖𝑇

(10)

𝑑λ𝑅𝑖𝑡
𝜕𝐻=−= 0.
𝑑𝑡
𝜕𝑅𝑖𝑡

Note that, the externality of infectious disease is dynamic, represented by λ𝐼𝑖𝑡 andthey changes while the production/consumption externalities are essentially static andaffect the economy through equations (3).

5Uniform Control
In the simple case where the authority can only control the activities of the residents
irrespective of their types, (2) is summarized by,
𝛽0𝑖𝑖𝑡 = 𝛽0 (𝑥𝑖𝑖𝑡 )2, 𝛽1𝑖𝑗𝑡 = 𝛽1 𝑥𝑖𝑗𝑡 𝑥𝑗𝑗𝑡 , 𝛽2𝑗𝑖𝑡 = 𝛽2 𝑥𝑖𝑖𝑡 𝑥𝑗𝑖𝑡(2’)

And (4) is rewritten as,
𝑋𝑖𝑡 = 𝜙𝑐 + 𝜙0 𝑁𝑖 𝑥𝑖𝑖𝑡 + 𝜙2 𝑁𝑗 𝑥𝑗𝑖𝑡

(4’)

The maximand is,𝑇𝑉 = ∫ 𝑒 −𝜌𝑡 {𝑁1 [𝑋1𝑡 (𝑥11𝑡 )𝜎 + 𝜃𝑋2𝑡 (𝑥12𝑡 )𝜎 ] − 𝑐𝐼1𝑡0(6’)+ 𝑁2 [𝑋2𝑡 (𝑥22𝑡)𝜎+ 𝜃𝑋1𝑡 (𝑥21𝑡)𝜎 ]− 𝑐𝐼2𝑡 }𝑑𝑡 − Φ(𝐼1𝑇 , 𝐼2𝑇 )(7) is rewritten to be:
𝐻 = 𝑒 −𝜌𝑡 {𝑁1[𝑋1𝑡 (𝑥11𝑡 )𝜎 + 𝜃𝑋2𝑡 (𝑥12𝑡 )𝜎 ] − 𝑐𝐼1𝑡+ 𝑁2 [𝑋2𝑡 (𝑥22𝑡 )𝜎 + 𝜃𝑋1𝑡 (𝑥21𝑡 )𝜎 ] − 𝑐𝐼2𝑡 }𝐼 {[𝛽 (𝑥2+λ1𝑡0 11𝑡 ) 𝐼1𝑡 + (𝛽1 𝑥12𝑡 𝑥22𝑡 + 𝛽2 𝑥11𝑡 𝑥21𝑡 )𝐼2𝑡 ](𝑁1 − 𝐼1𝑡 − 𝑅1𝑡 )− 𝛾1 𝐼1𝑡 }(7’)+λ𝐼2𝑡 {[𝛽0 (𝑥22𝑡 )2 𝐼2𝑡 + (𝛽1 𝑥21𝑡 𝑥11𝑡 + 𝛽2 𝑥22𝑡 𝑥12𝑡 )𝐼1𝑡 ](𝑁2 − 𝐼2𝑡 − 𝑅2𝑡 )− 𝛾2 𝐼2𝑡 }𝑅+λ1𝑡
𝛾1 𝐼1𝑡 + λ𝑅2𝑡 𝛾2 𝐼2𝑡

The optimality conditions are:6𝜕𝐻
𝜙0 𝑁𝑖𝜎= 𝑒 −𝜌𝑡 𝑁𝑖 (+) 𝑋 (𝑥 )𝜎𝜕𝑥𝑖𝑖𝑡
𝑋𝑖𝑡𝑥𝑖𝑖𝑡 𝑖𝑡 𝑖𝑖𝑡+ λ𝐼𝑖𝑡 (2𝛽0 𝑥𝑖𝑖𝑡 𝐼𝑖𝑡 + 𝛽2 𝑥𝑗𝑖𝑡 𝐼𝑗𝑡 )(𝑁𝑖 − 𝐼𝑖𝑡 − 𝑅𝑖𝑡 )𝐼+λ𝑗𝑡
𝛽1 𝑥𝑗𝑖𝑡 𝐼𝑖𝑡 (𝑁𝑗 − 𝐼𝑗𝑡 − 𝑅𝑗𝑡 ) = 0, 𝑖 = 1, 2,(8’)

𝜕𝐻
𝜙2 𝑁𝑖𝜎𝜎𝜎= 𝑒 −𝜌𝑡 [𝑁𝑖 𝜃 (+) 𝑋 (𝑥 ) + 𝜙𝑁𝑗 𝑁𝑖 (𝑥𝑗𝑗𝑡 ) ]𝜕𝑥𝑖𝑗𝑡𝑋𝑗𝑡𝑥𝑖𝑗𝑡 𝑗𝑡 𝑖𝑗𝑡
𝐼+λ𝐼𝑖𝑡 𝛽1 𝑥𝑗𝑗𝑡 𝐼𝑗𝑡 (𝑁𝑖 − 𝐼𝑖𝑡 − 𝑅𝑖𝑡 ) + λ𝑗𝑡
𝛽2 𝑥𝑗𝑗𝑡 𝐼𝑖𝑡 (𝑁𝑗 − 𝐼𝑗𝑡 − 𝑅𝑗𝑡 ) = 0,𝑖 = 1, 2, 𝑖 ≠ 𝑗.
Also, for the co-state variables,𝑑λ𝐼𝑖𝑡
𝜕𝐻=−𝑑𝑡𝜕𝐼𝑖𝑡= −λ𝐼𝑖𝑡 [𝛽0 (𝑥𝑖𝑖𝑡 )2 (𝑁𝑖 − 2𝐼𝑖𝑡 − 𝑅𝑖𝑡 ) − (𝛽1 𝑥𝑖𝑗𝑡 𝑥𝑗𝑗𝑡 + 𝛽2 𝑥𝑖𝑖𝑡 𝑥𝑗𝑖𝑡 )𝐼𝑗𝑡 − 𝛾𝑖 ]
(9’)
𝐼−λ𝑗𝑡
(𝛽1 𝑥𝑗𝑖𝑡 𝑥𝑖𝑖𝑡 + 𝛽2 𝑥𝑗𝑗𝑡 𝑥𝑖𝑗𝑡 )(𝑁𝑗 − 𝐼𝑗𝑡 − 𝑅𝑗𝑡 ) + λ𝑅𝑖𝑡 𝛾𝑖 + 𝑒 −𝜌𝑡 𝑐,𝑑λ𝑅𝑖𝑡𝜕𝐻=−= 0.
𝑑𝑡
𝜕𝑅𝑖𝑡

3.2 Comparing to the Static Cases
Now, consider static and/or decentralized cases to compare to the conditions derived
in the previous subsection. First, characterize the nature of externality using a staticmodel. Suppose, for a susceptible, the probability getting infected depends simply on
his/her behavior:𝑃̃𝑖𝑆 = 𝜂̃0 𝑥𝑖𝑖𝑆 + 𝜂̃1 𝑥𝑖𝑗𝑆 + 𝜂̃2 𝑥𝑖𝑖𝑆 ,(11)7

A self-interested resident maximizes the expected utility:
𝜎𝜎𝜎𝜎𝑆𝐸𝑈𝑖𝑆 = (1 − 𝑃̃𝑖𝑆 )[𝑋̅𝑖 (𝑥𝑖𝑖𝑆 ) + 𝜃𝑋̅𝑗 (𝑥𝑖𝑗) ] + 𝑃̃𝑖𝑆 [𝑋̅𝑖 (𝑥𝑖𝑖𝑆 ) + 𝜃𝑋̅𝑗 (𝑥𝑖𝑗𝑆 ) − 𝐶]

(12)

The first-order conditions are:𝜕𝐸𝑈𝑖𝑆𝜎̅ 𝑆 𝜎 (𝜂̃2 )𝐶 = 0,𝑆 = 𝑆 𝑋𝑖 (𝑥𝑖𝑖 ) − ̃0 + 𝜂
𝜕𝑥𝑖𝑖𝑥𝑖𝑖(13)𝜕𝐸𝑈𝑖𝑆𝜎̅ 𝑆 𝜎 ̃1 𝐶 = 0, 𝑖 = 1, 2, 𝑖 ≠ 𝑗.
𝑆 = 𝑆 𝑋𝑗 (𝑥𝑖𝑗 ) − 𝜂𝜕𝑥𝑖𝑗𝑥𝑖𝑗Next, consider the command-optimum of a static model of this type. Suppose the“actual” rate of infection, which takes into account the secondary or later infections, isrepresented by 𝜂s instead of 𝜂̃s:𝐼𝑖𝑆 = 𝑃𝑖𝑆 𝑆𝑖 = (𝜂0 𝑥𝑖𝑖𝑆 + 𝜂1 𝑥𝑖𝑗𝑆 + 𝜂2 𝑥𝑖𝑖𝑆 )𝑆𝑖 ,(14)

The plannner’s maximand is,
𝜎𝜎𝜎𝜎𝐸𝑉𝑖𝑆 = (𝑆𝑖 − 𝐼𝑖𝑆 )[𝑋𝑖 (𝑥𝑖𝑖𝑆 ) + 𝜃𝑋𝑗 (𝑥𝑖𝑗𝑆 ) ] + 𝐼𝑖𝑆 [𝑋𝑖 (𝑥𝑖𝑖𝑆 ) + 𝜃𝑋𝑗 (𝑥𝑖𝑗𝑆 ) − 𝐶](15)

The optimal conditions are:𝜕𝐸𝑉𝑖𝑆𝑆𝑖𝜎𝑆 𝜎(𝜂)𝐶 = 0,
𝑆 = 𝑆𝑖 (𝑋 + 𝑆 ) 𝑋𝑖 (𝑥𝑖𝑖 ) − 𝑆𝑖 0 + 𝜂2𝜕𝑥𝑖𝑖𝑥𝑖𝑖𝑖𝜕𝐸𝑉𝑖𝑆
𝜙𝑆𝑖𝜎
𝑆 𝜎
𝑆 𝜎
𝑆 = 𝑆𝑖 𝜃 ( 𝑋 + 𝑆 ) 𝑋𝑗 (𝑥𝑖𝑗 ) + 𝜙𝑆𝑗 𝑆𝑖 (𝑥𝑗𝑗 ) − 𝑆𝑖 𝜂1 𝐶 = 0,𝜕𝑥𝑖𝑗𝑥𝑖𝑗𝑗(16)𝑖 = 1, 2, 𝑖 ≠ 𝑗.

The first term in the parentheses in each equation of (16) corresponds toagglomeration economies and, without taking account such as in (13), activity levels tend
to be lower. The second term in the second equation as well as the last term in each8

equation, which is higher than that of (13) in per-capita terms, illustrate theexternalities caused by the infectious disease. Without considering them, activities tend
to exceed the optimal levels.

3.3 Comparing to the Dynamic Decentralized Case
Moreover, consider the case in which the model is dynamic but decentralized. Supposethat a resident decides the level of consumption irrespective of the externality he/shecauses. That is, he/she takes 𝑋s, 𝐼s and 𝑆s as given. Once infected in period 𝑡 = 𝜏, theprobability of getting recovered is 𝛾𝑖 in every period. Therefore, the expected loss is:𝑇𝐶̃𝑖𝜏 = ∫ 𝑒 −𝜌(𝑡−𝜏) 𝑒 −𝛾𝑖 (𝑡−𝜏) 𝑐𝑑𝑡 =𝜏𝑐(1 − 𝑒 −(𝛾𝑖+𝜌)(𝑇−𝜏) )𝛾𝑖 + 𝜌

(17)

Then, in each period 𝜏, each self-interested resident maximizes:
𝑇𝑍 𝜎𝑍 𝜎𝑉̃𝑖𝜏𝑍 = ∫ 𝑒 −𝜌(𝑡−𝜏) [𝑋̅𝑖𝑡 (𝑥𝑖𝑖𝑡) + 𝜃𝑋̅𝑗𝑡 (𝑥𝑖𝑗𝑡) ]𝑑𝑡
𝜏𝑆 𝐼𝑆 𝐼𝑆 𝐼
̅ 𝐼𝑖𝑡̅ 𝑥𝑖𝑖𝑡
̅ 𝐼𝑗𝑡
̅ 𝑥𝑖𝑗𝑡
̅ 𝐼𝑗𝑡
̅ 𝑥𝑖𝑖𝑡−(𝛽0 𝑆𝑖𝑡𝑥𝑖𝑖𝑡 + 𝛽1 𝑆𝑖𝑡
𝑥𝑗𝑗𝑡 + 𝛽2 𝑆𝑖𝑡
𝑥𝑗𝑖𝑡 )(18)𝑐(1 − 𝑒 −(𝛾𝑖+𝜌)(𝑇−𝜏) )𝛾𝑖 + 𝜌
In the case of an uninfected resident (i.e., 𝑍 = 𝑆,) evaluating at 𝜏 and differentiatingwith respect to 𝑥s,𝜕𝑉̃𝑖𝑡𝑆𝜎𝑆 𝜎−𝜌𝑡̅̅ ̅ 𝐼̅ ̅ 𝐼𝑆 =𝑒
𝑆 𝑋𝑖𝑡 (𝑥𝑖𝑖𝑡 ) − (𝛽0 𝑆𝑖𝑡 𝐼𝑖𝑡 𝑥𝑖𝑖𝑡 + 𝛽2 𝑆𝑖𝑡 𝐼𝑗𝑡 𝑥𝑗𝑖𝑡 )
𝜕𝑥𝑖𝑖𝑡
𝑥𝑖𝑖𝑡
𝑐𝑒 −𝜌𝑡 (1 − 𝑒 −(𝛾𝑖+𝜌)(𝑇−𝜏) ) = 0, 𝑖 = 1, 2, 𝑖 ≠ 𝑗.
𝛾𝑖 + 𝜌
9(19)𝜕𝑉̃𝑖𝑡𝑆𝜎𝑆 𝜎−𝜌𝑡
̅
̅ ̅ 𝐼
𝑆 =𝑒
𝑆 𝑋𝑗𝑡 (𝑥𝑖𝑗𝑡 ) − 𝛽1 𝑆𝑖𝑡 𝐼𝑗𝑡 𝑥𝑗𝑗𝑡
𝜕𝑥𝑖𝑗𝑡
𝑥𝑖𝑗𝑡
𝑐𝑒 −𝜌𝑡 (1 − 𝑒 −(𝛾𝑖+𝜌)(𝑇−𝜏) ) = 0, 𝑖 = 1, 2, 𝑖 ≠ 𝑗.
𝛾𝑖 + 𝜌Comparing (19) to (8), one may find the differences that correspond to the ones foundin the previous subsection. Also, one can easily show that the last term in each equationin (19) essentially correspond to 𝜆 s in (8). In words, the market (self-interested)allocation tends to underestimate the role of activities when the agglomeration
economies matter but overestimate them when secondary and inter-regional infections
may occur. To correct for those externalities, the corresponding rate of charges orphysical restrictions such as lock-down are necessary.

3.4 Externalities at Steady State: Simplest CaseIn addition, this subsection illustrates the steady state with a simple numericalexample. Consider, first, the following simplest SIS (discrete) setting in which 𝛽1 = 𝛽2 =0:̇ = −𝛽0𝑖𝑖𝑡 𝐼𝑖𝑡 𝑆𝑖𝑡 − 𝛾𝑖 𝐼𝑖𝑡
𝑆𝑖𝑡(20)𝐼𝑖𝑡̇ = 𝛽0𝑖𝑖𝑡 𝐼𝑖𝑡 𝑆𝑖𝑡 − 𝛾𝑖 𝐼𝑖𝑡 ,10

Figure 1 depicts the changes in 𝑆 and 𝐼

𝑆
𝐼
when 𝛽0 = 0.3, 𝑥𝑖𝑖𝑡
= 𝑥𝑖𝑖𝑡= 1 (withoutcontrol,) and 𝛾𝑖 = 0.2 in each region. By (20), the (non-zero) steady state in which 𝐼𝑖𝑡̇ =0 is characterized by the following equation:
𝑆𝑖𝑡 =𝛾𝑖,𝛽0𝑖𝑖𝑡(21)which equals 2/3 in the case of Figure 1. When the number of newly infected equals
the recovered, the steady state is achieved. When 𝛽0𝑖𝑖𝑡 can be controlled, for example, inthe case of the “Uniform Control” in subsection 3.1,
𝐼𝑖𝑡 = 𝑁𝑖 −𝛾𝑖.
𝛽0 (𝑥𝑖𝑖∗ )2(22)

100%
90%
80%
70%
60%
50%
40%
30%
20%10%

048121620
24
28
32
36
40
44
48
52
56
60
64
68
72
76
80
84
88
92
96
1000%

I

SFigure 1: Simple SIS model without ControlThus, around the vicinity of the steady state,Δ𝐼𝑖𝑡2𝛾𝑖≅Δ𝑥 .
Δ𝑥𝑖𝑖 𝛽0 (𝑥𝑖𝑖∗ )3 𝑖𝑖

(23)
11

which corresponds to 𝜂0 in the static case of subsection 3.2 and considerably large.
Inverse of (22) is,12𝛾𝑖1∗
𝑥𝑖𝑖 = () ,𝛽0 𝑁𝑖 − 𝐼𝑖𝑡(24)which correspond to 𝐼 ̇ = 0 curve in the phase diagram of Figure (2). In addition, in thisparticular case, for 𝑇 → ∞, (9) reduces to,𝑑λ𝐼𝑖𝑡
𝜕𝐻=−= −λ𝐼𝑖𝑡 [𝛽0 (𝑥𝑖𝑖𝑡 )2 (𝑁𝑖 − 2𝐼𝑖𝑡 ) − 𝛾𝑖 ] + 𝑒 −𝜌𝑡 𝑐𝑑𝑡
𝜕𝐼𝑖𝑡

(25)

Replacing λ𝐼𝑖𝑡 with λ𝐼𝑖𝑡 = 𝑒 −𝜌𝑡 μ𝐼𝑖𝑡 ,𝑑μ𝐼𝑖𝑡= −μ𝐼𝑖𝑡 [𝛽0 (𝑥𝑖𝑖𝑡 )2 (𝑁𝑖 − 2𝐼𝑖𝑡 − 𝑅𝑖𝑡 ) − 𝛾𝑖 − 𝜌] + 𝑐𝑑𝑡(26)The curve for 𝜇̇ = 0 is drawn such as in Figure 2.

𝑥𝜇̇ = 0𝐼̇ = 01𝛾 2( )𝛽01𝛾+𝜌 2()
𝑁𝛽0𝐼

Figure 2: An Example of the Phase Diagram around the Steady State

12

100%
90%
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60%
50%
40%
30%
20%
10%

048
121620
24
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68
72
76
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84
88
92
96
1000%

IIm

I*

SSmS*Figure 3: No control, market and optimal paths

Illustrate the change in the steady state when the economy is already in the situation
characterized by (21) or in Figure 1. Consider, for simplicity, the cases in which 𝑇 → ∞,Φ(𝐼1𝑇 , 𝐼2𝑇 ) → ∞, and 𝜙0 = 𝜙2 = 0. In the decentralized economy without travels (𝜃 = 𝛽1 =𝐼
𝛽2 = 0,) the infected residents do not impose restrictions on themselves (𝑥𝑖𝑖𝑡=1) and thecondition in (19) reduces to:𝑆
𝜎𝜙0 (𝑥𝑖𝑖𝑡)

𝜎−1

𝐼− 𝛽0 𝑆𝑖𝑡 𝐼𝑖𝑡 𝑥𝑖𝑖𝑡𝑐= 0.
𝛾𝑖 + 𝜌(19’)

Also, the optimal control at steady state maximizes the following with respect to (22):∞

𝑉𝑖 = ∫ 𝑒 −𝜌𝑡 {𝜙0 𝑁𝑖 (𝑥𝑖𝑖∗ )𝜎 − 𝑐 (𝑁𝑖 −0𝛾𝑖)} 𝑑𝑡𝛽0 (𝑥𝑖𝑖∗ )21𝛾𝑖= {𝜙0 𝑁𝑖 (𝑥𝑖𝑖∗ )𝜎 − 𝑐 (𝑁𝑖 −)}.
𝜌𝛽0 (𝑥𝑖𝑖∗ )213(6”)

The curves in Figure 3 indicate the paths without control, decentralized control by(19’) and optimal control by (6”), respectively, for 𝜎 = 0.5, 𝜙0 = 1, 𝑐 = 1.5, 𝑁𝑖 = 7, and𝜌 → 0.

80%
75%
70%
65%
60%
55%
50%
45%
40%
35%

0481216202428323640
4448525660646872768084889296
10030%

I1

I1+

S1

S1+

Figure 4: With Travels (no control and controlled paths)

With travels, taking into account the symmetry, the steady state (21) changes to:
𝑆𝑖𝑡 =𝛾𝑖,2𝛽0 (𝑥𝑖𝑖𝑡 ) + 𝛽1 𝑥𝑖𝑗𝑡 𝑥𝑗𝑗𝑡 + 𝛽2 𝑥𝑖𝑖𝑡 𝑥𝑗𝑖𝑡

(27)

The planner need to control both 𝑥𝑖𝑖 s and 𝑥𝑖𝑗 s so that to obtain an explicit solution
for the condition (6’) is somewhat difficult. Thus, this subsection considers the case inwhich only 𝑥𝑖𝑗 s, the travels, are controlled. Then, (6’) is rewritten as:∞∗ )𝜎 ]∗ )𝜎 ]𝑉 = ∫ 𝑒 −𝜌𝑡 {𝜙0 𝑁1 [1 + 𝜃(𝑥12− 𝑐𝐼1∗ + 𝜙0 𝑁2 [1 + 𝜃(𝑥21− 𝑐𝐼2∗ }𝑑𝑡014(28)=1∗ )𝜎 ]∗ )𝜎 ]{𝜙 𝑁 [1 + 𝜃(𝑥12− 𝑐𝐼1∗ + 𝜙0 𝑁2 [1 + 𝜃(𝑥21− 𝑐𝐼2∗ }𝜌 0 1

Moreover, the decentralized behavior is characterized by (19). Figure 4 presents thedifference between the path controlled by (28) and the one without a control (thedecentralized -path coincides with the latter as 𝑥𝑖𝑗 = 1 is the solution for the above
parameter values.

3.5 Externalities at a Particular Point: SIR CaseWhen SIR is assumed, for 𝛽0𝑖𝑖𝑡 = 0.3 and 𝛾𝑖 = 0.2, the changes in 𝑆, 𝐼 and 𝑅 aredepicted such as the light-colored curves in Figure 5. Note that there is no longer a steadystate because recovered people are not to be infected again. Instead, there is a local
maximum of 𝐼. Again, (22) and (23) hold around the vicinity of the corresponding point.
Deriving the optimal paths is extremely difficult because no “steady state” in this caseand the optimality conditions in (8’)-(9’) must be fully used.
Because 𝐼𝑖𝑡̇ = 0 when 𝐼𝑖𝑡 hits its maximum, this subsection considers the secondbest rule in which (27)-(28) are still applied for control. Then if the control starts when𝐼1𝑡 hits its maximum, the paths change to the darker ones in Figure5. Also, if the same
rule is applied form the beginning, the situation in Figure 6 is to occur and almost noinfection is to be seen in the second region.
15

100%

90%

90%

80%

80%

70%

70%

60%

60%

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0%

0%

I1

I1+

S1

S1+

R1

1
5
9131721252933374145
49
5357
6165
6973778185
899397101

159131721252933374145495357
6165
6973
77
8185
89
93
97
101

100%

R1+

I2+I2

S2+

S2

R2+

R2

Figure 5: SIR Case

90%

80%

80%

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70%

60%

60%

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0%

I1++

S1++

159131721252933
3741
45
49
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8185
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101

100%

90%

1
59131721252933374145
495357
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8185
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101

100%

R1++

I2++S2++

Figure 6: SIR Case (applied from the beginning)

16

R2++

4. Conclusion
This paper has modelled the optimal and market level of consumption in an economy
with two regions when infectious disease may spread. Typical SIS and SIR dynamic
model is extended to characterize the two-region economy. The corresponding dynamic
optimal control problem has been set up and the Hamiltonian is presented. Selfinterested behavior ignores agglomeration economy as well as the negative externalityof the disease such as the secondary and later infections and results in under- or overconsumption (activity) levels. In general, corresponding corrections such as charges orregulations are required. Some simple examples are presented to illustrate the steadystate and local extremum of SIS and SIR models, respectively.
The possible extensions of this paper includes 1) a detailed simulation with optimal
control of 𝑥s, specifying the paths of 𝜆s and using realistic estimates of the parameters,2) considering a case in which the interaction between two regions appears in more
prominent way (e.g., two peaks,) and 3) identifying the optimal levels of regulations.

17

Acknowledgements
The author is grateful to the comments by the attendants of the 2020 conference ofNorth American Regional Science Council, held online, in which I have presented theoriginal version of this study.

18

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