Economic dynamics of epidemiological bifurcations

David Aadland; David Finnoff; Kevin X.D. Huang

doi:10.1515/snde-2019-0111

Abstract

In this paper, we investigate the nature of rational expectations equilibria for economic epidemiological models, with a particular focus on the behavioral origins and dynamics of epidemiological bifurcations. Unlike mathematical epidemiological models, economic epidemiological models can produce regions of indeterminacy or instability around the endemic steady states due to endogenous human responses to epidemiological circumstance variation, medical technology change, or health policy reform. We consider SI, SIS, SIR and SIRS versions of economic compartmental models and show how well-intentioned public policy may contribute to disease instability, uncertainty, and welfare losses.

Keywords: bifurcation; economic epidemiology; endogenous disease risk; indeterminacy; rational expectations; self-fulfilling prophecies

JEL Classification: D1; I1

Corresponding author: David Aadland, Department of Economics, University of Wyoming, 1000 E. University Avenue, Laramie, WY, 82071, USA, E-mail: aadland@uwyo.edu

Funding source: National Institute of General Medical Sciences (NIGMS) at the National Institutes of Health

Award Identifier / Grant number: 1R01GM100471-01

Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: This publication was made possible in part by grant number 1R01GM100471-01 from the National Institute of General Medical Sciences (NIGMS) at the National Institutes of Health. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of NIGMS.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A. Derivation of the economic SIRS Euler equation with observable immunity

Here we derive the Euler equation for the economic SIRS model with observable immunity. To begin, note that Equations (9) and (10) imply

(A.1)VtR−VtIN=h+βEt[γ(Vt+1S−Vt+1R)+(1−v)(Vt+1R−Vt+1IN)] ,

while Equations (8) and (9) imply

(A.2)VtS−VtIN=ln (xt/x‾)+h+βEt[(1−pt)(Vt+1S−Vt+1IN)−v(Vt+1R−Vt+1IN)] .

Using Equation (11), we have

(A.3)Et(Vt+1S−Vt+1IN)=(βxtpx,t)−1 ,

for all t. Next, rearrange (A.2) as

(A.4)Vt+1R−Vt+1IN=1βv[ln (xt/x‾)+h]+1v(1−pt)Et(Vt+1S−Vt+1IN)−1βv(VtS−VtIN) .

Take E_t−1 on both sides of (A.4) and substitute (A.3) to get

(A.5)Et−1(Vt+1R−Vt+1IN)=1βvEt−1[ln (xt/x‾)+h]+1βvEt−1(1−ptxtpx,t)−1β2v(1xt−1px,t−1) .

Now rewrite Equation (A.1) as

(A.6)VtR−VtIN=h+βEt[γ(Vt+1S−Vt+1IN)+(1−v−γ)(Vt+1R−Vt+1IN)] .

Move (A.6) ahead one period, take E_t−1 of both sides, and set equal to (A.5) to get

(A.7)1βvEt−1[ln (xt/x‾)+h]+1βvEt−1(1−ptxtpx,t)−1β2v(1xt−1px,t−1)=h+βEt−1{γ(βxtpx,t)−1+(1−v−γ)(1βv[ln (xt/x‾)+h]+1βv(1−ptxtpx,t)−1β2v(1xt−1px,t−1))} .

Impose perfect foresight, move ahead one period, and rearrange to get

xt−1=βpx,t[ln (xt+1/x‾)+h+(1−v−pt+1)xt+1px,t+1−βΔt+2] ,

where

Δt+2=vγxt+2px,t+2+(1−v−γ)[ln (xt+2x‾)+1−pt+2xt+2px,t+2]+(1−γ)[h−1βxt+1px,t+1] .

Appendix B. SIRS economic epidemiological (EE) steady-state and matrix systems

Here, we describe the steady state EE system and the linearized EE matrix system used in the bifurcation and stability analyses. The endemic steady states solve time-invariant versions of (6), (7), and the Euler equation. The Euler equation either takes the form of (12) when an indicator variable set at ϕ = 1 (observable immunity) or the form of (13) when ϕ = 0 (unobservable immunity). The steady-state system can therefore be rewritten as three equations:

(B.1)in=p(1−in−r)/(v+μ)

(B.2)r=vin/(γ+μ)

(B.3)x−1=β[px(ln (x/x‾)+h−ϕβΔ)+(1−v−p)/x]

in three unknown variables (in, r, x), where the immunity externality is given by

Δ=1pxx[vγ+(1−v−γ)(1−p)−(1−γ)/β]+(1−v−γ) ln (x/x‾)+(1−γ)h .

Similar to Goenka, Liu, and Nguyen (2012), we also note the existence of an eradication steady state and focus on the local stability properties around the endemic steady states.

To analyze these transition dynamics, we linearize around the endemic steady states:

(B.4)inˆt+1=(1−v−μ−p)inˆt+(1−in−r)pˆt−prˆt

(B.5)rˆt+1=(1−γ−μ)rˆt+vinˆt ,

where hats (^) over the variables indicate deviation from one of the steady states. The linearized Euler equation is:

(B.6)pxxˆt+xpˆx,t=βpx(1−v−p−xpx)Etxˆt+1+βx(1−v−p)Etpˆx,t+1+βxpxEtpˆt+1+ϕβ2Et{px[vγ+(1−v−γ)(1−p−xpx)]xˆt+2+x[vγ+(1−v−γ)(1−p)]pˆx,t+2+[(1−v−γ)xpx]pˆt+2−[(1−γ)px/β]xˆt+1−[(1−γ)x/β]pˆx,t+1}

where

(B.7)pˆt=pininˆt+pxxˆt

(B.8)pˆx,t=[(1+ln [1−p])/x]pˆt−(px/x)xˆt

and

(B.9)pin=xλ(1−λin)x−1

(B.10)px=−ln(1−p)(1−p)/x .

In matrix form, the EE system can be written as

(B.11)zˆt=Jzˆt+1 ,

where zˆt=(xˆt,inˆt,rˆt)′ when ϕ = 0 or zˆt=(xˆt,inˆt,rˆt,xˆt+1,inˆt+1)′ when ϕ = 1, and J is the transition matrix.

Specifically, if we impose perfect foresight, the ϕ = 0 linearized EE matrix system can be written as:

(B.12)[01−v−μ−p−p0v1−γ−μpx00]⏟A[xˆtinˆtrˆt]+[1−in−r0000x]⏟B[pˆtpˆx,t]=[010001βpx(1−v−p−xpx)00]⏟C[xˆt+1inˆt+1rˆt+1]+[0000βxpxβx(1−v−p)]⏟D[pˆt+1pˆx,t+1]

and

(B.13)[−10−(1+ln (1−p))/x1]⏟F[pˆtpˆx,t]=−[pxpin0px/x00]⏟G[xˆtinˆtrˆt] .

When ϕ = 1, we have

[01−v−μ−p−p000v1−μ−γ00px00000001000001]⏟A[xˆtinˆtrˆtxˆt+1inˆt+1]+[s00000000x0000000000]⏟B[pˆtpˆx,tpˆt+1pˆx,t+1]=[0100000100βpx(1−v−p−xpx)−β2[(1−γ)px/β]00β2px[vγ+(1−v−γ)(1−p−xpx)]01000001000]⏟C[xˆt+1inˆt+1rˆt+1xˆt+2inˆt+2]+[00000000βxpxβx(1−v−p)−β2[(1−γ)x/β]β2(1−v−γ)xpxβ2x[vγ+(1−v−γ)(1−p)]00000000]⏟D[pˆt+1pˆx,t+1pˆt+2pˆx,t+2]

and

(B.14)[−1000−(1+ln(1−p))/x10000−1000−(1+ln(1−p))/x1]⏟F[pˆtpˆx,tpˆt+1pˆx,t+1]=−[pxpin000px/x0000000pxpin000px/x0]⏟G[xˆtinˆtrˆtxˆt+1inˆt+1] .

where

(B.15)J=(A−BF−1G)−1(C−DF−1G) .

We use the method of Blanchard and Kahn (1980) to analyze the nature of the rational expectation EE equilibrium. When ϕ = 0 the three-variable system contains one jump (xˆt) and two predetermined (inˆt and rˆt) variables. The system will exhibit saddle-path stability if there are two eigenvalues of J outside the unit circle, indeterminate multiple stable paths if there are no forward stable eigenvalues, and explosive paths if there is more than one forward-stable eigenvalue. When ϕ = 1 the five-variable system contains three jump (xˆt, xˆt+1 and inˆt+1) and two predetermined (inˆt and rˆt) variables. The fifth equation is an identity for inˆt+1with a zero eigenvalue. Considering the other four eigenvalues, the system will exhibit saddle-path stability if exactly two of the eigenvalues are outside the unit circle, indeterminate multiple stable paths if there are three or more eigenvalues outside the unit circle, and explosive paths if there is less than two eigenvalues outside the unit circle.

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Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/snde-2019-0111).

Received: 2019-09-11

Accepted: 2020-12-05

Published Online: 2020-12-30

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Economic dynamics of epidemiological bifurcations

Abstract

Appendix A. Derivation of the economic SIRS Euler equation with observable immunity

Appendix B. SIRS economic epidemiological (EE) steady-state and matrix systems

References

Supplementary Material

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