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.