Posts

2014-08-17: Feynman's missing method for third-orders?

2014-07-31: CIA spies even on congress

2014-07-16: Rehm on vaccines

2014-06-21: Kurtosis, 4th order diffusion, and wave speed

2014-06-20: Random dispersal speeds invasions

2014-05-06: Preservation of information asymetry in Academia

2014-04-16: Dual numbers are really just calculus infinitessimals

2014-04-14: More on fairer markets

2014-03-18: It's a mad mad mad mad prisoner's dilemma

2014-03-05: Integration techniques: Fourier--Laplace Commutation

2014-02-25: Fiber-bundles for root-polishing in two dimensions

2014-02-17: Is life a simulation or a dream?

2014-01-30: PSU should be infosocialist

2014-01-12: The dark house of math

2014-01-11: Inconsistencies hinder pylab adoption

2013-12-24: Cuvier and the birth of extinction

2013-12-17: Risk Resonance

2013-12-15: The cult of the Levy flight

2013-12-09: 2013 Flu Shots at PSU

2013-12-02: Amazon sucker-punches 60 minutes

2013-11-26: Zombies are REAL, Dr. Tyson!

2013-11-22: Crying wolf over synthetic biology?

2013-11-21: Tilting Drake's Equation

2013-11-18: Why $1^\infty != 1$

2013-11-15: Adobe leaks of PSU data + NSA success accounting

2013-11-14: 60 Minutes misreport on Benghazi

2013-11-11: Making fairer trading markets

2013-11-10: L'Hopital's Rule for Multidimensional Systems

2013-11-09: Using infinitessimals in vector calculus

2013-11-08: Functional Calculus

2013-11-03: Elementary mathematical theory of the health poverty trap

2013-11-02: Proof of the area of a circle using elementary methods

Elementary mathematical theory of the health poverty trap

In journal club, a few years ago, my students and I read an interesting paper on how infectious diseases can trap communities in poverty. In a Bonds et al, 2013 paper, the authors proposed theory for how poverty-traps could emerge from infectious disease transmission. It's an interesting idea, but we had a hard time following the presentation. In this note, I'll give an alternative presentation, with some notes on form. We can show that disease-induced poverty traps can only occur when increased income rapidly reduces infection risk.

Theory

Assume people in a homogeneous population have two health states, sick ($S$) and well ($I$). People also have an average annual income $M$. The rates at which they move between these states depends on the health states of others and their income. At the population level, the transitions between these two states are governed by \begin{align} \dot{S} &= - \beta(M) S I + \gamma(M) I, \\ \dot{I} &= \beta(M) S I - \gamma(M) I. \end{align} Without loss of generality, we take $S+I=1$. Also, we expect $\beta(M)$ to be decreasing in income, and $\gamma(M)$ to be increasing in income. The basic reproduction number $\mathscr{R}_0(M) = \beta(M)/\gamma(M)$ is a decreasing function of average income. From the SIS model, $I^* = \max \{ 0, 1-1/\mathscr{R}_0(M) \}$. Now, suppose individuals who are well earn income at rate $m_S$, and individuals who are sick earn income at rate $m_I$. At steady-state, then, the average income $M^*$ should satisfy \begin{gather} M^* = m_S \frac{ \gamma(M^*)}{\beta(M^*) I^* + \gamma(M^*)} + m_I \frac{ \beta(M^*) I^*}{\beta(M^*) I^* + \gamma(M^*)}, \end{gather} where $I^*$ is the equilibrium disease prevalence. After some algebra, we conclude that at steady-state, \begin{gather} \frac{M^*-m_I}{m_S-m_I} = \min \left\{ 1, \frac{1}{\mathscr{R}_0(M^*)} \right\}. \end{gather} So the equilibrium income should be someplace between the minimum and maximum possible values, like we'd expect intuitively ($M^* \in [ m_I, m_S ]$). If $\mathscr{R}_0(m_s) < 1 $, then the maximum income $M^* = m_s, S^*=1, I^*=0$ will be a stable steady-state. As a general rule, we can rule out the existence of poverty traps when infection risk is insensitive to changes in income level. This result takes two forms. First, if $1/\mathscr{R}_0(M^*)$ is concave, then there can not be more than 1 intersection point in the steady-state equation. This amounts to the following:

Theorem. If \[ \forall M : \mathscr{R}_0(M) < 1 \lor \frac{ \mathscr{R}_0(M) \mathscr{R}_0''(M) }{\mathscr{R}_0'(M)^2} > 2 \] then there is never a poverty-trap.

This condition is satisfied by any basic-reproductive-number-function that locallly resembles a power law with exponent between $-1$ and $0$. For example, if $\mathscr{R}_0(M) = k/(1+r \sqrt{M})$, there is no poverty trap. The boundary case is $\mathscr{R}_0(M) = k/(1+r \sqrt{M})$, which weakly satisfies the condition everywhere. The second situation is one where $1/\mathscr{R}_0(M)$ is convex, but does not change sufficiently fast to create multiple crossings.

Theorem. In general, if $1/\mathscr{R}_0(M)$ is convex and \[ \left. (m_I-m_S)\mathscr{R}'_0(M^*) \right|_{\mathscr{R}_0=1} < 1 \] then there is only one fixed point and no poverty trap.

So, if increased income is in-effective at reducing disease risk, there is no poverty trap.