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Syphilis Cycles

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Published/Copyright: June 19, 2013
The B.E. Journal of Economic Analysis & Policy
From the journal The B.E. Journal of Economic Analysis & Policy Volume 13 Issue 1

Abstract

Syphilis has re-emerged as a global public health issue. In lesser developed countries, millions of people are contracting the disease, which can be fatal without access to proper treatment. In developed countries, prevalence is on the rise and has cycled around endemic levels for decades. We investigate syphilis dynamics by extending the classic SIRS epidemiological model to incorporate forward-looking, rational individuals. The integrated economic-epidemiological model shows that human preferences over health and sexual activity are central to the nature of syphilis cycles. We find that low-activity individuals will behave in a manner that significantly dampen the cycles, while high-activity individuals will tend to exacerbate the cycles, a phenomenon we refer to as rational dynamic resonance. The model also provides insights into failed attempts by the U.S. government to eradicate syphilis from the U.S. population.

Keywords: syphilis; AIDS; disease; eradication; cycles; fatalism; dynamic resonance; SIRS
JEL Codes: D1; I1

Acknowledgement: We would like to thank the two anonymous reviewers, seminar participants at the Colorado State University, the annual meeetings for Society of Economic Dynamics, Peter Dasak, Kate Smith, Chris Jerde, Flavio Toxvaerd, Jason Shogren, Frank Caliendo, Tom Crocker, Chuck Mason, Sherrill Shaffer, and Dan Aadland for their insightful comments. This publication was made possible in part by grant number 1RO1GM100471-01 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.

Appendix

In this appendix, we present the technical details of the integrated EE model with and without AIDS. We start by presenting the joint syphilis–AIDS model and then present the details of the model without AIDS.

Syphilis–AIDS epidemiological model

The joint syphilis–AIDS (SIRS-SI) population model contains six mutually exclusive categories: susceptible to both diseases (s), infected with syphilis only (inS), infected with AIDS only (inA), infected with syphilis and AIDS (inSA), immune to syphilis (r), and immune to syphilis while infected with AIDS (rA). Each disease category is measured as a proportion of the overall population with the sum of the categories equal to one. The model collapses to a traditional SIRS model when and to a traditional SI model when .

Assuming that individuals independently choose xt partners and engage in a fixed number of sexual acts (a) with each partner, the probability that susceptible individuals become infected with syphilis or AIDS is

[27]
[27]
[28]
[28]

where is the probability of contracting disease from a single infected partner, and is the probability of contracting the disease from a single sexual act. The conditional probabilities for those infected with one disease are

[29]
[29]
[30]
[30]

where is the number of partners chosen by those infected with syphilis (AIDS). Individuals infected with syphilis or AIDS are allowed to have a different natural probability of infection, and , than those without a disease. As mentioned above, those with primary or secondary syphilis have an elevated probability of acquiring HIV (i.e. ). The dependence on the chosen number of partners distinguishes the analysis from standard mathematical epidemiology.

The transition matrix between the categories is shown in Table 3:

Table 3

Transition matrix for disease categories.

00
0000
0000
000001
00
0000

Using the transition probabilities and a 100% syphilis treatment rate, the equations of motion for the disease categories are

Value functions

The four value functions apply to individuals: (1) susceptible to both diseases, ; (2) infected with syphilis only, ; (3) infected with AIDS only, ; and (4) infected with syphilis and AIDS, , where is the vector of states. There are no value functions for those recovered (and immune) to syphilis, because we assume that the recovered stage cannot be observed by individuals. The value functions are

[31]
[31]
[32]
[32]
[33]
[33]
[34]
[34]
Euler equations

The necessary first-order conditions for s, inS, and inA individuals are

[35]
[35]
[36]
[36]
[37]
[37]

where the x subscript on the probabilities refers to the partial derivatives with respect to the appropriate x. The left side of the Euler equations is the marginal utility or benefit (MB) and the right side is the marginal disutility or cost (MC) associated with the chosen number of partners. Using eqs [31]–[34] to substitute out the optimized value functions, the Euler equations become

[38]
[38]
[39]
[39]
[40]
[40]

where

The second-order conditions for an optimal program require

for eqs [35–37]. Since the marginal benefits decline with x, an optimal program requires an upward sloping marginal cost curve (i.e. ), or if it slopes down, it must be locally flatter than the MB curve (i.e. ).

Expectations

We consider two types of expectations by individuals: naïve and rational. Under naïve expectations, the expectation of all future variables is set equal to the associated current value. Under rational expectations, E is the mathematical expectations operator conditional on all information dated time t and earlier. With rational expectations, individuals have complete information on the laws of motion for the aggregate disease variables and the optimal choices of other individuals.

Steady state

The endemic steady-state solves for nine variables, from the following nine equations:

Linearization

We start by linearizing the SIRS and SI epidemiological equations around the endemic steady state. Variables with hats refer to deviations from the steady state (e.g. )

The linearized equations for the probabilities (and the derivatives of the probabilities with respect to partners) are given by:

where

Summarizing, the linearized EE system isSIRS/SI system:

Probabilities:

Euler equations:

with coefficients

Linearized matrix system

The linearized EE system in matrix form is

and

where for our parameter choices, individuals with only AIDS always choose the maximum number of partners . This implies that for all t.

Writing the matrix EE system in compact form, we get

or

where .

Rational expectations equilibrium

We use the method of Blanchard and Kahn (1980) to solve for the rational expectations equilibrium (REE). The vector contains six predetermined and two jump variables. If J contains two forward-stable roots, the system displays saddle-path stability and a unique endemic REE. If there are less than two forward-stable roots of J, the steady state is a sink and the endemic REE is indeterminate. The equilibrium under naïve expectations is calculated by setting , , and .

Parameters and steady-state values

Table 4 shows the baseline parameter values and the implied steady-state values.

Table 4

Baseline parameters and steady-state values.

Parameters
ahSAhAhSh
0.960.20.05400055100.0230.00080.0240.023
Endemic steady-state values
xxSxAsinSinAinSArrApSpApp
1.9410.116100.5620.0430.0810.0280.1610.1250.0820.0140.3580.018
Syphilis eradication steady-state values
xxSxAsinSinAinSArrApSpAppR0
2.141–100.74500.25500000.0170–2.393

We now justify our choice of parameter values, which can be placed into epidemiological and economic categories.

Epidemiological parameters

  1. (probability of contracting syphilis with an infected partner, one act)

  2. (probability of contracting syphilis with an infected partner, all acts)

  3. (probability of contracting AIDS with an infected partner, one act)

  4. (probability of contracting AIDS with syphilis and an infected partner, one act)

  5. (syphilis treatment rate)

  6. (syphilis loss of host immunity rate)

  7. (birth/death rate)

For the syphilis parameters, Garnett et al. (1997) suggested that is a potentially “unbiased estimate” (p. 189) for the partner probability of syphilis transmission. If we assume that a susceptible individual has sexual acts with each partner, the implied probability of contracting syphilis from a single act is . For the AIDS parameters, Chesson and Pinkerton (2000) documented mean per act probabilities of AIDS transmission to be 0.001 for male-to-female transmission and 0.0006 for female-to-male transmission. We employ the average of these in our gender-neutral per-act AIDS transmission probability of . Chesson and Pinkerton (2000) also provided an estimate of the probability that an individual who has syphilis will contract AIDS from a single act with an infected partner (). The treatment parameter for syphilis v captures both the rate of diagnosis and treatment. The treatment effectiveness for syphilis appears to be close to 100% (Alexander et al. 1999), so that . Following Garnett et al. (1997), we assume an average duration of host immunity to syphilis of 5 years, implying a value of . The population is assumed to have a birth/death rate of as in Garnett et al. (1997).

Economic parameters

  1. (number of sexual acts per partner)

  2. (health parameter with syphilis and AIDS)

  3. (health parameter with AIDS only)

  4. (health parameter with syphilis only)

  5. (health parameter without syphilis or AIDS)

  6. (discount factor)

  7. (maximum number of partners per period)

Sexual acts per partner is set at . Chesson and Gift (2000) set the total number of sexual acts per year at 100. Smith (1994) cited a figure of 62 total sexual acts per year, on average across the adult population. Using our steady state of approximately two partners per year, the implied total number of sexual acts is a midpoint of these two estimates. We normalize the utility health parameter with syphilis and AIDS (hSA) to zero. The health parameter for individuals with AIDS but not syphilis is also set at zero. This captures the notion that the health risks of AIDS dominate those of syphilis. Contracting syphilis is still a concern to susceptible individuals, because it significantly increases the risk of contracting AIDS. The health parameter without AIDS or syphilis (h) or without AIDS but with syphilis (hS) is set at 5. This value produces dynamic dampening and is chosen to produce syphilis cycles with an approximate 10-year period under naïve expectations (Grassly, Fraser, and Garnett 2005). A discount factor of is standard for annual data and is consistent with a 4% real rate of return. The value of was inferred from a number of sources. Andrus et al. (1990) reported an average number of partners for those infected with syphilis of 6.3 partners during the infectious period. Koblin et al. (2003) found that in a non-HIV sample of approximately 4,300 homosexual men across six major U.S. cities, over half the sample report having more than 15 partners per year.

Dynamics and impulse response functions

The beginning of the AIDS epidemic in the early 1980s drastically changed the risks of sexual activity. Sexually active individuals were primarily concerned with AIDS, rather than syphilis or other STDs. Annual deaths due to AIDS in the U.S. jumped from 135 individuals in 1981 to a peak of over 48,000 in 1995 (CDC 2007). The annual mortality rate for AIDS has since declined to under 15,000 due to the introduction and widespread availability of effective antiretroviral therapies (Boily et al. 2005). The effect of AIDS on the dynamics of syphilis prevalence in the U.S. can be seen in Figure 1. Starting in 1990, the overall number of primary and secondary syphilis infections in the U.S. gradually fell over the decade and has been gradually rising since 2000. The nearly two decade U-shaped pattern in syphilis prevalence is significantly different than the 10-year oscillations marking the period between the introduction of penicillin and the beginning of the AIDS epidemic. To better understand the changing dynamics in syphilis prevalence, we explore the predictions of the joint syphilis–AIDS EE model.

Figures 7 and 8 show the dynamic responses to a 0.05 increase in the fraction of the population infected with syphilis only (inS) and a 0.05 increase in the fraction of the population infected with AIDS only (inA). Figure 7 uses the baseline parameters in Table 4 and displays dynamic dampening for both naïve and rational expectations. In a setting with a relatively high health parameter so that individuals choose fewer sexual partners, a one-time increase in syphilis prevalence has little impact on the optimal number of partners or the dynamics of syphilis prevalence, because AIDS, not syphilis, is the primary health concern. The primary impact of higher syphilis prevalence is to increase the risk of AIDS through the higher natural probability of infection . A one-time increase in AIDS prevalence leads to a greater initial reduction in the number of partners but monotonically returns to the steady state. The dynamics of syphilis infections are similar to those from the ME model. This similarity occurs because individuals are responding to a portfolio of risks, which is dominated by the lifetime consequences of contracting AIDS.

Figure 7 IRFs for the ME and EE syphilis–AIDS systems – rational dynamic dampening.Notes: EE fundamental parameters:
Figure 7 IRFs for the ME and EE syphilis–AIDS systems – rational dynamic dampening.Notes: EE fundamental parameters:
Figure 7

IRFs for the ME and EE syphilis–AIDS systems – rational dynamic dampening.

Notes: EE fundamental parameters:

Figure 8 IRFs for the ME and EE syphilis–AIDS systems – rational dynamic resonance.Notes: EE fundamental parameters: .
Figure 8 IRFs for the ME and EE syphilis–AIDS systems – rational dynamic resonance.Notes: EE fundamental parameters: .
Figure 8

IRFs for the ME and EE syphilis–AIDS systems – rational dynamic resonance.

Notes: EE fundamental parameters: .

Figure 8 uses identical parameter values except the health parameter, , is reduced to 1.95. This causes fatalism to set in for naïve individuals, leading to dynamic resonance. With an increase in either syphilis or AIDS prevalence, individuals choose a higher number of partners, as the marginal probability of AIDS infection declines. The higher number of partners exacerbates the initial increase in syphilis or AIDS prevalence. This interplay between the marginal probability of contracting AIDS and the chosen number of partners continues over time, amplifying and stretching out syphilis cycles. The cycles in syphilis prevalence spillover into AIDS dynamics through the higher natural probability of AIDS infection. This is rational dynamic resonance in the joint EE model, and it occurs in both SIRS and SI diseases. For individuals with rational expectations, the equilibrium path displays dynamic dampening of cycles (not shown). Further reductions in lead to either an indeterminate or an unstable equilibrium path under rational expectations.

Syphilis eradication

Now, consider the stability of syphilis eradication in the joint syphilis–AIDS model. Here, the chance of syphilis eradication is improved, because susceptible individuals will take fewer partners due to the fear of AIDS. To examine the stability of syphilis eradication, we calculate the basic reproductive number for syphilis around the eradication steady state. The basic reproductive number is given by

[41]
[41]

Note the similarities of to the single-disease model. The first term in the numerator measures the rate at which individuals from the susceptible pool are becoming infected. This term is weighted by s, the proportion of the susceptible syphilis population without AIDS, and involves rather than , because susceptible individuals will not take the maximum number of partners due to the risk of AIDS. The second term involves the proportion of individuals that are susceptible to syphilis but have AIDS, . These individuals will choose more partners, because they are already infected with AIDS and the health risks of syphilis are relatively low. As in the single-disease model, the denominator () captures the rate at which individuals are leaving the infected pool, through either treatment (100%) or death ().

To contrast the stability of syphilis eradication in the single and dual disease models, we calculate the necessary degree of altruism for successful eradication. We model altruism by allowing infected individuals to choose partners, where is the altruism parameter. Table 5 shows the degree of altruism needed for syphilis eradication to be locally stable.

Table 5

Altruism and stability of syphilis eradication.

ModelTypeNumber of partners
Fraction ofpopulation (%)Self-interestedchoiceRequiredaltruism
Syphilis onlys1001.75
Syphilis–AIDSs851.871.87
Syphilis–AIDSinA150.95

In the syphilis only model, the EE model shows that a high degree of altruism is necessary for eradication to be stable and keep the system from gravitating toward an endemic equilibrium: the susceptible population must reduce their number of partners by 83% (from to less than two partners per year). For the sexually active population under consideration, this is an extreme degree of altruism (Andrus et al. 1990). With the AIDS epidemic, individuals are primarily concerned with the risk of contracting AIDS, not syphilis. Those susceptible to AIDS will voluntarily take fewer partners, x 1.87 for our calibration, not out of concern for the general population but rather out of self-interest. The remaining portion of the population (those with AIDS) will need to reduce their number of partners to less than one partner per year () for syphilis eradication to be successful. Overall, this is a much smaller degree of required altruism, because the majority of the population voluntarily reduced the number of partners due to the risk of AIDS. Yet, those with AIDS are still required to take no more than one partner per year and display a high level of altruism for syphilis to be eradicated.

Syphilis epidemiological model

The syphilis (SIRS) population model contains three mutually exclusive categories: susceptible to syphilis (s), infected with syphilis (in), and immune to syphilis (r). We start by presenting the transition matrix between these categories in Table 6:

Table 6

Transition matrix for disease categories.

0
001
0

Using the transition probabilities and a 100% syphilis treatment rate, the equations of motion for the disease categories are

where the probability of contracting syphilis is

Value functions

The value functions apply to individuals: (1) susceptible to syphilis, and (2) infected with syphilis, . There is no value function for those recovered (and immune) to syphilis because we assume individuals cannot distinguish the susceptible state from the recovered and immune state. The value functions are

[42]
[42]
[43]
[43]

where the health parameter is normalized to zero for infected individuals.

Euler equations

Assuming an interior solution, the necessary first-order condition for susceptible individuals is

where

Using eqs [42] and [43] to substitute out the optimized value functions, the Euler equation becomes

[44]
[44]
Steady state

The endemic steady state solves for four variables, from the following four equations:

[45]
[45]
[46]
[46]
[47]
[47]
[48]
[48]
Linearization

We start by linearizing the SIRS epidemiological equations around the endemic steady state:

Next linearize the probabilities (and derivative of the probability with respect to the number of partners):

[49]
[49]
[50]
[50]

where

The linearized Euler equation is

[51]
[51]

Summarizing, the linearized EE system with eqs [49] and [50] substituted into eq. [51] is

SIRS system:

Probabilities:

Euler equation:

with coefficients

Linearized matrix system

The linearized EE system in matrix form is

[52]
[52]

The matrix system includes the restriction and the maximum choice of partners for those with syphilis, .

REE

The EE system contains one jump and two predetermined variables. The system will exhibit saddle-path stability if there is one forward-stable root for . Assuming one forward-stable root and using the method of Blanchard and Kahn (1980), we solve for a contemporaneous relationship between the jump variable and the two state variables:

[53]
[53]

where

refers to the (i, j) element of the inverse of the matrix of stacked eigenvectors for , and the eigenvalue of is the forward stable root. Using eq. [53], we then solve for the reduced-form representation:

[54]
[54]
Dynamics and impulse response functions

Figure 9 shows the dynamic dampening responses to a 0.05 increase in syphilis prevalence under naïve and rational expectations using the baseline parameter values from Table 2 in the paper. Lowering the health parameter to as in Figure 6 (which generates dynamic resonance with naïve expectations) produces an unstable REE. Additional increases in h move the REE from unstable to indeterminate to determinate with rational dynamic dampening.

Figure 9 IRFs for the ME and EE systems – rational dynamic dampening (solid naïve expectations, dashed rational expectations).Notes: The fundamental parameters in the EE system are set at , and . For comparison purposes, we set the steady-state number of partners (x) in the ME model equal to the endogenously solved for number of partners in the EE model. As a result, the steady-state prevalence is also equal in the ME and EE models.
Figure 9

IRFs for the ME and EE systems – rational dynamic dampening (solid naïve expectations, dashed rational expectations).

Notes: The fundamental parameters in the EE system are set at , and . For comparison purposes, we set the steady-state number of partners (x) in the ME model equal to the endogenously solved for number of partners in the EE model. As a result, the steady-state prevalence is also equal in the ME and EE models.

Figure 10 Syphilis prevalence IRF for the ME linear and nonlinear systems. All parameters and the number of partners are from the endemic dampening case shown in Table 2.
Figure 10

Syphilis prevalence IRF for the ME linear and nonlinear systems. All parameters and the number of partners are from the endemic dampening case shown in Table 2.

Contrasting the linear and nonlinear systems

Figures 10–12 show the comparison of the linear and nonlinear impulse response functions (IRFs) for the ME, EE dynamic dampening, and EE dynamic resonance cases. The IRFs are quite similar and support our use of the linearized system for moderate-sized initial shocks.

Figure 11 Prevalence and partner IRFs for dynamic dampening with EE linear and nonlinear models under naïve expectations (left graphs) and rational expectations (right graphs). All parameters and the number of partners are from the endemic dampening case in Table 2.
Figure 11

Prevalence and partner IRFs for dynamic dampening with EE linear and nonlinear models under naïve expectations (left graphs) and rational expectations (right graphs). All parameters and the number of partners are from the endemic dampening case in Table 2.

Figure 12 Prevalence and partner IRFs for dynamic resonance with EE linear and nonlinear models under naïve expectations. Parameter values are from Table 2 except for h = 4.54.
Figure 12

Prevalence and partner IRFs for dynamic resonance with EE linear and nonlinear models under naïve expectations. Parameter values are from Table 2 except for h = 4.54.

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  1. 1

    Syphilis is remembered by many for the infamous Tuskagee experiments where poor, Southern black men were misleadingly infected with the disease and studied by the U.S. Public Health Service over a period of 40 years starting in 1932 (Nakashima et al. 1996). In 1997, the U.S. government formally apologized for the incident.

  2. 2

    For more details on the plan, see http://www.cdc.gov/stopsyphilis/.

  3. 3

    Of course, eradication policies can still have a positive impact. Policies aimed at reducing the risk of infection for high-activity individuals, either through reductions in the number of partners or through increased protection, can lower long-run endemic equilibria and stabilize cycles.

  4. 4

    This integrated model is derived directly from the behavior of rational individuals. The resulting dynamic system closely resembles classic epidemiological models (Murray 2002) with one major difference. In the integrated model, the traditional infection parameters are not fixed but vary over time and depend on the optimally chosen number of sexual partners, the number of sex acts with each partner, the overall infection rate in the population, and the natural rates of infection. Consequently, predictions of individuals’ collective responses to changes in the risk of disease transmission (e.g. through education campaigns emphasizing prevention and treatment) will be more robust than predictions from traditional models with fixed parameters and no behavioral responses. For instance, policies designed to reduce the transmission of the disease may fail, if individuals choose to offset reductions in the risk of infection by engaging in increased amounts of sexual activity.

  5. 5

    This effect is in contrast to the effect of coherence resonance (see, for example, Dushoff et al. 2004). Coherence resonance can amplify cycles and is derived from the interaction of the mean infection period and the average duration of immunity. In the modeling of the effect, the contact rate is specified a priori by a sinusoidal function with no behavioral basis.

  6. 6

    Chesson, Dee, and Aral (2003) argue that the causality may also run in the other direction. They show that high rates of AIDS mortality in high-risk men were responsible, at least in part, for the decline in the prevalence of syphilis in the U.S. during the 1990s.

  7. 7

    The SIRS and SI models are traditionally modeled in continuous time, but the discrete time version is more convenient for specifying lead and lag relationships, selecting the timing of driving shocks, and for contrasting predictions of the model with the annually observed U.S. syphilis data. See Allen (1994) for a treatment of discrete-time mathematical epidemiology models. See Auld 2003 for an application of a discrete-time economic SI model and Lightwood and Goldman (1995) for an application of a discrete-time economic SIS model.

  8. 8

    The risk of contracting an STD can be manipulated by varying the level of protection or the number of partners. Geoffard and Philipson (1996) and Toxvaerd (2010) are examples of studies where the control variable is costly prevention, such as using prophylaxis. Kremer (1996) and Auld (2003) are examples where the control variable is the number of partners. Both methods capture the essential tradeoff that risk of infection can be reduced by costly behavior, either increased protection or taking fewer partners.

  9. 9

    The consequences and policies associated with the externalities imposed by infected individuals have been studied in depth for the SIS epidemiological model by Goldman and Lighwood (2002) and Gersovitz and Hammer (2004, 2005). Their work focuses on the design of optimal tax policies to encourage effective treatment and prevention of the disease.

  10. 10

    The marginal benefit and cost curves, along with the optimal choices, are shown in Figure 4 and discussed in further detail below.

  11. 11

    Simulations were also performed on the non-linear system using GAMS. Comparisons of the results to those from the linearized system are reported in the Appendix.

  12. 12

    The calibration exercise is described in the Appendix.

  13. 13

    If the eigenvalues have an imaginary part, then they come in complex conjugates with period equal to (Hamilton 1994)

    and persistence equal to

  14. 14

    This is similar to the behavioral response cited in Geoffard and Philipson (1996) where the hazard rate (probability of infection) decreases as prevalence increases. The implication of is not that the probability of infection must fall with an increase in prevalence, rather that the change in probability of infection is smaller than if individuals did not alter their choice of partners.

  15. 15

    The critical probability, , is found by taking the cross partial derivative of p with respect to x and in, setting the expression equal to zero, and solving for p (Kremer 1996). The relevant equation is , which reduces to or .

  16. 16

    For purposes of comparison, we set x in the ME model equal to the endogenous solution for x from the EE model. This also implies that p and in will be equal across the two models.

  17. 17

    For the baseline parameter values, the elasticity of partner change with respect to prevalence is . Furthermore, if we hold x fixed at its steady-state value, the probability of infection at the steady state is 0.23, increasing to 0.33 with the five percentage point increase in prevalence.

  18. 18

    For the baseline parameter values, the elasticity of partner change with respect to prevalence is . Furthermore, if we hold x fixed at its steady-state value, the probability of infection at the steady state is 0.56, increasing to 0.67 with the five percentage point increase in prevalence.

  19. 19

    In their ME model, Grassly, Fraser, and Garnett (2005) implicitly chose the number of partners per year to be 14.5. Breban et al. (2008) found that cycles only occur if individuals take more than 9.8 new partners per year. We find a much lower threshold in the EE model due to the behavioral responses.

  20. 20

    To derive the transition matrix around the eradication steady state, evaluate eq. [23] at the eradication steady state. The Euler equation for x is not relevant, because when the system is near the eradication boundary, individuals will optimally choose for all t.

  21. 21

    The other eigenvalue will be less than one in magnitude, because our calibrations always satisfy .

  22. 22

    Alternatively, the stability threshold [26] for eradication can be interpreted in terms of the basic reproduction number , which using L’Hôpital’s rule reduces to . The standard result in the epidemiological literature is that eradication is locally stable if is less than one (Anderson and May 1991). The intuition is straightforward – for eradication to be stable, the rate at which people are entering the infection pool () must be less than the rate at which people are leaving the infected pool ().

Published Online: 2013-06-19
Published in Print: 2013-07-01

©2013 by Walter de Gruyter Berlin / Boston

Articles in the same Issue

  1. Masthead
  2. Masthead
  3. Contributions
  4. Women Rule: Preferences and Fertility in Australian Households
  5. Can Land Reform Avoid a Left Turn? Evidence from Chile after the Cuban Revolution
  6. Incentive Effects of Parents’ Transfers to Children: An Artefactual Field Experiment
  7. Reclassification and Academic Success among English Language Learners: New Evidence from a Large Urban School District
  8. Fairness, Search Frictions, and Offshoring
  9. The Incentive Effect of Equalization Grants on Tax Collection
  10. Why Have Labour Market Outcomes of Youth in Advanced Economies Deteriorated?
  11. A Commitment Theory of Subsidy Agreements
  12. The Effects of Transactions Costs and Social Distance: Evidence from a Field Experiment
  13. Syphilis Cycles
  14. Impact of Voucher Design on Public School Performance: Evidence from Florida and Milwaukee Voucher Programs
  15. Topics
  16. Outsourcing and Innovation: An Empirical Exploration of the Dynamic Relationship
  17. Economies of Scope, Entry Deterrence and Welfare
  18. Can Horizontal Mergers Without Synergies Increase Consumer Welfare? Cournot and Bertrand Competition Under Uncertain Demand
  19. Institutions and information in multilateral bargaining experiments
https://doi.org/10.1515/bejeap-2012-0060
Keywords for this article
syphilis; AIDS; disease; eradication; cycles; fatalism; dynamic resonance; SIRS

Articles in the same Issue

  1. Masthead
  2. Masthead
  3. Contributions
  4. Women Rule: Preferences and Fertility in Australian Households
  5. Can Land Reform Avoid a Left Turn? Evidence from Chile after the Cuban Revolution
  6. Incentive Effects of Parents’ Transfers to Children: An Artefactual Field Experiment
  7. Reclassification and Academic Success among English Language Learners: New Evidence from a Large Urban School District
  8. Fairness, Search Frictions, and Offshoring
  9. The Incentive Effect of Equalization Grants on Tax Collection
  10. Why Have Labour Market Outcomes of Youth in Advanced Economies Deteriorated?
  11. A Commitment Theory of Subsidy Agreements
  12. The Effects of Transactions Costs and Social Distance: Evidence from a Field Experiment
  13. Syphilis Cycles
  14. Impact of Voucher Design on Public School Performance: Evidence from Florida and Milwaukee Voucher Programs
  15. Topics
  16. Outsourcing and Innovation: An Empirical Exploration of the Dynamic Relationship
  17. Economies of Scope, Entry Deterrence and Welfare
  18. Can Horizontal Mergers Without Synergies Increase Consumer Welfare? Cournot and Bertrand Competition Under Uncertain Demand
  19. Institutions and information in multilateral bargaining experiments
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