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Epidemic Geography: A Theory of International Tradeand Disease Transmission

Tuotuo Yu ∗

Paris School of Economics

January 18, 2013

Abstract

In this paper, I combine a Dixit-Stiglitz trade model with an SIS epidemic model to

study how international trade can facilitate the transmission of infectious diseases. I

then use this model to analyze the impact of trade barriers and vaccination programs

on disease prevalence, with emphasis on their international spillover effects. It is shown

that in equilibrium state, the disease’s prevalence rate first declines and then rises with

trade costs. This implies raising trade barriers is not always an effective way of curbing

disease transmission, even if we do not consider its economic costs. As for vaccination

programs, coordination between different countries is required to overcome the free

rider problem and achieve global eradication. International transfers, if used properly,

can largely facilitate the coordination and reduce the costs of vaccination programs

for both countries’ sake. I thus conclude that countries should opt for cooperative

strategies (coordinated vaccination, transfers) rather than antagonist strategies (trade

barriers) to fight epidemics in the global era.

Keywords: Dixit-Stiglitz model, SIS model, diseases transmission, epidemics, dy-

namic system, stability analysis, dynamic analysis, trade costs, SPS measures, vacci-

nation, coordination, free rider

JEL Classification Numbers: F12, I18

∗Ph.D. candidate, address: B107, 48 Boulevard Jourdan, 75014 Paris, email: [email protected]. The authorthanks Professor Thierry Verdier, Professor Pierre-Yves Geoffard and Professor Akiko Suwa-Eisenmann fortheir very helpful suggestions. She also owes thanks to many anonymous seminar participants who havegiven their precious comments.

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1 Introduction

Epidemics have shaped the trajectory of many civilizations in the past and they are

still changing the course of some societies today (Diamond, 1999). In pre-industrial times,

cholera, measles and smallpox were some of the major limits to population growth (Hays,

1998). Today, malaria and HIV/AIDS are still hampering the social and economic develop-

ment in some countries of the world (Jamison et al., 2006).

Many factors contribute to the spread of infectious diseases among humans, such as

malnutrition, crowded living conditions, bad hygienic habits and high mobility of people.

While the first three factors are generally caused by poverty and can thus be alleviated by

economic growth (Bloom and Canning, 2000), the last problem - human mobility can only

be exacerbated by the ongoing globalization process. For example, after the outbreak of

SARS in southern China in 2003, the virus spread to a dozen of countries and regions all

over the world within a matter of weeks, due to the fast movement of international tourists

and business travelers, and the efficiency of modern transport technology (Kimball, 2006).

On the sub-Saharan Africa continent, road transportation is a major cause of the rapid

propagation of HIV. (Wawer et al., 1991; Tanser et al., 2000; Oster, 2012)

Economic exchanges have played an important role in the transmission of infectious

diseases since early history. The Black Death which wiped out a quarter of the population

of Europe in the 14th century was carried by rats on trading vessels linking Europe and Asia

(Alchon, 2003). In 1789, soon after the arrival of the first fleet of English settlers in Sydney,

smallpox broke out among the Aboriginals and killed an estimated 50 to 70 percent. 1

Given the scale of this problem, theoretical studies on the relationship between trade

and epidemics are astonishingly scarce. Both trade and epidemic models are abundant, but

nobody has ever tried to combine the two, at least to my knowledge. In this paper, I try to

1There are still polemics over who introduced smallpox to Australia. Some put the blame on Macassanfishermen who had been visiting the Northern coast of Australia regularly since the early 18th century(Campbell, 2002). However, most scholars still believe the English settlers to be the real “wrongdoers”(Mear,2008).

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fill this gap by constructing a hybrid model of trade and epidemics. The model consists of

two countries trading with and infecting each other at the same time. The basic idea is that

the more the two countries trade, the more people cross borders every day, and the higher

the risk of cross-border contagion in case of disease outbreak. Therefore, When studying

disease prevalence in one country, we also need to consider the situation in the other. In

other words, countries are linked by economic exchanges into an integrated epidemic system.

Two problems of policy relevance are then addressed. First, trade barriers not only affect

economic welfare but also disease prevalence. Intuitively, the higher the trade costs, the

smaller the bilateral trade flows, the lower the international mobility of humans, and the

lower the risks of cross-border contagion. This is the reasoning behind the use of trade

barriers as an instrument of disease control. However, it is not clear to what extent trade

barriers are efficient in curbing disease transmission. Moreover, high trade costs imply high

economic losses. It is thus important to weight the benefits of trade barriers against their

costs.

Second, when one country launches a vaccination campaign, it has to take into account

the reactions of its neighbor. This is because diseases and vaccines have international exter-

nalities. On the one hand, the vaccination program in one country not only benefits its own

citizens but also people from across the border. On the other, no country can eradicate the

disease alone if its neighbor remains in an endemic situation, because the disease may come

back in the future. Therefore, the best solution is to carry out an international vaccination

program and eradicate the disease once and for all. However, due to the public goods nature

of global eradication, the international vaccination program can be seriously hampered by

free rider problems, especially if some countries lack the capacity to act. This is why modern

medical technologies have failed to eradicate most traditional infectious diseases, with the

only exception of smallpox (Barrett, 2007).

My analysis gives interesting answers to these two problems. First, raising trade barriers

is not always an effective way against disease transmission. In fact, the equilibrium prevalence

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rate is a non-monotonic function of trade costs; it first declines and then augments as trade

costs rise. This is because when trade costs are relatively low, cross-border contagion is

only partially substituted by within-border contagion, and the overall exposure to infection

sources falls as trade costs rise; when trade costs are relatively high, however, cross-border

contagion is more than compensated by within-border contagion, and the overall exposure

rate augments. Therefore, raising trade barriers is only useful to a certain extent, above which

it can become counterproductive. Apart from this problem, the economic losses resulted from

trade barriers and its long term sustainability raise other issues.

As for the vaccination program, international cooperation is needed to overcome the

free rider problem. In fact, there is a lot of room for cooperation if at least one country

is determined to eradicate the disease. Under certain circumstances, the disease/vaccines

externality can turn to a cost saving factor. This is to say, it can be less costly for one country

to donate some vaccines to its neighbor for free than to use them exclusively on its own

population, even from a short term perspective. Put it the other way, direct subsidization

(aids and transfers) can be more efficient than indirect subsidization (international “leakage”

of vaccines’ benefits).

Methodologically, I focus on equilibrium analysis and comparative statics. Most of my

results are derived from stability analysis of dynamic systems. The bifurcation problem is

also discussed. Corollaries, propositions and their proofs are all put in appendices to ensure

fluent reading. Some other analysis, such as asymmetric trade costs, are not presented here

but are available upon request.

The rest of this paper is organized as follows. Section 2 reviews some recent literature.

Section 3 presents the benchmark model with two countries of equal size and studies the

relationship between trade costs and equilibrium prevalence rate. Section 4 analyzes the

problem of global disease eradication and discusses the efficiency of different cost sharing

schemes and international transfers. Section 5 expands the above analysis to two countries

of different sizes. Section 6 concludes.

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2 Review of literature

On the empirical side, my paper is best supported by Oster (2012), which established a

strong relationship between exports and new HIV infections in Sub-Saharan Africa. Many

other studies have also found that the spread of HIV/AIDS in Africa is greatly facilitated by

road transportation. To cite a few, Wawer et al. (1991) analyzed the geographical distribution

of HIV infections in a rural district of Uganda and found that the disease spreads from main

road trading centers to intermediate trading villages and then to rural agricultural villages.

Similarly, Tanser et al. (2000) found that the distance of homestead to roads is negatively

correlated with HIV prevalence in rural South Africa. Public policies are set in line with

these research. In 2002, the World Bank launched the Abidjan-Lagos Transport Corridor

(ALCO) project with the aim of augmenting HIV awareness among truck drivers, sex workers

and local habitants along a major transport route in Eastern Africa. The first stage of this

project is believed to be successful.

On the theoretical side, my paper is closest to Barrett (2003), which analyzes the free

rider problem in global disease eradication and highlights the necessity of international coop-

eration. Continuing in this direction, I go one step further by studying the cost structure of

the eradication program and exploring different ways of cost sharing. In so doing, I find that

international transfers have a large cost saving potential even from the donor country’s point

of view. This result suggests that international cooperation may become easier to achieve if

its economic benefits are made clear to public authorities.

I also borrow from a large body of literature on disease control and prevention. Geoffard

and Philipson (1997) discussed the deficiencies of private vaccine provision and pointed out

that infectious diseases can only be eradicated through universal public programs. Accord-

ing to this result, I concentrate on public rather than private vaccination programs when

analyzing the disease eradication problem. Morton and Wickwire (1974) made a dynamic

cost-benefit analysis of epidemic control policies and found that the optimal policy is of the

“bang-bang” type, i.e., vaccinating the population all at once instead of doing so gradually.

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Therefore, when trying to minimize the costs of disease eradication, it suffices to look at

“one-shot” strategies, which largely simplifies my analysis.

3 Benchmark

In this section, I present a simple model of “epidemic geography” with two countries of

equal size. This model is composed of two parts: the first part is a Dixit-Stiglitz model of

trade between two countries, and the second is the SIS epidemic model.

I choose the SIS model for two reasons. First, it is simple; consisting of only one equa-

tion, it is more adapted to the two-country analysis than the commonly used SIR model.

Second, it is well suited for analyzing human-to-human diseases, on which this paper is built

concentrated. 2As for the trade model, I use Dixit-Stiglitz to avoid the issue of comparative

advantage or resource endowment, which is not relevant to my analysis. The DS model is

also preferable to the simple gravity model in the sense that it can account for both within-

country and cross-country transactions. However, the conclusions drawn from the DS model

are also valid for other trade models.

I then study the impact of trade barriers to the equilibrium prevalence rate. In this

paper, the term “trade barriers” should be understood in the ad valorem sense. It refers

to traveling alerts, border medical exams and quarantines, and other measures related to

human diseases. What is special about this type of trade barriers is that they tend to affect

human mobility and trade flows in both directions. For example, a traveling alert can cut

tourist flows to a country at risk and reduce that country’s “exports”; meanwhile, it can

make business travelers from other countries to cancel or delay their trips, which can hurt

their own countries’ exports. Medical exams and quarantines cause extra costs and time

delay on both entering and returning traffic flows, making people less willing to cross the

2Human to human diseases (like HIV/AIDS) are different from vector-borne diseases (like malaria) andanimal-to-human diseases (like BSE). Some diseases are first transmitted from animals to humans and thendirectly among humans, but I only study their second stage

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border. Considering these facts, I decide to model them as a bilateral rising of trade costs. 3

3.1 Trade model

There are two countries, Home and Foreign, with equal population size 1. There is only

one sector, composed of n tiny firms in each country. All firms produce with use the same

Increasing-Returns-to-Scale (IRS) technology:

q = l − f

where q is the quantity produced, l the amount of labor employed by each firm and f the

fixed cost in terms of labor. The labor efficiency is normalized to 1. The representative home

consumer’s utility function is of the Constant-Elasticity-of-Substitution (CES) type:

U =

(n∑i=1

cσ−1σ

Hi +n∑

i∗=1

c∗σ−1

σFi

) σσ−1

i and i∗ are home or foreign firm indexes. cHi and c∗Fi are respectively the home consumer’s

demands of goods produced by home and foreign firm i and i∗. σ is the elasticity of substi-

tution.

All home and foreign firms being identical, I remove the firm index and simplify the

utility function to:

U = nσ

σ−1

(c

σ−1σ

H + cσ−1σ

F

) σσ−1

(3.1)

Same for the foreign consumer (denoted by *):

U∗ = nσ

σ−1

(c∗σ−1

σH + c

∗σ−1σ

F

) σσ−1

(3.2)

Country sizes being equal, home and foreign wage rates must also be equal, and I take it

3The results do not change much if trade costs rise unilaterally. The proof can be given upon request.

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as the numeraire. The out-of-factory prices of home and foreign goods should be equal too,

as they are just the same marginal costs times the same markup:

p = p∗ =σ

σ − 1

Likewise, the sizes of home and foreign firms are the same:

l = l∗ = σf

q = q∗ = (σ − 1)f

The labor market equilibrium entails:

n = n∗ =1

σf

The consumers’ budget is divided between home and foreign goods, and the demands

for each depend on their relative price. The out-of-factory prices being equal, the relative

price is determined only by trade costs. Note the iceberg trade costs by τ , and the relative

demand of the home consumer is:

cHcF

= τσ

Finally, the home consumer’s budget constraint writes:

n(p · cH + τp∗ · cF ) = 1

Combining the last two equations, I get:

cH =(σ − 1)f

1 + τ 1−σ, cF =

(σ − 1)f

τ + τσ

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Similarly, for the foreign consumer, there is:

c∗H =(σ − 1)f

τ + τσ, c∗F =

(σ − 1)f

1 + τ 1−σ

According to Walrus’ law, the goods markets clear automatically.

Plugging cH , cF , c∗H and c∗F into (3.1) and (3.2), I get consumer utilities in terms of trade

costs τ :

U = U∗ = [n(σ − 1)f ]σ

σ−1(1 + τ 1−σ

) 11−σ

Conform to economic intuition, consumer utilities decline with trade costs. Figure 1

shows the percentage utility loss compared to free trade (τ = 1) for different values of

τ ∈ [1, 3]. It can be seen that the utility loss is an increasing and concave function of τ .

Besides, the lower the elasticity of substitution σ, the greater the utility loss for a given τ .

Figure 1: Utility loss compared to free trade (τ = 1)

3.2 Epidemiology

In the SIS model, the population is divided in two compartments: Susceptible (S) and

Infectious (I). Let k be the contagion rate and γ the recovery rate. Under autarky, the

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epidemic dynamics in each country can be described by the following two equations:

dI

dt= kSI − γI

dS

dt= γI − kSI

where I + S = 1, ∀t.

kSI is the number of new infections per period: S · I is the total number of contacts

between infectious and susceptible people, and k is the probability of the latter getting

infected per contact. γI is the number of recovered individuals per period. The difference

between the two is the just net change in the number of infectious individuals, dI/dt. People

become susceptible again as soon as they recover, which explains the second equation. 4

Considering S = 1−I and dI/dt+dS/dt = 0, the above two equations can be summerized

into one:

dI

dt= I[k(1− I)− γ] (3.3)

where 0 ≤ I ≤ 1.

When γ/k > 1, (3.3) has only one equilibrium I = 0. This equilibrium is also stable

because dI/dt < 0 for any 0 < I ≤ 1. When γ/k < 1, however, the problem is more complex,

because there are two equilibriums I = 0 and I = 1− γ/k. To analyze their stability, I look

at the signs of (3.3) in different intervals:

dI

dt=

> 0, if 0 < I < γ/k

< 0, if γ/k < I ≤ 1

Therefore, I = 0 is not stable and I = 1− γ/k is stable when γ/k < 1. To sum up, the only

4A more realistic scenario would be that recovered individuals gain immunity against the disease fora limited or unlimited period of time. This treatment would require that I add a new compartment, theRecoverd/Removed just as the SIR model suggests. To keep the analysis tractable in a two country setting,I opt for the simpler SIS model.

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Figure 2: Evolution of infectious diseases

stable equilibrium of (3.3) is:

Iau = max (0, 1− γ/k)

Figure 2 shows that starting from the same initial prevalence rate (I0 = 0.2), the disease with

γ/k = 1.5 tends to disappear naturally (I∞=0), while the disease with γ/k = 0.5 persists in

the long term (I∞ = 0.5).

3.3 Combining trade with epidemiology

To establish the link between trade and epidemics, I use an economic quantity to proxy

the level of human contact, and this quantity is consumption.

To understand why consumption measures human contact, one should first admit that

consumption is a proxy of overall economic activities. At the macro level, consumption

is just the other side of production. At the micro level, household consumption roughly

corresponds to its production: the more one works and earns, the more one can consume.

Therefore, my use of consumption as a measure of overall economic activities is justified.

It remains to show the relation between economic activities and disease transmission.

Just think of an ordinary person’s daily economic activities: taking a crowded bus to work,

shaking hands with colleagues and clients, waiting in queue in front of a supermarket counter,

etc. These occasions are just where most people get infected. Infection may not be directly

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caused by economic activities, but it is more or less the byproduct of human contact motivated

by economic reasons.

Another advantage of using consumption to proxy risky contact is that it can also ac-

count for cross-border contagion. The international mobility of people is even more directly

encouraged by economic considerations than local mobility. For example, to consume the

special “goods”of tourism, one must move physically to a foreign country and make direct

contact with foreign people. As for the trade of normal goods, i.e., merchandises, it is always

conducted by business travelers, expatriates, etc. In general, the more two countries trade

with each other, the more closely their people are interconnected. 5

Figure 3 sums up the above discussion on the relation between consumption and conta-

gion: to consume local and foreign goods, one have to make contact with local and foreign

people, and this contact exposes her to epidemic risks coming from both sides of the border.

Figure 3: How consumption is linked to contagion

In line with this idea, I define consumption related exposure as the number of infectious

workers “embedded”in a given quantity of consumer goods. By consuming one additional

unit of goods, the consumer needs to make contact with some additional workers who are

5Not every kind of international mobility is economic in nature, for example, there are military deploy-ment and pilgrim. However, even these activities can bring economic exchanges along with them.

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involved in its production and distribution; among these workers, some are infectious, and

contacting with them increases the risk of contagion. The more she consumes, the more

“embedded”workers she needs to contact with, and the higher the risk of contagion for her.

In my model, the number of workers “embedded”in each unit of goods is the same for all

varieties, which is just l(∗)/q(∗), the employment level of each firm divided by its production.

The proportion of workers producing these goods who are infectious depends on the specific

origin of the goods, which is I for home and I∗ for foreign. A home consumer consumes

n · cH units of home goods and n∗cF units of foreign goods, so her total level of exposure to

infectious individuals is:

E = ncH ·l

q· I + n∗cF ·

l∗

q∗· I∗

Plugging in n(∗), l(∗), q(∗), cH and cF calculated in the trade model, E becomes:

E = αI + βI∗ (3.4)

where:

α =1

1 + τ 1−σ, β =

1

τ + τσ

Following the same procedure, I can get the effective exposure level for the foreign con-

sumer:

E∗ = αI∗ + βI (3.5)

It can be seen that E and E∗ are just the weighted averages of I and I∗. The weights, α

and β, which I call respectively internal and external contagion coefficients, are proportional

to the consumption of local and imported goods by construction. Intuitively, the more

local/imported goods one consumes, the more contact she makes with local/foreign people,

and the larger impact the local/foreign epidemic situation has on her.

In order to put E and E∗ into the epidemic system, let us first look at the epidemic

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equation in autarky, which I recopy here for convenience:

dI

dt= k(1− I)I − γI

To account for risks originating from both countries, I should replace the second I on the

right hand side by E. This is to say, each susceptible individual now makes contact with E

infectious people instead of I. The home epidemic equation then becomes:

dI

dt= k(1− I)E − γI

Plugging (3.4) into it, I get:

dI

dt= k(1− I)(αI + βI∗)− γI (3.6)

Similarly, on the foreign side, there is:

dI∗

dt= k(1− I∗)(αI∗ + βI)− γI∗ (3.7)

(3.6) and (3.7) make up a nonlinear dynamic system. It is symmetric because I assume equal

country sizes. This assumption would be relaxed later to allow for more general analysis.

3.4 Equilibrium prevalence rate

According to Proposition 2 (Appendix A) and Proposition 5 (Appendix B), the only

stable equilibrium of this dynamic system is:

Isym = I∗sym = max

(0, 1− γ

k(α + β)

)

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To compare it with the autarkic equilibrium:

Iau = I∗au = max(

0, 1− γ

k

)

I need to look at the value of α + β.

As mentioned earlier, α and β are proportional to the demands of local and foreign

varieties. They are uniquely determined by trade costs τ :

α =1

1 + τ 1−σ, β =

1

τ + τσ

α increases and β decreases with τ . This is intuitive: when trade costs augment, the price

of imported goods augments relatively to that of local goods, and consumers substitute

the former by the latter; consequently, they substitute the “embedded”contact with foreign

people by contact with local people.

α + β corresponds to the total consumption level. It first decreases and then increases

with τ (Figure 4). The first part of the story is easy to explain: the higher the iceberg

trade costs, the higher the proportion of goods “melts down”during transportation, and the

lower the total amount received by final consumers. However, when τ augments further,

the story turns the other way: trade flows are cut down drastically, so the total loss during

transportation falls, although the per unit loss rate is even higher; consequently, the total

consumption rises again. In fact, when τ approaches infinity (autarky), there would be no

trade nor loss at all, and the total amount of goods received by consumers is the same as

under free trade.

The equilibrium prevalence rate Isym, which is an increasing function of α+β, also shows

a non-monotonous pattern (Figure 5). As trade costs rise, the equilibrium prevalence rate

first falls and then rises. Interestingly, it is of the same level under free trade and com-

plete autarky. At first glance, this may seem counter-intuitive, because autarky is supposed

to isolate one country from the rest of the world and protect it from the invasion of in-

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Figure 4: α, β and α+ β as functions of τ (σ = 4)

fectious diseases. However, epidemic autarky is an unstable state; once the diseases make

their way into the country by chance, intensive within-border exchanges will accelerate their

transmission and push up the prevalence rate to the free trade level.

It should be noted that the “melting-down” of goods during transportation does not

necessarily reduce consumers’ utility. Compared to autarky, opening up to trade can bring

in more varieties, for which consumers are ready to make some sacrifice on quantity. In other

words, the simple arithmetic sum of consumption is not equivalent to the aggregated sum

nor the consumer utility, which also takes into account the love of variety. In fact, when

trade costs rise, the arithmetic sum (total amount) can go up and down, but the aggregated

sum (consumer utility) always declines.

The expression “melting-down” should not be taken literally. Goods do not actually

disappear during transportation. In fact, when trade costs rise, some production and con-

sumption activities would no longer take place, and finally it seems as if goods evaporated.

6

6Some people asked me why governments cannot just dump goods into oceans to fight epidemics. Myanswer is yes they can, but a better solution is to prevent those goods from being produced at the first place,and this is just what trade barriers do.

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3.5 Political implications

The equilibrium prevalence rate is a non-monotonic function of trade costs. This somehow

put into question the effectiveness of raising trade barriers as a means of fighting epidemics.

Such measures may be helpful if the initial trade costs are low. However, when the initial

trade costs are already high, further augmenting them only worsens the situation. The

reasons is that although trade barriers can curb external transmission, they also facilitate

internal transmission by encouraging the substitution of foreign goods by local ones. At low

trade costs, the first effect dominates, and disease prevalence declines as trade costs rise; at

high trade costs, on the contrary, the second effect dominates, and disease prevalence rises.

Even within the “effective interval”, augmenting trade costs may not be a desirable

solution from the cost-benefit point of view. Suppose the two countries are initially in free

trade (τ = 1), then a disease with γ/k = 0.90 breaks out. The public authorities could raise

trade costs to τ = 1.45 and reduce the prevalence rate by no more than 8 percentage point

(Figure 5). Meanwhile, consumers would suffer from a 12% utility loss per period. Only a

disease of high severeness can justify such huge costs.

If trade barriers can eradicate the disease permanently within a short period of time and

then be removed without causing a resurgence, their use would be more justifiable. Whether

this can be done depends on the characteristics of the disease, γ/k. If γ/k is high enough, it

is possible to reach zero prevalence after a certain period of time with properly chosen level

of trade costs. For example, when γ/k = 0.95, it suffices to raise τ to about 1.2 for a certain

period of time, and the disease would die out naturally (Figure 5). 7 However, if γ/k is low,

trade barriers can only temporarily reduce the prevalence rate, which would rebound as soon

as the trade barriers are removed. This is to say, trade barriers can only eradicate a very

limited range of infectious diseases. In our case, the threshold is around γ/k ∈ (0.90, 0.95).

Below this threshold, the disease cannot be eradicated by simply imposing trade barriers; it

7Of course, τ can be raised to higher levels to speed up the eradication. To choose the cost-minimizingtrade costs, a dynamic analysis is needed.

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Figure 5: Equilibrium infection rate I(∗)sym for different trade costs

τ at γ/k = 0.90 and γ/k=0.95

can only be temporarily controlled.

In reality, trade barriers are seldom used alone to fight epidemics. Most of the time they

are implicitly coupled with other preventative measures. For example, when WHO issues

a traveling alert upon a certain epidemic, it also raises people’s awareness and motivates

them to seek prevention. As for medical exams and quarantines, their main objective is

to isolate infectious individuals and protect the susceptible; traffic delays are just their

side effects. Taken together, it is hard to say that these “trade barriers” are ineffective. To

better understand this problem, we should also analyze the effectiveness of other preventative

measures, notably the vaccination.

4 Vaccination and global disease eradication

Eradicating infectious diseases in two interconnected countries is intrinsically more com-

plicated than doing so in two separated countries. First, vaccination programs carried out by

one country have positive spillovers to the other, which could reduce the latter’s incentive to

contribute. Second, if the disease persists in the second country due to the lack of adequate

vaccination, its neighbor would constantly face the risk of cross-border contagion. Therefore,

local elimination is inseparable from global eradication, whose success depends largely the

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coordinated actions from both sides.

In this section, I study how the vaccination programs should be coordinated in the two

countries. First, I try to specify the set of local optimal strategies, i.e., for each quantity of

vaccines chosen by one country, at least how much should be chosen by the other in order to

achieve global eradication. Then I try to find the global optimal strategy which minimizes

the total costs and analyze its feasibility under different cost sharing schemes.

4.1 Minimal vaccination requirements

Suppose a 100% effective vaccine exists for the disease in question. 8 Home and foreign

governments vaccinate respectively θ and θ∗ percentage of their population. The vaccination

reduces the number of susceptible individuals from 1− I(∗) to 1− θ(∗)− I(∗), so the epidemic

system becomes:

dI

dt= k(1− θ − I)(αI + βI∗)− γI

dI∗

dt= k(1− θ∗ − I∗)(αI∗ + βI)− γI∗

θ and θ∗ are bounded between 0 and 1 by definition.

The question is to find the minimum values of θ and θ∗ which can eradicate the disease,

i.e., make the system converge to I = I∗ = 0. 9 According to Proposition 6 (Appendix C),

θ and θ∗ must satisfy the following conditions:

θ + θ∗ ≥ 2 (1− γ/k) , if τ = 1 (4.1)

8In reality vaccines are never effective to 100%. However, they can be discounted to calculate theequivalent coverage. For example, 60% of the population have taken a 80% effective vaccine, then theequivalent coverage is 60% × 80% = 48%. In this way, different preventative measures of different effectiverates can be added up to obtain a total equivalent coverage.

9It should be noted that we cannot have I = 0, I∗ > 0 in equilibrium unless θ = 1. This is to say, ifForeign remains in an endemic situation, the only way to achieve local elimination in Home is to vaccinateeverybody. While leaving this option open, we focus on the global eradication solution.

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[θ −

(1− α

α2 − β2· γk

)][θ∗ −

(1− α

α2 − β2· γk

)]≥(

β

α2 − β2· γk

)2

, if τ > 1

(4.2)

Both terms in brackets on the left hand side of (4.2) should be positive.

Figure 6 illustrates the “minimum vaccination frontier” implied by these conditions.

Under free trade (τ = 1) , the frontier is a straight line, as home and foreign vaccines are

perfect substitutes.

Under autarky (τ =∞), the frontier takes an angled shape. In fact, when τ =∞, α = 1

and β = 0, and (4.2) becomes:

[θ −

(1− γ

k

)] [θ∗ −

(1− γ

k

)]> 0

which is equivalent to two independent conditions:

θ > 1− γ/k, θ∗ > 1− γ/k

This is to say, there is no substitutability between home and foreign vaccines.

When 1 < τ <∞, home and foreign vaccines are imperfect substitutes, and the frontier

becomes a hyperbola.

4.2 Cost sharing problem

Let us focus on the most interesting case of 1 < τ <∞, where home and foreign vaccines

are imperfect substitutes. It can be seen from Figure 6 that the total cost θ+θ∗ is minimized

at the tangent point of the hyperbola and the −1 sloped line:

θmin = θ∗min = max

(0, 1− 1

α + β· γk

)

Two remarks should be made. First, θmin and θ∗min are always smaller than 1, implying

the global eradication is always achievable. Second, θmin and θ∗min are just the equilibrium

20

Figure 6: Minimum vaccination frontier for different trade costs(symmetric size)

prevalence rates θmin and θ∗min. Similar to the relation between I(∗)sym and τ , θ

(∗)min first declines

and then rises with τ . This is to say, there are some substitutability between trade barriers

and vaccines when trade costs are not too high. When τ is at some intermediate level and:

γ/k ≥ α + β

The disease can die out naturally and there is no need for vaccines. In fact, γ/k ≥ α + β

implies: θmin = θ∗min = 0

Now suppose for various reasons (free rider problem, budget shortfall, etc.), Foreign can

only provide its population with θ∗ < θ∗min vaccines. Unable to force it to contribute more,

Home must augment its own vaccine coverage to θ > θmin to compensate for Foreign’s

shortfall. The question is how much θ should be for any given θ∗.

The answer is simply given by (4.2):

[θ −

(1− α

α2 − β2· γk

)][θ∗ −

(1− α

α2 − β2· γk

)]=

(β

α2 − β2· γk

)2

21

However, this equation cannot always be satisfied. Remember that θ should be no greater

than 1, which implies θ∗ should at least be:

θ∗min = max(

0, 1− γ

αk

)(4.3)

Otherwise global eradication is impossible. 10

If Home’s objective is only to eliminate the disease locally, it should contribute at least:

θ =

1, if θ∗ < θ∗min

max

(0, 1− α

α2 − β2· γk− β2

[(1− θ∗)(α2 − β2)− α · γ/k](α2 − β2)

)if θ∗ ≥ θ∗min

(4.4)

This two expressions can be grouped up as:

θ = min

[1,max

(0, 1− α

α2 − β2· γk− β2

[(1− θ∗)(α2 − β2)− α · γ/k](α2 − β2)

)]

4.3 International transfers

In the previous subsection, I assume that each country is responsible for vaccinating its

own population and there is no international transfers. However, if I allow for transfers, the

story can be very different. The intuition is that the marginal utility of vaccine decreases

with its coverage. As the coverage goes up, fewer people remain susceptible, so an infectious

person can cause fewer secondary infections. The social costs of each new infection go down,

so do the social benefits of each new dose of vaccine. Therefore, if the vaccine coverage is

already high in Home but remains low in Foreign, it could be “profitable” to shift some

vaccines abroad. In so doing, home can increase the utility of its vaccines. Although this

utility is initially absorbed by foreign people, it can be “fed back”to home people through

trade.

10The same condition should also be satisfied when τ = 1 or τ =∞.

22

If Home is determined to eradicate the disease globally through transfers, it has to choose

the optimal amount of vaccines to donate to Foreign. This amount is just θmin − θ∗, θmin

being the “tangent point” and θ∗ the amount provided by Foreign itself. The reason is the

following: Home’s cost is (1) the total amount of vaccines needed to eradicate the disease,

minus (2) the amount contributed by Foreign. (2) being fixed, Home’s minimization problem

falls exclusively on (1). As shown earlier, (1) is minimized at the tangent point, and:

(θ + θ∗)min = 2

(1− γ

k(α + β)

)

Home’s cost of implementing this optimal transfer strategy is:

θtrans = 2

(1− γ

k(α + β)

)− θ∗

And the transfer amounts to:

θH→F = 1− γ

k(α + β)− θ∗

To verify that the transfer strategy indeed saves money for Home, i.e.,

θtrans ≤ θ

Two cases should be studied:

1. θ∗ < θ∗min = 1− γ/αk. In this case, the disease cannot be eradicated without transfer.

However, Home can adopt the “autarkic” strategy by setting θ = 1 and ignoring what

happens on the other side of the border. The cost differential between the transfer and

23

autarkic strategy is:

θtrans − θ = 1− 2γ

k(α + β)− θ∗

≥ 0, if θ∗ ≤ 1− 2γ

k(α + β)

< 0, if 1− 2γ

k(α + β)< θ∗ < 1− γ

k

2. θ∗ ≥ θ∗min = 1− γ/kα. In this case, there is always:

θtrans < θmin

Apart from solving the complex inequality, there is a more convenient way of finding

this relationship. In Figure 7, the downward curved hyperbola represents the minimum

vaccination frontier, and point A represents the initial situation, with Home contribut-

ing θ and Foreign θ∗ ≥ θ∗min. By giving Foreign θmin−θ∗ (represented by the horizontal

segment) and cutting its own vaccines by θ − θmin (represented by the vertical wide

line), Home can still have the disease eradicated, but at a lower cost (Point B).

To summarize, there is:

θtrans > θmin, if θ∗ ≤ 1− 2γ

k(α + β)

θtrans ≤ θmin, if θ∗ ≥ 1− 2γ

k(α + β)

(4.5)

This is to say, if Foreign contributes very little, Home would be better off in the short

term by resorting to the autarkic strategy; if foreign contribution is not that small, Home

would better help it out.

From Foreign’s point of view, there is some room for strategic non-cooperation: as long

as it is in Home’s profit to help it, it can keep decreasing its contribution until reaching the

“breaking point”θ∗ = 1 − 2γ/k(α + β). However, this breaking point can never be reached

24

Figure 7: cost differential: transfer vs. autarky (symmetric size)

if it is negative. 11 Therefore, foreign’s optimal choice is:

θ∗opt = max (0, 1− 2γ/k(α + β))

Foreign can contribute even less if Home is willing to bear short term losses to achieve global

eradication for the sake of future generations.

From Home’s point of view, there seems to be no effective way of limiting this strategic

non-cooperation except for negotiating with Foreign. This is another issue which I prefer to

leave for the future.

11Figure 7 happens to illustrate such as a situation.

25

5 Different country sizes

5.1 Equilibrium analysis

In this section, we expand the above analysis to two countries of different sizes. Let

foreign population still be 1 and home population be L > 1. Following the same procedure

as before, I can obtain home and foreign consumer’s demands:

cH =(σ − 1)f

L+ (τ/w)1−σ, cF =

(σ − 1)f

(τ/w) + L(τ/w)σ

c∗H =(σ − 1)f

L(τw) + (τw)σ, c∗F =

(σ − 1)f

1 + L(τw)1−σ

where w is the home/foreign wage ratio; it is given by the following implicit function:

L

L+ (τ/w)1−σ+

τ

L(τw) + (τw)σ= 1

and:

∂w

∂L≥ 0,

∂w

∂τ≥ 0

The equalities are reached only when w = τ = 1. The proof is given in Appendix D.1.

Figure 8: Home/foreign wage ratio for different trade costs (L=2,5, 10)

26

In the presence of trade barriers, consumers of the larger home country enjoy a higher

economic welfare than their smaller foreign country counterparts. Not only do they receive

a higher wage, but also do they face a lower aggregate price level.

The epidemic system writes:

dI

dt= k(1− I)(αHI + βHI

∗)− γI

dI∗

dt= k(1− I∗)(αF I∗ + βF I)− γI∗

where:

αH =L

L+ (τ/w)1−σ, βH =

1

(τ/w) + L(τ/w)σ

αF =1

1 + L(τw)1−σ, βF =

L

L(τw) + (τw)σ

Figure 9 and 10 show αH , βH , αF and βF for different trade costs τ . Like in the case

of equal country size, both αs increase and both β decrease with τ . For the larger home

country, the “internal coefficient” αH is always higher than the “external coefficient” βH ,

because local consumption always exceeds imports. For the smaller home country, however,

αF is first lower and then higher than βF as τ rises. This is because the smaller economy

initially relies a lot on its larger neighbor for imports but gradually turns inward as trade

costs rise. In other words, the two countries pose asymmetric epidemic risks to each other,

and this asymmetry tends to increase with country size disparity and decrease with trade

costs.

It is easy to verify α2H ≥ βH · βF and α2

F ≥ βH · βF all τ , L and w. According to

Proposition 2 and Proposition 5, the epidemic system converges to zero equilibrium if:

γ

k≥ 1

2

(αH + αF +

√(αH − αF )2 + 4βHβF

)(5.1)

Otherwise the steady state is above zero.

27

Figure 9: Home’s internal and external contagion coefficients αHand βH for different trade costs τ

The three lines in Figure 11 correspond to the right hand side of (5.1), respectively for

L = 2, L = 5 and L = 10. They are the “borders” between zero and non-zero steady states:

when γ/k (presented on the vertical axe) is below these lines, the system converges to a non-

zero equilibrium; when γ/k is above, the system converges to zero. It can be seen that the

larger the size disparity (L), the higher the border line, and the more difficult to eliminate

the disease by raising trade costs. This is because bilateral trade flows between disparate

countries are smaller than those between similar countries for any given τ ; consequently,

trade barriers are less likely to make a big difference on trade volume and disease prevalence

for the former.

The equilibrium prevalence rates are solved numerically and shown in Figure 12 - 14 for

L = 2, L = 5 and L = 10. Like in the case of same country sizes, they first decline and

then rise with trade costs. Moreover, the equilibrium prevalence rate is always higher in the

larger home country, because home consumers consume more and make more risky contact.

The difference between home and foreign prevalence rate also increases with size disparity.

28

Figure 10: Foreign’s internal and external contagion coefficients αFand βF for different trade costs τ

Figure 11: Border line between zero and non-zero steady state

Figure 12: Home and foreign prevalence rates I and I∗ for γ/k =0.90 and γ/k = 0.95 at L = 2

29



30

Figure 15: Minimum vaccination frontier for different trade costs(asymmetric size)

5.2 Vaccination problem

Home and foreign vaccinate respectively θ and θ∗ percentage of their population. Then

the epidemic system becomes:

dI

dt= k(1− θ − I)(αHI + βHI

∗)− γI

dI∗

dt= k(1− θ∗ − I∗)(αF I∗ + βF I)− γI∗

According to Proposition 6, θ and θ∗ should satisfy the following conditions to achieve global

eradication:

Lθ + θ∗ ≥ (L+ 1)(1− γ/k), if τ = 1 (5.2)[θ − (1− αF

αHαF − βHβF· γk

)

] [θ∗ − (1− αH


)

]≥ βHβF

(αHαF − βHβF )2· γ

2

k2, if τ > 1

(5.3)

Both terms on the left hand side of (5.3) should be positive.

Figure 15 shows the shapes of minimum vaccination frontiers under different trade costs.

When τ = 1, the two-country economy is identical to a single country of size L + 1, and

31

home and foreign vaccines are perfect substitutes. In this case, the minimum vaccination

frontier is a straight line, and the disease can be eradicated as long as the overall coverage

ratio (Lθ + θ∗)/L exceeds 1− γ/k.

Under complete autarky (τ =∞), αH = αF = 1, βH = βF = 0, and (5.3) becomes:

[θ − (1− γ

k)] [θ∗ − (1− γ

k)]≥ 0

This is equivalent to two independent conditions:

θ ≥ 1− γ/k, θ∗ ≥ 1− γ/k

There is no substitutability between home and foreign vaccines.

When 1 < τ <∞, home and foreign vaccines are imperfect substitutes, and the minimum

vaccination frontier becomes a hyperbola. The total cost Lθ + θ∗ can be minimized at its

tangent point with the −L sloped straight line:

θmin = 1−αF −

√βHβF/L


θ∗min = 1− αH −√βHβF · L


Three remarks should be made. First, like in the case of symmetric country size, θmin

and θ∗min are both inferior to 1, meaning that global eradication is always possible. 12

Second, θmin is larger than θ∗min, for any τ > 1 and L > 1. 13 This is consistent with the

fact that the larger home country experiences a higher disease prevalence than the smaller

foreign country in equilibrium. Intuitively, the more disease-prone needs more vaccines.

Third, unlike the asymmetric-country case, here free-riding can be legitimate from the

12θmin < 1 and θ∗min < 1 because αF −√βHβF /L and αH −

√βHβF · L are increasing functions of τ and

take value zero when τ = 1.13See Appendix D.2 for detail.

32

cost minimizing point of view. This is because when:

γ/k ≥ αHαF − βHβFαH −

√βHβF · L

θ∗min is negative. In this case, only the suboptimal strategy:

θsub = 1− αFαHαF − βHβF

· γk

+βHβF

(αHαF − βHβF )(αHαF − βHβF − αH · γ/k)· γ

2

k2

θ∗sub = 0

is implementable. 14

If both θmin and θ∗min are negative, there is no need for vaccines and the disease can die

out naturally. In fact, θmin ≤ 0 and θmin ≤ 0 imply:

γ

k≥ 1

2

[αH + αF +


]

which is just the stability condition of zero equilibrium in natural state.

In sum, the minimum total cost is:

(θ + θ∗)min =

L+ 1− LαF + αH − 2√βHβF · L

αHαF − βHβF, if γ/k < Φ1

L ·(

1− αFαHαF − βHβF

· γk

+βHβF

(αHαF − βHβF )(αHαF − βHβF − αH · γ/k)· γ

2

k2

)if Φ1 ≤ γ/k < Φ2

0, if γ/k ≥ Φ2

where:

Φ1 =αHαF − βHβFαF −

√βHβF/L

, Φ2 =(αH + αF +


)/2

14θsub is smaller than 1 because θ∗sub ≥ θ∗min implies θsub ≤ θmin ≤ 1.

33

Figure 16 and 17 show the minimum overall coverage ratio (Lθ + θ∗)min/(L + 1) for

γ/k = 0.90 and γ/k = 0.95. It can be seen that in both cases, the curves become more

and more “shallow” as size disparity rises. In other words, the sensitivity of the minimum

vaccination rate with respect to trade costs declines as L goes up. The intuition is that the

more disparate the country sizes, the smaller the impact of trade barriers on cross-border

mobility, and the more heavily does the global eradication rely on vaccines. This fact is

well illustrated by Figure 17: when the two countries are of comparable sizes (L = 1 and

L = 2), trade barriers can eradicate the disease alone; however, when their sizes differ too

much (L = 5 and L = 10), trade barriers are not enough and vaccines are needed.

Figure 16: Minimum overall coverage rate (γ/k = 0.90)

Figure 17: Minimum overall coverage rate (γ/k = 0.95)

34

Then let us consider cost sharing problems. If Foreign can only provide 0 < θ∗ < θ∗min

vaccines, Home would need to come up with θ > θmin. There are two different cases:

• θ∗ < 1 − γ/αFk. In this case, the disease can only be eliminated in Home by setting

θ = 1; Foreign remains in an endemic situation.

• 1− γ/αFk ≥ θ∗ < θ∗min. In this case, global eradication can be achieved with:

θ = 1− αFαHαF − βHβF

· γk− βHβF

(αHαF − βHβF )[(1− θ∗)(αHαF − βHβF )− αH · γ/k]

Now consider transfers:

• If θ∗ < 1− γ/αFk, Home should donate:

θH→F = θ∗min − θ∗ = 1− αH −√βHβF · L

αHαF − βHβF· γk− θ∗

and its cost amounts to:

Lθtrans = Lθmin + θ∗min − θ∗ = L+ 1− LαF + αH − 2√βHβF · L

αHαF − βHβF− θ∗

The net savings are:

Lθ − Lθtrans =LαF + αH − 2

√βHβF · L

αHαF − βHβF+ θ∗ − 1

≥ 0, if 1− LαF + αH − 2

√βHβF · L

αHαF − βHβF· γk≤ θ∗ < 1− γ/αFk

< 0, otherwise

• If 1− γ/αFk ≤ θ∗ < θ∗min, there is always:

Lθ − Lθtrans > 0

35

This is illustrated by Figure 18. Initially at point A, Home can move the equilibrium

to point B by donating θ∗min − θ∗ to Foreign and cutting its own coverage by θ− θmin.

The net savings L(θ− θmin)− (θ∗min − θ∗) is positive, as the slope of AB is larger than

−L. It is not profitable for the smaller Foreign to donate to Home, as illustrated by

the line CB, whose slope is smaller than −L.

Figure 18: Cost differentials: transfer vs. autarky (asymmetricsizes)

To sum up, transfer is profitable from Home’s point of view if:

1− LαF + αH − 2√βHβF · L

αHαF − βHβF· γk≤ θ∗ < 1− αH −

√βHβF · L


According to this condition, I can get the range of γ/k under which it is profitable for Home

to donate even if Foreign contributes nothing (θ∗ = 0):

αHαF − βHβFLαF + αH − 2

√βHβF · L

≤ γ

k<

αHαF − βHβFαH −

√βHβF · L

36

When:

γ

k<

αHαF − βHβFLαF + αH − 2

√βHβF · L

It is better for Home to adopt the “autarkic” solution from the short term point of view if

Foreign does not cooperate. When:

γ

k≥ αHαF − βHβFαH −

√βHβF · L

Either is Foreign a “legitimate free rider” or there is no need for vaccines.

Figure 19 shows the “free lunch line”:

(γ/k)0 ≡αHαF − βHβF

LαF + αH − 2√βHβF · L

When γ/k ≥ (γ/k)c, Foreign can get disease eradication for free: either through Home’s

transfers, or by free-riding (legitimately) on Home’s efforts, or by waiting the disease to die

out naturally. It can be seen that as the size disparity rises, it becomes easier and easier for

the smaller country to get the “free lunch”.

Figure 19: Free lunch line for Foreign (γ/k)0

37

6 Conclusion

In this paper, I combine a Dixit-Stiglitz trade model with an SIS epidemic model to study

how international trade can facilitate the transmission of infectious diseases. I then use this

model to analyze the impact of trade barriers and vaccination programs, with emphasis on

their international spillovers. It is shown that in equilibrium state, the disease’s prevalence

rate first declines and then rises with trade costs. This implies raising trade barriers is

not always an effective way of curbing disease transmission, even if we do not consider the

economic losses it may cause. As for vaccination programs, coordination between different

countries is required to overcome the free rider problem and achieve global eradication.

International transfers, if used properly, can largely facilitate the coordination and reduce the

costs of vaccination programs for both countries’ sake. I thus conclude that countries should

opt for cooperative strategies (coordinated vaccination, transfers) rather than antagonist

strategies (trade barriers) to fight epidemics in the global era.

I also expand the model to asymmetric countries. It is found that the bigger country

always experiences a higher disease prevalence, which poses a greater danger on its smaller

neighbor. The smaller country, nevertheless, is in a better position to benefit from the

international externality of vaccination programs. In some cases, its bigger neighbor may

have incentives subsidize it directly (through transfers) or indirectly (by tolerating its free-

riding behavior). The more disparate the countries sizes, the larger such incentives.

Some moderation should be made to my results. In fact, this paper is just a tentative

exploration; by no means is it comprehensive nor deep-going. One of the major drawbacks

is the lack of dynamic cost-benefit analysis. Moreover, there are a lot of problems on the

modeling choices. For example, I choose to use a simple SIS epidemic model, which may

be not suitable for most real world diseases. Frankly speaking, I sacrificed facticity for

tractability. I thus advice readers to make adaptations to specific situations when using it

to solve real world problems.

Last but not least, I do not consider the elasticity of consumers’ behavior to disease

38

prevalence. In reality, consumers can spontaneously reduce their consumption and other

economic activities to avoid infection, as the economic epidemiology literature suggests.

This spontaneous reaction can be modeled as a response to the rising “shadow price” of

consumption, which is determined by the disease’s severeness and prevalence rate. More

precisely, the more severe the disease and the higher its prevalence rate, the higher the

shadow price of risky consumption, and the more consumers would like to hold back. This

is surely a piste worth exploring, but I prefer to leave it for the future due to the length of

this paper.

39

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41

A Equilibrium

To make the results easy to adapt to different situations, I analyze the following gener-

alized system:

dI

dt= k(X − I)(AI +BI∗)− γI (A.1)

dI∗

dt= k(Y − I∗)(CI∗ +DI)− γI∗ (A.2)

where 0 ≤ I ≤ 1, 0 ≤ I∗ ≤ 1. All parameters, X, Y , A, B, C, D are positive. Moreover,

0 < X ≤ 1, 0 < Y ≥ 1 and AC > BD.

Corollary 1 I = I∗ = 0 is the only equilibrium point on the border of I× I∗ ∈ [0, 1]× [0, 1].

Proof. I = I∗ = 0 is an equilibrium point because:

dI

dt|I=I∗=0 =

dI∗

dt|I=I∗=0 = 0

Then I show that any point on the left border I = 0, 0 < I∗ ≤ 1 is not equilibrium.

According to (A.1):

dI

dt|I=0,0<I∗≤1 = kX ·BI∗ > 0

This is not an equilibrium, because I tends to augment above 0. Similarly, any point on the

lower border I∗ = 0, 0 < I ≤ 1 is not equilibrium neither.

It remains to analyze the right and upper border. For the right border (I = 1, 0 ≤ I∗ ≤

1), there is:

dI

dt= k(X − 1)(A+BI∗)− γ ≤ −γ < 0

I will diminish below 1. In fact, if I replace I = 1 by I = I, ∀I ≥ X, the last inequality

still holds. Therefore, I can exclude the whole area X ≤ I ≤ 1, 0 ≤ I∗ ≤ 1 from equilibrium

analysis. Same for 0 ≤ I ≤ 1, Y ≤ I∗ ≤ 1

42

Proposition 2 I0 = I∗0 = 0 is always an equilibrium point of the dynamic system, and there

may be another equilibrium 0 < I1 < 1,0 < I∗1 < 1 if:

γ

k<

1

2

[AX + CY +

√(AX − CY )2 + 4BDXY

]

otherwise I0 = I∗0 = 0 is the only equilibrium.

Proof. According to Corollary 1, I0 = I∗0 = 0 is the only border equilibrium. Any other

equilibrium, if exists, must be interior, i.e., it must satisfy:

k(X − I)(AI +BI∗)− γI = 0 (A.3)

k(Y − I∗)(CI∗ +DI)− γI∗ = 0 (A.4)

By transforming (A.3) and (A.4), I get:

I∗(I) =1

B

[γ

k(X − I)− A

]I (A.5)

I(I∗) =1

D

[γ

k(Y − I∗)− C

]I∗ (A.6)

These two functions are well defined on 0 ≤ I < X and 0 ≤ I∗ < Y . As for the intervals

X ≤ I ≤ 1 and Y ≤ I∗ ≤, they can be excluded according to Corollary 1.

The question becomes whether (A.5) and (A.6) intersect above zero. There are three

different situations, depending on the values of X, Y , A, B, C, D and γ/k:

1. γ/k ≤ min(AX,CY ). In this case, I∗(I) is a strictly increasing function on I ∈

[X − γkA, X) and:

I∗(X − γ

kA) = 0, lim

I→XI∗(I) = +∞

43

Also, I(I∗) is a strictly increasing function on I∗ ∈ [Y − γkC, Y ) and:

I(Y − γ

kC) = 0, lim

I∗→YI(I∗) = +∞

Therefore, I∗(I) and I(I∗) must have one and only one intersection in [X − γkA, X)×

[Y − γkC, Y ) (Figure 21).

2. min(AX,CY ) < γ/k < max(AX,CY ). Suppose AX < CY . In this case, I∗(I) and

I(I∗) are strictly increasing on I ∈ [0, X) and I∗ ∈ [Y − γkC, Y ). Moreover, there is:

I∗(0) = 0, limI→X

I∗(I) = +∞

I(1− γ

kC) = 0, lim

I∗→YI(I∗) = +∞

The two curves have one and only one intersection in [0, X)× [Y − γkC, Y ) (Figure 22).

3. γ/k ≥ max(AX,CY ). In this case, I∗(I) and I(I∗) are strictly increasing on I ∈ [0, X)

and I∗ ∈ [0, Y ), and:

I∗(0) = 0, limI→X

I∗(I) = +∞

I(0) = 0, limI∗→Y

I(I∗) = +∞

Whether the two curves intersect above zero depends on their slopes at I = I∗ = 0.

• If I∗′(0) < 1/I

′(0), there is one and only one intersection above zero (Figure 23).

Taking the first derivatives of I∗(I) and I(I∗) at I = 0 and I∗ = 0, I get:

1

B

( γ

kX− A

)· 1

D

( γ

kY− C

)< 1

⇒(γk

)2− (AX + CY )

γ

k+ (AC −BD)XY < 0

44

The solution of this quadratic inequality is:

1

2

[AX + CY −


]<γ

k

<1

2

[AX + CY +


]Merging it with the precondition γ/k ≥ max(AX,CY ), I obtain:

max(AX,CY ) ≤ γ

k<

1

2

[AX + CY +


]

• If I∗′(0) ≥ 1/I

′(0), there is no intersection above zero (Figure 24). The merged

condition writes:

γ

k≥ 1

2

[AX + CY +


]

Finally, to sum up, the system has one and only one non-zero equilibrium if:

γ

k<

1

2

[AX + CY +


]

Otherwise (0,0) is the only equilibrium (Figure 20).

Figure 20: Gray bars correspond to the existence of non-zero equi-librium, and the white bar to the non-existence.

45

Figure 21: Intersection (case 1)

Figure 22: Intersection (case 2)

46

Figure 23: Intersection (case 3-1)

Figure 24: No intersection (case 3-2)

47

B Stability

I recopy the dynamic system here for convenience:

dI

dt= k(X − I)(AI +BI∗)− γI

dI∗

dt= k(Y − I∗)(CI∗ +DI)− γI∗

Corollary 3 As long as I and I∗ are constrained to be positive, bifurcation does not make

the zero equilibrium unstable.

Proof. Bifurcation happens when the following two curves:

I∗(I) =1

B

[γ

k(X − I)− A

]I

I(I∗) =1

D

[γ

k(Y − I∗)− C

]I∗

are tangent at the zero point. As shown before, this happens when:

γ/k =(AX + CY +


)/2

Figure 25 shows the phrase graph. It can be seen that (0, 0) is a saddle-node bifurca-

tion. However, the “divergence” occurs in the 3rd quadrant, while I and I∗ are positive by

definition. Therefore, (0, 0) is still stable.

Corollary 4 If the following dynamic system:

dI


dI∗


has a non-zero equilibrium, then this equilibrium must be stable.

48

Figure 25: Bifurcation and stability

Proof. Denote the non-zero equilibrium by I1 and I∗1 (0 < I1 < 1, 0 < I∗1 < 1). They must

satisfy:

k(X − I1)(AI1 +BI∗1 )− γI1 = 0

k(Y − I∗1 )(CI∗1 +DI1)− γI∗1 = 0

These two equations can be rearranged as:

−AkI1 −BkI∗1 + AXk − γ = −BXk · I∗1/I1 (B.1)

−CkI∗1 −DkI1 + CY k − γ = −DY k · I1/I∗1 (B.2)

To study the stability of (I1, I∗1 ), I take the Jabobian matrix:

J1 =

−2AkI1 −BkI∗1 + AXk − γ Bk(X − I1)

Dk(Y − I∗1 ) −2CkI∗1 −DkI1 + CY k − γ

(B.3)

49

Inserting (B.1) and (B.2) into (B.3):

J1 =

−BXk · I∗1/I1 − AkI1 Bk(X − I1)

Dk(Y − I∗1 ) −DY kI1/I∗1 − CkI∗1

To know the signs of the two Eigen values, I need only to look at the signs of their sum

and product, which are respectively the trace and determinant of J1:

Trace(J1) = −BXkI∗1/I1 − AkI1 −DY kI1/I∗1 − CkI∗1 < 0

∆(J1) = (−BXkI∗1/I1 − AkI1)(−DY kI1/I∗1 − CkI∗1 )−BDk2(X − I1)(Y − I∗1 )

= (AC −BD)k2I1I∗1 +BCXk2I∗21 /I1 + ADY k2I21/I

∗1 +BDk2(XI1 + Y I∗1 )

> 0

15 The trace being negative and the determinant positive, the two Eigen values must be

negative real numbers or conjugate-complexes with negative real parts. Therefore, (I1, I∗1 ) is

either a stable node or a stable focus, depending on the sign of Trace(J1)2− 4∆(J1) (Figure

26).

Proposition 5 The dynamic system has one and only one stable equilibrium. If the non-

zero equilibrium I1 > 0, I∗1 > 0 exists, then it is stable. If it does not exist, then the zero

equilibrium is stable.

Proof. First, I analyze the stability of I0 = I∗0 = 0:

J0 =

kAX − γ kBX

kDY kCY − γ

Trace(J0) = k(AX + CY )− 2γ

≥ 0, if

γ

k≤ AX + CY

2

< 0, ifγ

k>AX + CY

2

15AC −BD ≥ 0 because A2 ≥ BD and C2 ≥ BD.

50

Figure 26: Classification of equilibria of a two-dimensional dynamicsystem according to the trace (τ) and the determinant (∆) of theJacobian matrix (provided by Eugene M. Izhikevich)

∆(J0) = (kAX − γ)(kCY − γ)− k2 ·BDXY> 0, if

γ

k>

1

2

[AX + CY +


]or

γ

k<

1

2

[AX + CY −


]≤ 0, otherwise

The stability condition is:

Trace(J0) < 0 & ∆(J0) > 0

⇐⇒ γ

k>

1

2

[AX + CY +


](B.4)

According to Corollary 3, the zero equilibrium is stable when:

γ

k=

1

2

[AX + CY +


]

51

Therefore, the stability condition of zero equilibrium is:

γ

k≥ 1

2

[AX + CY +


]

This is just the condition for the non-existence of (I1, I∗1 ) (Proposition 2). If this condition

is satisfied, I0 = I∗0 = 0 is the only equilibrium and it is stable. If this condition is violated,

the non-zero equilibrium I1 > 0, I∗1 > 0 exists; according to Corollary 4, it is also stable.

C Vaccination

Proposition 6 To make the system:

dI


dI∗


stable at I = I∗ = 0, X and Y must satisfy the following conditions:

AX + CY ≥ γ

k, if AC = BD[

C

AC −BD· γk−X

] [A

AC −BD· γk− Y

]≥ BD

(AC −BD)2· γ

2

k2, if AC > BD

(C.1)

Both terms in brackets on the left hand side of the second equation should be positive.

Proof. According to Proposition 5, I0 = I∗0 = 0 is the only stable equilibrium if:

γ

k≥ 1

2

[AX + CY +


](C.2)

If γ/k ≥ (A + C +√

(A− C)2 + 4BD)/2, this condition holds automatically for X = 1

and Y = 1, which are just their maximum values.

52

If γ/k < (A+C +√

(A− C)2 + 4BD)/2, X and Y need to be smaller than 1 to satisfy

(C.2). To look for their boundaries, I go back to the original stability conditions Trace(J0) <

0 and ∆(J0) ≥ 0:

AX + CY < 2 · γk

(C.3)

(AC −BD)XY − AX · γk− CY · γ

k+(γk

)2≥ 0 (C.4)

(C.4) can take two different shapes on the AX - CY plan, depending on the value of

AC −BD:

• if AC = BD, the term X · Y disappears, and (C.4) becomes a straight line:

AX + CY ≤ γ

k(C.5)

(C.5) is more binding than (C.3). So (C.5) is just the unique stability condition when

AC = BD.

• if AC > BD, (C.4) can be rearranged as:

[AX − AC

AC −BD· γk

] [CY − AC

AC −BD· γk

]≥ ABCD

(AC −BD)2·(γk

)2(C.6)

This is a pair of symmetric hyperbola on the AX - CY plan (Figure 27). The position

of this hyperbola with respect to the straight line Trace(J0) = 0 can be determined in

the following way:

– First calculate the intercept of the tangent line of the upper hyperbola with slope

-1 (the green line). At the tangent point, there is:

AX − AC

AC −BD· γk

= CY − AC

AC −BD· γk

=

√ABCD

AC −BD· γk

⇒ AX + CY = 2 · AC +√ABCD

AC −BD· γk> 2 · γ

k

53

so the upper hyperbola lies above trace(J0) = 0.

– Then calculate the intercept of the tangent line of the lower hyperbola with slope

-1 (the purple line). At the tangent point, there is:

AX − AC

AC −BD· γk

= CY − AC

AC −BD· γk

= −√ABCD

AC −BD· γk

⇒ AX + CY = 2 · AC −√ABCD

AC −BD· γk< 2 · γ

k

so the lower hyperbola lies below trace(J0) = 0.

In sum, the straight line Trace(J0) = 0 lies between the hyperbola ∆(J0) = 0, and the

stability area corresponding to trace(J0) < 0 and ∆(J0) ≥ 0 is just the lower dark

gray area, whose frontier is given by:

[C

AC −BD· γk−X

] [A

AC −BD· γk− Y

]=

BD

(AC −BD)2· γ

2

k2(C.7)

where the two terms in brackets on the left hand side should both be positive.

D Asymmetric countries

D.1 Implicit funtion

First, it is evident that:

F (w;L, τ) =L

L+ (w/τ)σ−1+

τ

L(τw) + (τw)σ

is a strictly decreasing function of w, ∀L, τ . Moreover, I have:

F (0;L, τ)→ +∞, F (+∞;L, τ) = 0

54

Figure 27: Stability area on the AX - CY plan

So there is a unique 0 < w <∞ which satisfies F (w;L, τ) = 1, ∀L, τ .

As long as L ≥ 1 and τ ≥ 1, I have:

F (w;L, τ)|wσ=τσ−1 =L

L+ w−1+

w−1

L+ w2σ−1 ≤ 1

To make F (w;L, τ) = 1, there must be w ≤ τ (σ−1)/σ. The equality is reached when w = τ = 1

Then I take the partial derivative of F on L:

∂F

∂L=

(w/τ)σ−1

(L+ (w/τ)σ−1)2− τ 2w

(L(τw) + (τw)σ)2

Plugging in F (w;L, τ) = 1 and re-arranging, I get:

∂F

∂L=

(w/τ)σ−1(1− wσ/τσ−1)(L+ (w/τ)σ−1)2

≥ 0

55

Therefore:

∂w

∂L= −∂F/∂L

∂F/∂w≥ 0

The equality is reached when when w = τ = 1.

Following the same procedure, I can show ∂w/∂τ ≥ 0. First, w ≤ L1/(2σ−1) because:

F (w;L, τ)|w2σ−1=L =wσ

wσ + τ 1−σ+

w−σ

wσ + τσ−1≤ 1

Then I take the partial derivative of F over τ :

∂F

∂τ=L(σ − 1)wσ−1τ−σ

(L+ (w/τ)σ−1)2− (σ − 1)(wτ)σ

(L(wτ) + (wτ)σ)2

Plugging in F (w;L, τ) = 1 and re-arranging, I get:

∂F

∂τ= (σ − 1)wσ−1τ−σ · L− w2σ−1

(L+ (w/τ)σ−1)2≥ 0

Therefore:

∂w

∂τ= − ∂F/∂τ

∂F/∂w≥ 0

Equality is reached only when w = τ = 1.

D.2 Vaccination

To show θmin ≥ θ∗min, where:

θmin = 1−αF −

√βHβF/L


θ∗min = 1− αH −√βHβF · L


It suffices to show:

αH −√βHβF · L ≥ αF −

√βHβF/L

56

which is equivalent to:

αH − αF√βHβF

≥√L−

√1

L

First, notice that F (w;L, τ) = 1 implies:

αH +τ

L· βF = 1, αF + Lτ · βH = 1

Therefore:

αH − αF = τ

(LβH −

1

LβF

)and:


= τ

(L

√βHβF− 1

L

√βFβH

)Then I would like to show:

βHβF≥ 1

L

which implies:

L

√βHβF− 1

L

√βFβH≥√L−

√1

L

and:


≥ L

√βHβF− 1

L

√βFβH≥√L−

√1

L

Dividing βH by βF , I get:

βHβF

=1

L· L(τw) + (τw)σ

(τ/w) + L(τ/w)σ

Therefore, βH/βF ≥ 1/L is equivalent to:

L(τw) + (τw)σ ≥ (τ/w) + L(τ/w)σ (D.1)

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Re-arrange F (w; τ, L) = 1, I get:

(τ/w) + L(τ/w)σ = Lτ + wσ−1τσ (D.2)

Combining (D.1) and (D.2), I get:

L(τw) + (τw)σ ≥ Lτ + wσ−1τσ

which holds for any w ≥ 1.

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