Infectious diseases and endogenous fluctuations

Econ Theory (2012) 50:125–149DOI 10.1007/s00199-010-0553-y

RESEARCH ARTICLE

Infectious diseases and endogenous fluctuations

Aditya Goenka · Lin Liu

Received: 26 June 2009 / Accepted: 11 June 2010 / Published online: 1 August 2010© Springer-Verlag 2010

Abstract This paper develops a framework to study the economic impact of infec-tious diseases by integrating epidemiological dynamics into a discrete time one sectorgrowth model. An infectious disease with SIS dynamics affects the labor force and theinfected individuals are too ill to work. The susceptible (healthy) individuals choosehow much labor to supply so that the effective labor supply comprises of the proportionof healthy individuals (extensive margin) and their labor supply (intensive margin).The epidemiology of disease transmission is modeled explicitly and the global dynam-ics of the economic variables is studied. Depending on how infectious the disease is,the disease may be eradicated or become endemic. If the disease is infectious enough,cycles and chaos emerge in the economy. A leading example illustrates the model: weshow how the system dynamics change as the parameters vary and how the intensivemargin responds to changes in the extensive margin due to the spread of diseases. Weshow how the fluctuations can be stabilized via disease control methods.

We would like to thank Murali Agastya, Russell Cooper, Roger Farmer, Charles Kahn, Saqib Jafarey,Takashi Kamihigashi, Cuong Le Van, Karl Shell, two anonymous referees of this journal, and seminarparticipants at the NUS Macro Brown Bag Workshop; 2007 Australasian Workshop on MacroeconomicDynamics, Adelaide; 2007 European General Equilibrium Workshop, Warwick; WISE, Xiamen; 2007Australasian Macroeconomics Workshop, Melbourne; 2007 Econophysics Colloquium, Ancona; 2007Winter School, Delhi School of Economics; SWIM 2008, Auckland; University of Paris I; and CityUniversity London for helpful comments and suggestions. The usual disclaimer applies.

A. Goenka (B)Department of Economics, National University of Singapore,AS2, Level 6, 1 Arts Link, Singapore 117570, Singaporee-mail: [email protected]

L. LiuDepartment of Economics, University of Rochester,Harkness Hall, Rochester, NY 14627, USAe-mail: [email protected]

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126 A. Goenka, L. Liu

Keywords Epidemiology · Two-dimensional chaos · Stabilization of chaos ·Infectious diseases · Growth

JEL Classification E32 · I10 · E63 · E13 · D90

1 Introduction

Epidemics of infectious diseases have led to an increasing awareness of the need tostudy their impact on the economy. This paper studies the effect of infectious diseaseson economic variables in a dynamic economic framework by modeling the diseasetransmission explicitly, as in the epidemiology literature. By modeling the dynamicsof disease transmission, new insights on their effects emerge. We show that varyingparameters and looking at steady states can be misleading as the disease dynamicsare a source of non-linearity: as the infectivity of the disease increases the nature ofsteady states change and endogenous fluctuations can emerge.

There is a growing empirical literature on the effects of infectious diseases on eco-nomic variables. This literature tries to measure the effect of diseases on per capitaGDP or economic growth. Some papers find the effect of control of diseases to belarge (Bloom et al. 2009), while others find the effect is modest (Ashraf et al. 2009) orthere might even be an adverse effect due to the dilution effect of a larger populationand increase in dependency ratio (Acemoglu and Robinson 2007; Young 2005). Theunderlying theoretical models in these papers largely look at steady state behavior witha fixed savings rate and exogenous labor supply. As we would expect savings behaviorand labor supply (Thirumurthy et al. 2007) to change in response to changes in diseaseincidence, it is important to incorporate these into the dynamic model. Moreover, thesepapers do not simultaneously model both capital accumulation and the epidemiologi-cal structure of the diseases.1 One of the key insights of the epidemiology literature isthat variations in infectivity change the dynamic properties of diseases. Thus, it is notsufficient to know how steady states change in the economic model as the dynamicproperties of the economy may not be invariant to changes in the disease incidence.We show that the incidence of disease not only has level effects but also can causeeconomic fluctuations.

In order to model the disease transmission explicitly we integrate the epidemiologyliterature (see Anderson and May 1991; Hethcote 2000) into dynamic economic anal-ysis. In this paper, we examine the effect of the canonical epidemiological structure forrecurring diseases—SI S dynamics—in a discrete time growth model. SI S dynam-ics characterize diseases where upon recovery from the disease there is no subsequentimmunity to the disease. This covers many major infectious diseases such as flu, tuber-culosis, malaria, dengue, schistosomiasis, trypanosomiasis (human sleeping sickness),typhoid, meningitis, pneumonia, diarrhoea, acute haemorrhagic conjunctivitis, strepthroat and sexually transmitted diseases (STD) such as gonorrhea, syphilis, etc. (seeAnderson and May 1991).

1 The model in Delfino and Simmons (2000) is an exception but it also uses fixed savings behavior. Gersovitzand Hammer (2004) model disease dynamics but not capital accumulation.

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Infectious diseases and endogenous fluctuations 127

Infectious diseases affect the economy mainly through three channels: laborproductivity (Thirumurthy et al. 2007; Weil 2007), human capital accumulation(Bell et al. 2003; Bleakley 2007) and population size (Kalemli et al. 2000; Young2005). A decrease in the first two will have adverse effects on economic outcomes,but a decrease in the population size may have a positive effect contingent on thedependency ratio through increases in capital per capita. For the diseases mentionedabove the major impact is making infected individuals ill and reducing labor produc-tivity. For several of these, disease related mortality is low for adults. In this paper, weconcentrate on such diseases and focus on the effect of diseases on labor productivity.Thus, we abstract away from disease related mortality and also do not consider anyeffect on human capital accumulation (see Goenka and Liu 2010 for inclusion of learn-ing-by-doing human capital). Focussing on the labor productivity channel enables usto obtain a two-dimensional dynamical system and permits an analysis of the globaldynamics of the economy.

We model labor supply as elastic so that there are variations in both the extensivemargin (number of healthy workers) and the intensive margin (labor supply of anindividual healthy worker). The effect of the disease is modeled as affecting laborproductivity—individuals who have the disease are incapacitated from working. Thisaffects the participation rate—number of workers who are healthy and can work—andthus, affects the dynamics of capital, individual labor supply (number of hours workedby a healthy individual), output, and consumption. Depending on the infectivity of thedisease, there can be a disease-free steady state so that there are no long run effectson the economy; the disease may be endemic and reduce the steady state capital andconsumption; or if the disease is infectious enough, cycles and chaos emerge. We showin a parameterized economy with standard assumptions on preferences, variations inthe extensive margin are partially offset by variations in the intensive margin: if thenumber of healthy workers increases, the labor supply of a healthy worker decreases,but the total number of hours worked increases. The emergence of chaos and cyclesdoes not depend on the parameters of the economy but on the disease dynamics, andthus, non-linear dynamics are possible for a wider range of models and parametersthan otherwise thought (see, e.g. Goenka et al. 1998; Majumdar 1994; Majumdar andMitra 1994; Nishimura et al. 1994).

Furthermore, as there are endogenous fluctuations in the economy, we use one ofthe control methods (Ott et al. 1990) to demonstrate how chaos can be stabilized. Weinterpret this method as either a low efficacy vaccination programme or isolation ofinfective individuals. Since individuals are risk averse, chaos stabilization might havea positive welfare effect through consumption smoothing.

The plan of the paper is as follows. Section 2 develops the disease transmissiondynamics, and Sect. 3 the economic model. The equilibrium dynamics are analyzedin Sect. 4. In Sect. 5 the leading example is given, Sect. 6 contains the discussion ofstabilizing chaos, and Sect. 7 concludes.

2 Disease transmission dynamics

In this paper, we study the canonical deterministic discrete time SI S model (Allen1994) which means individuals can move from the susceptible (healthy and suscep-

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Fig. 1 The transfer diagram forthe SIS epidemiology model

S I

tible to the disease) class to the infective (infected and capable of transmitting thedisease) class and then back to susceptible class upon recovery (see Fig. 1). Contract-ing the disease does not confer immunity against subsequent infections. This modelis applicable to infectious diseases where upon recovery from an infection there isno immunity to a subsequent infection or to diseases which mutate rapidly so thatpeople will be susceptible to a newly mutated strain of the disease even if they haveimmunity to the old one. We abstract away from all the demographics in the modeland assume the population consists of a continuum of individuals of mass N . The totalpopulation is categorized into two classes: the susceptible and the infective based onindividual’s health status. There is homogeneous mixing in the population. Let St andIt denote the number of the susceptible and the infective individuals, respectively, andst = St/N and it = It/N denote the susceptible and infective fractions. Let α be thecontact rate, and γ be the recovery rate. Then α It

N St is the number of new cases perunit time. This is the canonical frequency-dependent incidence or standard incidencemodel. Unlike the mass incidence model where the dynamics governing infectionsis α It St , what is important for transmission is not number of infectives but relativefrequency of infectives in the population. The epidemiology literature tends to use thestandard incidence model as the contact rate seems to be related only very weakly tothe population size for human diseases (Anderson and May 1991). γ It is the numberof people who upon recovery move out of infective class into susceptible class. Hence,the disease dynamics are given by the following difference equations:

St+1 = St + γ It − αIt

NSt

It+1 = It + αIt

NSt − γ It

St , It ≥ 0, ∀ t; S0, I0 > 0 given, with S0 + I0 = N .

Since St + It = N , one of the above equations is redundant. Therefore, in terms ofthe proportion of susceptibles, st , we can simplify the dynamics to:

st+1 = γ (1 − st )+ st (1 − α(1 − st )).

Denote this mapping as st+1 = g2(st ).

Assumption 1 α and γ are constant positive scalars with γ ≤ 1 and α ≤ (1 +√γ )2.

Lemma 1 ∀s0 ∈ [0, 1], 0 ≤ st ≤ 1.

Proof Showing 0 ≤ st ≤ 1,∀t , is equivalent to proving 0 ≤ g2(s) ≤ 1, ∀s ∈ [0, 1].We notice that g2(0) = γ and g2(1) = 1. Notice γ ≤ 1 implies g2

max ≤ 1. Moreover,

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g2min = γ − α

(1−γ−α

2α

)2, when s = − 1−γ−α

2α . Then 0 < γ ≤ 1 and 0 < α ≤(1 + √

γ )2 imply g2min ≥ 0. ��

Lemma 2 The mapping g2 is topologically conjugate to the logistic map xt+1 =p(xt ) = μxt (1 − xt ) with 0 ≤ μ ≤ 4.

Proof Define μ = 1 + α − γ , and the map ψ(x) = α1+α−γ (1 − x). Clearly, ψ is a

homeomorphism. Furthermore, we can verify that p ◦ ψ = ψ ◦ g2.

p ◦ ψ = (1 + α − γ )α

(1 + α − γ )(1 − x)

(1 − α

(1 + α − γ )(1 − x)

)

= α(1 − x)

(1 + α − γ )[1 + α − γ − α(1 − x)]

= α

(1 + α − γ )[1 − γ + γ x − x + αx(1 − x)].

In addition,

ψ ◦ g2 = α

(1 + α − γ )[1 − γ (1 − x)− x(1 − α(1 − x))]

= α

(1 + α − γ )[1 − γ + γ x − x + αx(1 − x)].

Thus, g2 and p are topologically conjugate with μ = 1 + α − γ . In addition, if0 ≤ μ ≤ 4, the mapping p lies entirely in the interval [0,1], which is consistent withst ∈ [0, 1]. ��

As we know, mappings that are topologically conjugate are completely equivalentin terms of their dynamics. Hence, we can deduce the dynamics of st in terms of theparameters α, γ from the well-examined logistic map, as shown in Table 1, withμ = 1 + α− γ . Typically, the map has both stable and unstable orbits, and in Table 1the stable orbits are reported.

Table 1 The dynamics of the logistic map

μ Attractors

0 ≤ μ ≤ 1 x∗ = 0

1 < μ ≤ 3 x∗ = μ−1μ

3 < μ ≤ 1 + √6 x∗

t = μ+1+√(μ−3)(μ+1)2μ , x∗

t+1 = μ+1−√(μ−3)(μ+1)2μ

1 + √6 < μ ≤ μ∞ Cycles of period 2r , r = 2, 3, 4, . . .

μ∞ < μ ≤ 4 Chaotic attractor

μ∞ is the accumulation point of cycles of period 2r (r = 2, 3, 4, . . .) andμ∞ = 3.57 . . .. Source: Weisstein(2009)

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Fig. 2 Dynamics of the SISmodel in the parameter space(α, γ )

Lemma 3 If 0 < α ≤ γ , the disease-free steady state s∗ = 1 is stable; if γ < α ≤γ + 2, there is a stable endemic steady state s∗ = γ

α; If γ + 2 < α ≤ √

6 + γ , there

is a stable cycle of period 2; if√

6 + γ < α ≤ γ + μ∞ − 1, there are stable cyclesof period 2r , r = 2, 3, 4, . . .; if γ + μ∞ − 1 < α ≤ γ + 3, the system is chaotic.

Proof As the map g2 is topologically conjugate to the logistic map p(xt ), from agiven value of μ we can find the corresponding value of α = μ − 1 + γ which willgive rise to the equivalent dynamics. For this value of α the stable orbit can thus bedetermined. The steady states are calculated in the usual way of letting st = g2(st )

[see Ott 2002, Section 2.2 for a discussion of the logistic map, and Weisstein (2009)for the bifurcation points of the logistic map]. ��

Define R0 = α/γ as the basic reproductive rate. In the epidemiology literature, itis the key parameter which determines whether the disease is eradicated in the longrun (when R0 ≤ 1) or becomes endemic (when R0 > 1). However, knowing the valueof R0 is not sufficient to fully understand the dynamics. As we see from Lemma 2,the difference in magnitude of α and γ determines the parameter μ which in turndetermines the dynamics of the system. The dynamics of the SIS model are shown inFig. 2 and this combines the content of Lemmas 1 and 3. The combination of α andγ has to be below the solid line (Lemma 1) and the basic reproductive rate, R0 is aray from the origin. From Lemma 3, below the ray α = γ , in area A there is a stabledisease-free steady state, and above this ray, the disease is endemic. In area B there is astable endemic steady state, in area C there is a stable cycle of period 2, in area D thereare stable cycles of period 2r and in area E the system exhibits chaos. Thus, we cansee that we need sufficiently large γ and α for period-doubling and chaotic behavior.

3 The economy

The economy consists of a one sector growth model with elastic labor supply. Thereare a continuum of individuals of mass N . We assume infective individuals cannot

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work, and susceptible individuals supply their labor elastically.2 Thus, in period t , thelabor supply from infective individuals it is 0, while the labor supply from susceptibleindividuals st is lt ∈ [0, 1]. Therefore, the effective labor supply in each period isst lt ∈ [0, st ]. This can be interpreted as the effective labor supply consisting of boththe extensive margin st and the intensive margin lt . Denote the production function asf (kt , st lt ), with the depreciation rate of capital, δ ∈ (0, 1].Assumption 2 The production function f (k, sl) : �2+ → �+ is a C2 function andsatisfies:

1. f is strictly increasing in both k and sl, and is concave in (k, sl);2. f satisfies Inada conditions, i.e. limk→0 f1 = ∞, limk→∞ f1 = 0 and liml→0 f2 =

∞;3. f (0, sl) = f (k, 0) = 0.

We assume that for each individual the instantaneous utility function is time additiveseparable and depends on consumption c, and leisure, 1 − l.

Assumption 3 The individual utility function u(c, 1 − l) : �+ × [0, 1] → � is a C2

function and satisfies:

1. u is strictly increasing in both c and 1 − l, and strictly concave in (c, 1 − l);2. u satisfies Inada conditions, i.e. limc→0u1 = ∞ and liml→1u2 = ∞;3. u is additively separable in consumption and leisure, i.e. u12 = 0.

Let m(c, 1 − l) denote the marginal rate of substitution between leisure and con-sumption, i.e. m(c, 1 − l) = u2

u1. Additive separability implies that m(c, 1 − l) is

increasing in c.Let the consumption and leisure of the susceptible and infective in period t be

denoted by cst , 1 − ls

t and cit , 1, respectively. In this paper, we study the social plan-

ner’s problem. The objective of the social planner is to maximize the weighted averageof individual utilities which is discounted at the rate β with 0 < β < 1. As the utilityfunction of each individual is additively separable, the optimal allocation will havefull insurance of consumption, i.e. cs

t = cit = ct . Thus, we do not have to keep track

of individual health histories and we can show the results analytically. Denote thewelfare function by U (ct , 1 − lt , st ) = st u(ct , 1 − lt )+ (1 − st )u(ct , 1).

Lemma 4 The social planner’s objective function U (c, 1 − l, s) : �3+ → � is a C2

function and for each fixed s it satisfies:

1. U is strictly increasing in both c and 1 − l, and strictly concave in (c, 1 − l);2. U satisfies Inada conditions, i.e. limc→0U1 = ∞ and liml→1U2 = ∞;3. U is additively separable in consumption and leisure, i.e. U12 = 0.

Proof From the assumption on u, we have

U1 = su1(c, 1 − l)+ (1 − s)u1(c, 1) > 0

U2 = su2(c, 1 − l) > 0

2 Instead of assuming infective individuals cannot work at all, we can alternatively assume they can worka fixed fraction of the labor endowment.

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U11 = su11(c, 1 − l)+ (1 − s)u11(c, 1) < 0

U22 = su22(c, 1 − l) < 0

U12 = U21 = su12(c, 1 − l) = 0

U11U22 − U12U21 > 0.

The Inada conditions also follow. ��Hence, for the social planner’s objective function, the marginal rate of substitution

M(c, 1 − l, s) = U2U1

is also an increasing function of c. This is important to show themonotonicity of the optimal capital stock.

The social planner’s maximization problem is given by:

max{ct ,lt ,kt+1}

∞∑t=0

β tU (ct , 1 − lt , st )

s.t

kt+1 − (1 − δ)kt + ct ≤ f (kt , st lt ), ∀t

st+1 = γ (1 − st )+ st (1 − α(1 − st )), ∀t

0 ≤ lt ≤ 1, ∀t

kt ≥ 0, ct ≥ 0, ∀t; k0 > 0, 0 < s0 < 1 given.

The state variables are kt and st ; the control variables are ct , lt and kt+1.

4 Equilibrium dynamics

Define F(kt , st lt ) = f (kt , st lt )+(1−δ)kt . Since limk→0 F1 = ∞, we have kt > 0,∀t ,if k0 > 0. Since limk→∞F1 < 1 there is a maximum sustainable stock ks withF(ks, s) = ks for each fixed s, and ks is increasing in s by the implicit function the-orem. Hence kmax with F(kmax, 1) = kmax is the maximum sustainable stock for alls ∈ [0, 1]. Therefore, we have (kt , st ) ∈ X = [0, kmax] × [0, 1] ⊂ �2 and the set Xis closed and bounded.

We use standard dynamic programming methods and the Bellman equation for theplanning problem is:

V (kt , st ) = max{ct ,lt ,kt+1}[U (ct , 1 − lt , st )+ βV (kt+1, st+1)

]

s.t

kt+1 + ct ≤ F(kt , st lt ), ∀t

st+1 = γ (1 − st )+ st (1 − α(1 − st )), ∀t

0 ≤ lt ≤ 1, ∀t

kt ≥ 0, ct ≥ 0, ∀t; k0 > 0, 0 < s0 < 1 given.

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We can isolate the choice of consumption and labor supply from the investment choiceby decomposing the planning problem into a static problem and a dynamic problem(see for example, Aiyagari et al. 1992). The static problem is given by:

W (k, k′, s) = max{c,l}∈B(k,k′,s)

U (c, 1 − l, s)

where

B(k, k′, s) = {(c, l) ∈ �2+ : 0 ≤ l ≤ 1, c + k′ ≤ F(k, sl)}.

Lemma 5 W is continuous and bounded, and the maximizers c(k, k′, s) and l(k, k′, s)are continuous functions; W is strictly concave in (k, k′) for each fixed s; W is con-tinuously differentiable and W1 > 0,W2 < 0,W12 > 0,W22 < 0 for each fixed s.

Proof See Appendix. ��Based on the properties of function W above, now we turn to the dynamic problem,

which is given by:

V (kt , st ) = max{kt+1,st+1}∈A(kt ,st )

[W (kt , kt+1, st )+ βV (kt+1, st+1)

]

where

A(kt , st )={(kt+1, st+1)∈�2+ : kt+1 ≤ F(kt , st ), st+1 =γ (1−st )+st (1−α(1−st ))}.

Let the optimal capital investment kt+1 = g1(kt , st ). Note that st+1 is governed bythe mapping g2(st ).

Lemma 6 V is continuously differentiable. For each fixed s, V is strictly concave inkt and g1 is a continuous function.

Proof See Appendix. ��Given (k0, s0), the sequence (kt , st )

∞t=0 defined by (kt+1, st+1) = (g1(kt , st ),

g2(st )) is the unique solution and satisfies

W2(kt , kt+1, st )+ βV1(kt+1, st+1) = 0

V1(kt , st ) = W1(kt , kt+1, st )

st+1 = g2(st ) = γ (1 − st )+ st (1 − α(1 − st ))

limt→∞β

t W1(kt , kt+1, st )kt = 0

(1)

The dynamical system we want to study has two state variables kt , st . Therefore,the dynamical system is given by the pair (X, g), where X = {[0, kmax] × [0, 1]} isthe state space and g = (g1, g2) the law of motion is a two-dimensional continuousmapping.

Lemma 7 The sequence {kt }∞t=0 is strictly monotonic for every fixed s.

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Proof From Eq. (1), we have W2(kt , g1(kt , st ), st )+ βV1(g1(kt , st ), g2(st )) = 0 bysubstitution into the policy function. By the implicit function theorem, g1 is differ-entiable in k for fixed s and by the chain rule, W21 + W22g1

1 + βV11g11 = 0. Thus,

g11 = − W21

W22+βV11. Since W21 > 0,W22 < 0 and V11 < 0, we have g1

1(k, s) > 0,

i.e. g1 is strictly increasing in the first argument. Hence, for every fixed s, if k1 >

k0, k2 = g1(k1, s) > g1(k0, s) = k1. By induction kt+1 > kt , ∀t . Similarly ifk1 < k0, kt+1 < kt , ∀t , and if k1 = k0, kt+1 = kt , ∀t . Thus, the sequence {kt }∞t=0 isstrictly monotonic for every fixed s. ��

4.1 Fixed points

Definition 1 Let � be a set and � ⊂ �n . The map h : � → � is said to have a fixedpoint, x∗, if h(x∗) = x∗.

Proposition 1 If 0 < α ≤ γ + 2, the system achieves a unique stable fixed point(k∗, s∗) ∈ X.

Proof From Lemma 3, if 0 < α ≤ γ , the sequence {st }∞t=0 will converge to s∗ = 1and if γ < α ≤ γ + 2 the sequence {st }∞t=0 will converge to s∗ = γ

α< 1 when t

is sufficiently large. Moreover, for fixed s∗ the sequence {kt }∞t=0 is strictly monotonic(Lemma 7) and bounded (as kt ∈ [0, kmax]). By the monotone convergence theorem(MCT) {kt }∞t=0 converges to some fixed point k∗. Due to the uniqueness of the limitand continuity of the mapping g, s∗ = g2(s∗) and k∗ = g1(k∗, s∗). In addition, as theset X is closed, (k∗, s∗) ∈ X . ��

4.2 Cycles

Definition 2 Let � be a set and � ⊂ �n . The map h : � → � is said to have cyclesof period r > 1 if hr (x) = x .3

Proposition 2 If γ + 2 < α ≤ γ + μ∞ − 1, the system has cycles of period 2r , r =1, 2, 3, . . .. Moreover, for each α, there is a unique periodic orbit which is attracting.

Proof Consider the case of r = 1 first. If γ + 2 < α ≤ γ + √6, from Lemma 3, the

sequence {st }∞t=0 has a unique attracting cycle of period 2. So for any t ≥ T, {st }∞t=0takes value from the set O(g2) = {s1, s2}, and we substitute the value of st into themapping g1 for t > T :

kT +1 = g1(kT , s1)

kT +2 = g1(kT +1, s2) = g1(g1(kT , s1), s2)

kT +3 = g1(kT +2, s1) = g1(g1(kT +1, s2), s1)

kT +4 = g1(kT +3, s2) = g1(g1(kT +2, s1), s2)

kT +5 = g1(kT +4, s1) = g1(g1(kT +3, s2), s1)

· · ·3 Note hr (x) = h[hr−1(x)].

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We define a new sequence {kT +2 j }∞j=0 = {kT , kT +2, kT +4, . . .} with kT +2( j+1) =g1(g1(kT +2 j , s1), s2). If kT < kT +2, g1(kT , s1) < g1(kT +2, s1) since g1(k, s1)

is increasing in k for fixed s1, and thus, by the same reasoning kT +2 =g1(g1(kT , s1), s2) < g1(g1(kT +2, s1), s2) = kT +4. Hence, by induction kT +2 j <

kT +2( j+1), ∀ j . Similarly, if kT > kT +2 we have kT +2 j > kT +2( j+1), ∀ j , and if kT =kT +2 we have kT +2 j = kT +2( j+1), ∀ j . Hence, the sequence {kT +2 j }∞j=0 is strictlymonotonic. Moreover, since {kT +2 j }∞j=0 is a subsequence of {kt }∞t=0, {kT +2 j }∞j=0is bounded. By the MCT, {kT +2 j }∞j=0 will converge to some fixed point k∗

1 . By

the uniqueness of the limit and continuity of the mapping g1(g1(k, s1), s2), k∗1 is

determined by k∗1 = g1(g1(k∗

1 , s1), s2) and k∗1 ∈ [0, kmax]. Similarly, we can

define the sequence {kT +1+2 j }∞j=0 = {kT +1, kT +3, kT +5, . . .} with kT +1+2( j+1) =g1(g1(kT +1+2 j , s2), s1). This sequence, {kT +1+2 j }∞j=0, will converge to some fixed

point, k∗2 , determined by k∗

2 = g1(g1(k∗2 , s2), s1) and k∗

2 ∈ [0, kmax].Therefore, by construction we decompose the sequence {kT + j }∞j=0 into two sub-

sequences, {kT +2 j }∞j=0 consisting all the even terms and converging to k∗1 , and

{kT +1+2 j }∞j=0 consisting all the odd terms and converging to k∗2 . Thus, {kT + j }∞j=0

will fluctuate between k∗1 and k∗

2 for sufficient large j . Moreover {kT + j }∞j=0 is thetail of the sequence {kt }∞t=0, so {kt }∞t=0 has a cycle of period 2 eventually. Hence, thesystem has a cycle of period 2, which is attracting.

If γ + √6 < α ≤ γ + u∞ − 1, from Lemma 3 the sequence {st }∞t=0 has a cycle

of period 2r , r = 2, 3, 4, . . .. Moreover for each α, there is a unique attracting peri-odic orbit. Following the same argument above we can show the sequence {kt }∞t=0has a cycle of the same period as {st }∞t=0. Hence, the system has a cycle of period2r , r = 1, 2, 3, 4, . . .. ��

4.3 Chaos

A key feature of chaotic orbits is the sensitive dependence on initial conditions—twoorbits of nearby initial points diverge as they are iterated over time. The Lyapunovexponents are a way to study the divergence of orbits on an attractor and other invariantsets. For uni-dimensional systems if the Lyapunov exponent is positive, orbits from twoinfinitesimally close initial conditions (on the real line) will diverge exponentially overtime and there is sensitive dependence on initial conditions. For higher dimensionalsystems, the local behavior of the system may depend on the direction: nearby pointsmay be moving apart in one direction but moving together in another. The definitionof higher dimensional chaos that we use (see Alligood et al. 1997, Ch. 5) is equippedto deal with such issues. This definition also relies on the Lyapunov exponent. Notethat while it is well known that g2 on its own exhibits chaos (Ott 2002, Section 2.2) asit is topologically conjugate to the logistic map and has a positive Lyapunov exponent,we need to study the behavior of the entire system, i.e. g = (g1, g2).

Definition 3 Let h be a smooth map on� ⊂ �n , and J T = DhT (x0) be the Jacobianof hT (x) evaluated at an orbit starting from the initial condition x0. Let λT

q (q =1, . . . , n) be the eigenvalues of J T . The qth Lyapunov number of x0 is defined by

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Lq = limT →∞(λTq )

1/T if this limit exists. The qth Lyapunov exponent of x0 is

q = limT →∞

1

Tln

∣∣∣λTq

∣∣∣ .

The set of all Lyapunov exponents = {1, 2, . . . , n} is called the Lyapunovspectrum of a dynamical system. Using the concept of Lyapunov exponents, we cangive the definition of chaotic orbits of higher dimensional maps (Alligood et al. 1997,p. 196). The conditions 1 and 3 below are standard and hold for the conventionaldefinitions of one-dimensional chaos. Condition 2 is to rule out quasi-periodic orbitswhich are predictable.

Definition 4 Let h be a map on � ⊂ �n , and let {x0, x1, . . .} be a bounded orbit ofh. The orbit exhibits multi-dimensional chaos if

1. It is not asymptotically periodic,2. No Lyapunov exponent is exactly zero, and3. At least one Lyapunov exponent is positive.

Proposition 3 If γ + μ∞ − 1 < α < γ + 3, the dynamical system exhibits multi-dimensional chaos.

Proof Firstly, the orbit O(g) = {(k0, s0), (k1, s1), . . .} is bounded since (kt , st ) ∈ X .Secondly, it is a well-known result that beyond the accumulation point of cycles ofperiod 2r , that is μ∞ < μ ≤ 4 (μ∞ = 3.570 . . .), the logistic map p has a chaoticattractor with an infinite number of embedded unstable periodic orbits. Since the map-ping g2 and p are topologically conjugate and the Lyapunov exponent is invariant withrespect to topological conjugacy, g2 has the same Lyapunov exponent as the logisticmap, and which is positive when γ+μ∞−1 < α < γ+3 (from Lemma 3). It turns outg2

, the Lyapunov exponent of g2, is one of the Lyapunov exponent of mapping g aswell, that is,g

2 = limT →∞ 1T ln |λT

2 | = g2> 0 because the Jacobian of the system is

given by J = ( g11 g1

20 g2

2

). The second Lyapunov exponent is g


1 |.Now λT

1 = ∏Tt=1 |g1

1(kt , st )|. For this Lyapunov exponent to be exactly equal to zero,it must be that g1

1(kt , st ) = 1, ∀t. As g11(kt , st ) = − W21

W22+βV11, we can see that this

will not be true in general. Thirdly, the orbit O(g2) is not asymptotically periodic asthe dynamics are governed by the well-known logistic map. Thus, the orbit O(g) isnot asymptotically periodic as well. Therefore, the system exhibits multi-dimensionalchaos. Note, we exclude the case α = γ + 3 which is equivalent to μ = 4 in thelogistic map, as when μ = 4 the orbit is dense in [0, 1]. Hence, st maybe equal to zeroand the economy shuts down. ��

In summary, the monotonicity of capital accumulation (for fixed s) implies that thedynamics of the system are fully determined by the disease transmission dynamics.Non-linearity in the disease dynamics can induce fluctuations in the economy. Thedynamics of the system in the parameter space are summarized in Fig. 2 and we find

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that sufficiently large contact rate and recovery rate can generate complex dynamicsin the economy.

Intuitively, when the contact rate is low (relative to the recovery rate), that is 0 <α ≤ γ , the inflow into the infective class is small while the outflow is relativelylarger. Thus, the number of infective people is decreasing which leads to an evensmaller inflow as less susceptible people are infected. Therefore, the number of infec-tive people will keep declining and eventually becomes zero. This is the disease-freesteady state, and is the same as the one sector neo-classical growth model where thedisease is absent. When the contact rate is not as low, that is γ < α ≤ γ + 2, theinflow into the infective class is larger than the outflow and the system reaches the sta-ble steady state with a fixed proportion of infective people. This is called the endemicor disease-persistent steady state and has a lower level of physical capital stock andeconomic performance compared with the disease-free case. This can easily be shownby the comparative statics of the steady states. The larger the proportion of healthypeople, the higher the capital per capita. The disease-free and endemic steady stateshave been the focus in the economic literature. As we abstract away from change inpopulation size, the disease prevalence has an adverse effect on the economy throughits impact on labor productivity.

When both the contact rate and recovery rate are high, cycles and chaos may emergein the economy and looking at steady states is no longer justified as the economy mightnot even have a locally stable steady state. If the contact rate is high, a large numberof susceptible people will get infected and move into the infective class. Thus, thenumber of susceptible people will decline and the number of infective people willrise. Next period many people will move from the infective class to the susceptibleclass as a result of recovery from the disease. Thus, the number of susceptible peoplewill increase while the number of infective people will drop. Therefore, the numberof people in each class will fluctuate sharply, and eventually they may settle down toa cycle of period 2r , r = 1, 2, 3, 4, . . . or exhibit chaotic behavior. Which situation isrealized depends on the relative rate of flows in and out of the classes, that is on themagnitudes of α and γ . Moreover, the oscillation in the number of susceptible peoplecauses the fluctuations of the capital stock, and consequently, causes endogenous eco-nomic fluctuations.

5 The leading example

In this section we give the leading example of a parameterized economy by introducingfunctional forms for the utility function and production function so that a closed formsolution can be derived and the effects of stabilization policy can be studied. As before,we assume each individual’s utility is separable in consumption and leisure. Due to fullinsurance, the consumption is the same across the health status. The utility functionsfor the infective and the susceptible, respectively, are ui (ct , 1 − lt ) = log(ct )+ v(1)and us(ct , 1− lt ) = log(ct )+v(1− lt )where v is a twice continuously differentiable,strictly increasing and concave function. Normalize v(1) = 0. Thus, the social plan-ner’s objective is U (ct , 1−lt , st ) = st (log(ct )+v(1−lt ))+(1−st )(log(ct )+v(1)) =log(ct )+ stv(1 − lt ). As in the literature, we adopt a Cobb–Douglas production func-

123


tion yt = F(kt , st lt ) = Akσt (st lt )1−σ , 0 < σ < 1, with full depreciation of capital.The social planner’s problem is:

V (kt , st ) = max{ct ,lt ,kt+1}[log(ct )+ stv(1 − lt )+ βV (kt+1, st+1)

]

s.t.

ct + kt+1 ≤ F(kt , st lt ), ∀t

st+1 = γ (1 − st )+ st (1 − α(1 − st )), ∀t

0 ≤ lt ≤ 1, ∀t.

Then the first order conditions are:

1

ct− βV1(kt+1, st+1) = 0 (2)

−stv′(1 − lt )+ βV1(kt+1, st+1)F2(kt , st lt )st = 0 (3)

V1(kt , st ) = βV1(kt+1, st+1)F1(kt , st lt ). (4)

Combining Eqs. (2) and (4), we have

1

ct= β

1

ct+1F1(kt+1, st+1lt+1) = σβ

1

ct+1

yt+1

kt+1= σβ

1

ct+1

yt+1

yt − ct. (5)

That is ytct

= σβyt+1ct+1

+ 1. By iteration we have ct = (1 − σβ)yt and kt+1 = σβyt .

Combining Eqs. (2) and (3), we have stv′(1− lt ) = 1

ctF2(kt , st lt )st = 1

ct(1−σ) yt

lt,

that is lt stv′(1 − lt ) = 1−σ

1−σβ . We see that lt depends only on st , and by the implicit

function theorem, there is a unique lt with ∂lt∂st

= − ltv′(1−lt )st [v′(1−lt )−ltv′′(1−lt )] < 0. Let effec-

tive labor supply nt = st lt , and ∂nt∂st

= lt + st∂lt∂st

= l2t v

′′(1−lt )ltv′′(1−lt )−v′(1−lt )

> 0. Thus, asthe proportion of susceptible people increases the individual labor supply decreasesbut the total effective labor increases, that is the change in intensive margin partiallyoffsets the change in extensive margin.

For the purpose of simulation we assume v(1 − lt ) = ϕ log(1 − lt ) with ϕ > 0.Then, the labor supply is

lt = 1 − σ

(1 − σβ)ϕst + 1 − σ, (6)

and effective labor is:

nt = (1 − σ)st

(1 − σβ)ϕst + 1 − σ. (7)

123


Lemma 8 In the parameterized economy, there exists a unique optimal solution{(kt , st )}∞0 satisfying

kt+1 = g1(kt , st ) = σβAkσt

((1 − σ)st

(1 − σβ)ϕst + 1 − σ

)1−σ

st+1 = g2(kt , st ) = γ (1 − st )+ st (1 − α(1 − st )),

with initial point (k0, s0).

Proof Omitted. ��

Proposition 4 If 0 < α ≤ γ , the system achieves the disease-free steady state

(k∗, s∗) =((βσ A)

11−σ

(1−σ

(1−σβ)ϕ+1−σ), 1

). If γ < α ≤ γ + 2, the system achieves

the endemic steady state (k∗, s∗) =((βσ A)

11−σ

((1−σ)γ /α

(1−σβ)ϕγ /α+1−σ),γα

). If γ + 2 <

α ≤ γ+u∞−1, the system has cycles of period 2r , r = 1, 2, 3, 4, . . .. If γ+μ∞−1 <α < γ + 3, the system is chaotic.

Proof Most results follow from the propositions in the previous section. The only thingthat is shown here is g


1 | < 0. This not only gives an explicitcomputation of the Lyapunov exponent, but also confirms Proposition 3 on the exis-tence of chaos. We note the eigenvalues of the system are λ1 = βσ 2 Akσ−1n1−σand λ2 = 1 − α − γ + 2αs. Suppose the trajectory of mapping g is O(g) ={(k0, s0), (k1, s1), . . . , (kT −1, sT −1)}. Then one of the eigenvalues of the matrixJ T is

λT1 = βσ 2 Akσ−1

0 n1−σ0 · βσ 2 Akσ−1

1 n1−σ1 . . . βσ 2 Akσ−1

T −1n1−σT −1

= (βσ 2 A)T(

n0n1 . . . nT −1

k0k1 . . . kT −1

)1−σ(8)

Furthermore, by the mapping g1, we have

k1 = βσ Akσ0 n1−σ0

k2 = βσ Akσ1 n1−σ1

· · ·kT −1 = βσ AkσT −2n1−σ

T −2.

Multiplying the above equations together and rearranging, we get

(n0n1 . . . nT −2)1−σ = kT −1

kσ0(k1 . . . kT −2)

1−σ (βaσ)−T .

123


Substituting this into the Eq. (8), and we have

g1 = lim

T →∞1

Tln |λT

1 |

= limT →∞

1

Tln

∣∣∣σ T kσ−20 kσT −1n1−σ

T −1

∣∣∣

= limT →∞

[ln σ + (σ − 2)

ln k0

T+ σ

ln kT −1

T+ (1 − σ)

ln nT −1

T

]

≤ limT →∞

[ln σ + (σ − 2)

ln k0

T+ σ

ln kmax

T

]

= ln σ < 0.

��As the model is abstract, the goal of simulation in this paper is not to match any

observed data, but rather to show the importance of examining dynamics, i.e. how thedynamics change as we vary the epidemiology parameters. The parameter values arechosen by following economic literature conventions: A = 1, β = 0.99, σ = 0.36and ϕ = 1. For epidemiological parameter values, the literature primarily estimatesR0 = α

γ(see Sect. 2) as this determines whether a disease becomes endemic or not.

This can vary considerably across diseases and even for the same disease across dif-ferent populations (Dietz 1993). The R0 for some diseases are: 1.2–7.5 for the 1918strain of influenza (Mills et al. 2004; Vynncky et al. 2007); H1N1 1.4–3.2 (ECDC2009); measles 15–20 (Griffin 2001), etc. We know from Lemmas 2 and 3 that it is thedifference between α and γ (i.e. μ = 1 +α− γ ) that is important for the dynamics ofthe system and knowing R0 is not sufficient for us to pin down the dynamics. Thus,we hold γ constant and vary α as in a sense we are doing a sensitivity analysis of howthe dynamics of the economy change as μ is varied. We select γ = 1 arbitrarily sinceit varies a lot among different infectious diseases and γ = 1 implies a high recoveryrate and generates the full range of dynamics.

We use the bifurcation diagram (Fig. 3) to illustrate changes in the state variablesas the infectivity of the disease increases. We plot the bifurcation diagram (secondpanel for st and third panel for kt ) and one of the Lyapunov exponents (first panel)against the parameter α with initial conditions k0 = 0.1 and s0 = 0.3. By Lemma 1,α lies in the region (0, 4). Let α vary from 2.9 to 4 by α = 0.01 (Note for α ≤ 2.9there is only a steady state and hence, the range 0 < α ≤ 2.9 is suppressed.) We cansee on the first panel that as α increases the Lyapunov exponent remains negative, andwhen α > 3.57 . . . it can become positive. Correspondingly on the second panel asα increases the steady state of the susceptibles bifurcates into a 2-period cycle, thena 4-period cycle, etc. and when α > 3.57 . . . the orbit is chaotic. In the third panel,capital stock exhibits same pattern of dynamics as the susceptibles. This is becausewe have shown in Lemma 7 the dynamics of kt are monotonic for a given s, and inPropositions 1–3 that the kt dynamics inherit properties of st dynamics.

We also show the attractors (stable orbits) for the susceptibles, investment, con-sumption, and effective labor supply when α = 3.4, α = 3.5 and α = 3.9 in Fig. 4.These illustrate how for different values of α, the control variables vary in the long

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3 3.2 3.4 3.6 3.8 4−2

−1

0

1

Γ2g

3 3.2 3.4 3.6 3.8 40

0.02

0.04

0.06

0.08

α

kt

3 3.2 3.4 3.6 3.8 40

0.5

1

st

Fig. 3 Lyapunov exponent and bifurcation diagrams for susceptibles and capital

run with the variation in st (the x-axis). The values of α are chosen according toProposition 4 so that for α = 3.4 there is a 2-period cycle; when α = 3.5 there is a4-period cycle and when α = 3.9 there is chaos. We observe that there are positiverelationships between st and investment as well as st and consumption. The mech-anism is that an increase in st acts like an increase in productivity. As productivityincreases so does investment, and consumption (as it is a normal good). From Eq. 6we can see that there is a negative relationship between st and individual labor supply(as leisure is normal), but from Eq. 7 a positive relationship between st and effectivelabor supply. As st increases, there is an increase in the extensive margin, and thisallows the planner to compensate for disutility of labor by decreasing the intensivemargin. In this framework, the “productivity shock” is deterministic and this allowsthe planner to adjust the capital stock in response to changes in st . The labor-leisurechoice leads to changes in intensive margin of labor supply and affects the level of kt

but not the dynamics.

6 Stabilization of the economy

Since the economic fluctuations essentially come from the oscillation of disease preva-lence, stabilizing the disease prevalence will stabilize the fluctuations in the economy.

123


0 0.5 10

0.02

0.04

0.06

0.08

st

kt+1

(c) α=3.9

0 0.5 10

0.05

0.1

st

ct

0 0.5 10

0.2

0.4

0.6

0.8

st

nt

0 0.5 10

0.02

0.04

0.06

0.08

st

kt+1

(a) α=3.4

0 0.5 10

0.05

0.1

st

ct

0 0.5 10

0.2

0.4

0.6

0.8

st

nt

0 0.5 10

0.02

0.04

0.06

0.08

st

kt+1

(b) α=3.5

0 0.5 10

0.05

0.1

st

ct

0 0.5 10

0.2

0.4

0.6

0.8

st

nt

Fig. 4 Chaotic attractor of investment, consumption, effective labor supply and susceptibles

This will presumably eliminate volatility in consumption, and as agents have convexpreferences this may lead to an increase in overall welfare. In this paper we con-sider “local” stabilization of the disease, that is the inherent dynamics of the economyare not changed but the stabilization is to a given trajectory of the dynamics. Thisis different from optimal disease control methods which would change the inherentdynamics of the disease (see Goenka et al. 2010 for such a model). Two methods areconsidered—vaccination and isolation of infective individuals. These methods are themost commonly discussed forms of disease control and are likely to be of low costas they perturb the dynamics only for some periods of time. It turns out that both areisomorphic to the Ott et al. (1990) (OGY) method in control theory.

The OGY method is used to stabilize a chosen unstable periodic orbit embedded inthe chaotic attractor by applying a time dependent parametric feedback control. Thereare typically an infinite number of unstable periodic orbits in a chaotic system, and eachof them gives rise to different dynamics. Thus, by selecting a given orbit, we can in prin-ciple change the behavior of the system via unstable orbits in the neighborhood of thegiven orbit to meet a variety of different goals. This method is based on the observationthat there always exist stable manifolds around periodic points. Thus, we can adjust theparameters to push the state close to the stable manifold of the desired periodic point.

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Vaccination is a widely discussed disease control strategy and for several diseaseseffective vaccines exist, e.g. mumps, measles, rubella, poliomyelitis, for others eitherthey do not exist, e.g. gonorrhea, malaria, etc. or are of limited efficacy, e.g. influenza.For example, an influenza pandemic results when antigenic shift leads to a new highlyvirulent influenza subtype for which there is little or no immunity in the population.Thus, a vaccine against a particular strain of influenza may have no effect in preventinginfection against the new strain of influenza. For diseases with SIS dynamics, effectivevaccines generally do not exist. Thus, we consider a vaccine which is imperfect so thata person who is vaccinated may not have full immunity. The vaccine has effectivenessεt so that the susceptibility is now αt = (1 − εt )α (Bowman and Gumel 2006). Thevaccine is not very effective—or εt is very low εt ∈ [0, ε], so that α is perturbed onlyby a small amount.

Another important method of disease control is isolation which applies to peoplewho are known to be ill with contagious diseases. Isolation prevents susceptibles fromcoming in contact with the infectives, and thus, prevents transmission of the disease.Given that the cost of the stabilization should be small, we assume at each time periodt, εt proportion of the infective people It are isolated. Thus, the susceptible people St

come into contact with only It (1 − εt ) infective people and the number of new casesper unit time, according to frequency-dependent incidence, is α It (1−εt )

N St . Owing tothe limited resource constraint, εt lies in the range [0, ε].

Thus, under the two above disease controlling methods, the disease dynamics arenow:

St+1 = St + γ It − αIt (1 − εt )

NSt

It+1 = It + αIt (1 − εt )

NSt − γ It ,

and the dynamics of the susceptible population is now given by:

st+1 = γ (1 − st )+ st (1 − αt (1 − st )), αt ≡ α(1 − εt ). (9)

The question of what should be the efficacy of the vaccine or alternatively howmany people should be isolated can be addressed by the OGY method. Assume atwo-dimensional system given by xt+1 = g(xt ;α), where xt = (kt , st ) and α is theaccessible parameter. We change the parameter around its nominal value α0 withinthe range α ≤ αt ≤ α0. We observe the system has an unstable fixed point x∗(α0) =(k∗(α0), s∗(α0)), which has one stable manifold and one unstable manifold. We waittill the state of the system falls within a specified region around the desired fixed point.Once the trajectory falls into the region, we modifyαt so as to bring xt+1 close to the sta-ble manifold of the unperturbed map by αt = αt −α0 = − λ2

v2·wv2 · (xt − x∗), where

w = ( ∂g∂α

)(x=x∗,α=α0)

, λ2 is the eigenvalue of the Jacobian J = ( ∂g∂x

)(x=x∗,α=α0)

with modulus greater than one, and v2 is the contravariant basis of the correspondingeigenvector e2.4 If there is explicit knowledge of the map, the control equation can be

4 v2 is determined by v2 · e1 = 0 and v2 · e2 = 1, where e1 and e2 are the eigenvectors of the matrix J .

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50 100 150 2003.895

3.9

3.905

αt

t50 100 150 200

0

0.5

1

st

t

50 100 150 2000

0.02

0.04

0.06

0.08

0.1

kt

t50 100 150 200

0

0.05

0.1

ct

t

Fig. 5 Stabilization of fluctuations via the OGY method

determined straightforwardly as above. However, no model of dynamics is required forOGY method because the information needed to apply OGY method can be obtainedfrom the trajectory as well.

Based on the dynamics given by the leading example, we choose the nominal valueα0 = 3.9, αt ∈ [α0 − 5%, α0] or εt ∈ [0, 1.28%]. We want to stabilize the chaotictrajectory to the fixed point which in this case is (k∗(α0), s∗(α0)) = (0.041, 0.2564).Then the equation for controlling is given by: αt = αt − α0 = −9.96517(st −0.2564). That is, if the trajectory falls into the specified region, control is switchedon with αt = α − 9.96517(st − 0.2564), and αt = α0 otherwise. Or equivalentlycontrol is switched on with εt = 1 − αt/α0 = 9.96517(st − 0.2564)/α0, and εt = 0otherwise. Thus, by using the OGY stabilization method, we are able to control chaosto the fixed point as shown in Fig. 5.

We report here simulation results of total welfare along a trajectory, ϒ =∑t=T −1t=0 β tU (ct , 1 − lt , st ) of T = 500 periods.5 We see that the system is stabi-

lized in around 100 periods. The total welfare along a trajectory without and withthe stabilization policy are ϒ = −316.867 and ϒ ′ = −315.224, respectively. As weare evaluating welfare along trajectories from the initial conditions, the transitionaldynamics are taken into account. We see there is a gain to welfare from stabilizingthe economy. We also calculate the welfare measure due to Lucas (1987) which is thepercentage increase (κ) in the original consumption stream that will generate the sameutility as in the stabilized stream. Let ct , lt , st and c′

t , l ′t , s′t be the consumption-labor

streams before and after stabilization, respectively.

5 If we further increase T there is little change in welfare due to discounting.

123


t=T −1∑t=0

β t [log((1 + κ)ct )+ st log(1 − lt )] =t=T −1∑

t=0

β t [log(c′t )+ s′

t log(1 − l ′t )]

t=T −1∑t=0

β t log(1 + κ) =t=T −1∑

t=0

β t [log(c′t )+ s′

t log(1 − l ′t )]

−t=T −1∑

t=0

β t [log(ct )+ st log(1 − lt )]

= ϒ ′ − ϒ

⇒ κ = exp((ϒ ′ −ϒ)/(

t=−1∑t=0

β t ))− 1

Then this welfare gain by stabilization is equivalent to 1.67% increase in consumptionstream.

7 Conclusion

This paper integrates SIS dynamics into a growth model with elastic labor supply.Thus, it provides a framework to study the effects of an infectious disease on thedynamics of economic variables. Firstly, diseases adversely affect the economy byincapacitating people from working. They act as a “productivity shock” and if oneknows the dynamics of the disease, one can predict their effect over time. Secondly,doing comparative statics of steady states by varying infectivity of the disease can bemisleading as cycles and chaos can emerge if the disease becomes infective enough.Thirdly, if the dynamics of the disease is well understood then one can potentiallymeasure the economic cost of their outbreak. Fourthly, we see how the traditionalmethods of vaccination and isolation can stabilize the disease fluctuations as they areisomorphic to the OGY method in control theory. Finally, we see that even if the dis-ease remains endemic and is not eradicated, there may be substantial gains in welfarefrom stabilizing it.

In this paper the disease transmission is exogenous in that the parameters do notdepend on economic variables. One way to interpret the model is a developing econ-omy with no effective public health policies in place. The difference in economic andpublic health institutions determines the epidemiology parameters, but in the absenceof an effective policy, these are largely exogenous. In another paper, Goenka et al.(2010) we examine the continuous time formulation of a similar model. In that paperthere are two types of capital—physical capital and health capital. Increases in the lat-ter reduces disease transmission and increases the recovery rate from the disease. Thus,public policy directly affects the epidemiology parameters and provides a frameworkfor considering the global optimality of stabilization. However, the dynamics becomeshigh dimensional and it is difficult to characterize the global dynamics.

In this paper, we do not model individual choices regarding exposure to diseasesand thus changing the dynamics of the disease (see Geoffard and Philipson 1997;

123


Kremer 1996). This literature looks at rational choices of individuals in the face of thediseases and how these choices may affect the spread of the disease. These choices arevery important in case of diseases generated by one-to-one contact such as STDs. Theyare less applicable to the many of the diseases considered in this paper where short ofquarantine (isolation of the healthy population) there may be no way to avoid exposureto the disease. Our ongoing research project is extending the simple model to incor-porating disease related mortality, more complex epidemiological process, rationalepidemiology models, and to embed these into an endogenous growth framework(Goenka and Liu 2010).

Appendix

Proof of Lemma 5

Proof Since U is continuous, and the correspondence B(k, k′, s) is continuous andcompact-valued, by the Theorem of the Maximum the function W is continuous andthe maximizers are non-empty, compact-valued and upper hemi-continuous. More-over, since the objective function is strictly concave, the maximizers are unique andhence, continuous functions. Since (k, k′, s) ∈ [0, kmax] × [0, kmax] × [0, 1] which iscompact, W is bounded.

Fix s and choose any λ ∈ [0, 1], let (c1, l1) be the maximizer for (k1, k′1, s) and

(c2, l2) be the maximizer for (k2, k′2, s). (λc1 + (1−λ)c2, λ(1− l1)+ (1−λ)(1− l2))

is feasible for (λk1 + (1 − λ)k2, λk′1 + (1 − λ)k′

2, s) and

W (λk1 + (1 − λ)k2, λk′1 + (1 − λ)k′

2, s) ≥ U (λc1 + (1 − λ)c2, λ(1 − l1)

+ (1 − λ)(1 − l2), s)

> λU (c1, 1−l1, s)+(1−λ)U (c2, 1−l2, s)

= λW (k1, k′1, s)+ (1 − λ)W (k2, k′

2, s)

by strict concavity of U . Thus, W is strictly concave for each fixed s.Let (c∗, l∗) be the optimal consumption and labor supply. By the Envelope Theo-

rem, W is continuously differentiable and

W1 = U1 F1 |(c∗,l∗)> 0

W2 = −U1 |(c∗,l∗)< 0.

Given the Inada conditions, we can rule out the corner solution. So the optimalsolution c∗ = c(k, k′, s) and l∗ = l(k, k′s) is determined by

F2(k, sl)s − M(c, 1 − l, s) = 0

F(k, sl)− c − k′ = 0.

123


By the implicit function theorem c(k, k′, s) and l(k, k′, s) are differentiable in theneighborhood of (k, k′, s) and

⎡⎢⎢⎢⎣

∂c

∂k

∂c

∂k′

∂l

∂k

∂l

∂k′

⎤⎥⎥⎥⎦ = −

[−M1 F22s2 + M2−1 F2s

]−1 [F21s 0F1 −1

]

= 1

[F2 F21s2 − F1(F22s2 + M2) F22s2 + M2

F21s − M1 F1 M1

],

where = M1 F2s − M2 − F22s2 = M1 M − M2 − F22s2 > 0 by strictly concavityof U . So we have ∂l

∂k′ = M1 > 0. Then,

W22 = −∂U1(c, 1 − l, s)

∂k′

= −[U11∂c

∂k′ − U12∂l

∂k′ ]

= − 1

[U11 F22s2 + M2U11 − U12 M1]

= − 1

[U11 F22s2 + U11U22 − U12U21

U1]

< 0

and

W12 = ∂W1

∂k′ = −∂W2 F1

∂k′ = −W22 F1 − W2 F12sM1

> 0.

��

Proof of Lemma 6

Proof Since X is a convex subset of �2+ and the correspondence A : X → X is non-empty, compact-valued and continuous, the function W is continuous and bounded,and 0 < β < 1, by Theorem 4.6 in Stokey and Lucas (1989), there exists a unique valuefunction which solves the dynamic programming problem and the policy correspon-dence is compact valued and u.h.c. Since W is strictly increasing in k and for each fixeds, A is monotone in k in the sense that k1 ≤ k2 implies A(k1, s) ⊂ A(k2, s), V is strictlyincreasing in k (Stokey and Lucas 1989, Theorem 4.7). Moreover since for each fixeds, A is convex in k in the sense that for any 0 ≤ λ ≤ 1 and k1, k2 ∈ [0, kmax], (k′

1, s) ∈A(k1, s) and (k′

2, s) ∈ A(k2, s) implies (λk′1 + (1 −λ)k′

2, s) ∈ A(λk1 + (1 −λ)k2, s),and W is strictly concave in (k, k′), V is strictly concave and the policy function is acontinuous function (Stokey and Lucas 1989, Theorem 4.8). In addition since W iscontinuously differentiable and there is an interior solution (Stokey and Lucas 1989),V is continuously differentiable. ��

123


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