# Why stigma's used to help, but don't any more

### by Tim Reluga in collaboration with Rachel Smith and David Hughes

Originally posted May, 2014. Last updated April 20, 2016.

Hi there. This page presents our little theory of the interaction of stigmas and disease prevalence. Please let me (T. Reluga) know if you have comments or questions about it.

Stigmas are a primal phenomena, ubiquitous in human societies past and present. Some evolutionary biologists have argued that stigmatization in response to disease is an adaptive behavior, that may help people and communities reduce the risks they face from infectious diseases and increase reproductive success. At the same time, cultural anthropologists and social critics have argued that stigmatization has strong negative impacts on community health. Smith and Hughes proposes to resolve this conflict by hypothesizing that stigmas had individual and group-evolutionary benefits early in human evolution may now be mal-adaptive.

In our new manuscript, we have proposed the first theory of the interactions between disease and stigmas that gives numerical predictions for how stigmas can increase or decrease prevalence. This web page is designed to let you interact with our model and see how the predictions change depending on the specific social and biological properties of a system.

Stigmatization can be thought of as a stepping-stone process. First, a person gets infected by some natural process, be it exposure to a disease reservoir, indirect transmission by a vector, or direct transmission from another infected person. The infection event itself, however, is almost never observed -- the infected person and the population have to wait for some informative signal like symptoms to reveal the new infection state, and the infected person might take action to conceal these symptoms as long as possible. Once the infection is revealed and becomes public knowledge, the population at large can impose a stigma onto the infected person.

To represent this process, each person is categorized in one of four possible states: susceptible, cryptically infected, infected and labelled, or stigmatized. The probabilities that a person is in each of these states is represented respectively by $S(t)$, $C(t)$, $L(t)$, and $Z(t)$, where $t$ represents time, measured in years. A person can also leave the system because of death. Each individual is initially healthy and in the susceptible state. The transitions between states are governed by a continuous-time Markov chain summarized in state diagram below.

Each of the 3 infected states (cryptic, labelled, and stigmatized) can have a different mortality rate and transmission rate. For example, labelled individuals can openly seek treatment without fear of revealing their infection, since everybody already knows, while crypticly infected people can not. The rate of infection then depends on the number of infected people.

From our theory, you can project the prevalence of disease depending on the rates of infection, stigmatization, and labelling. The basic result is that stigmatization decreases prevalence when the average number of transmission events by stigmatized people is less than the average number by labelled people in the absence of stigmatization. But it can also go the opposite way -- faster stigmatization can increase prevalence sometimes when stigmatized people transmit more than labelled people who have not been stigmatized.

The interactive plot below allows you to change the parameters for yourself and see how, for different parameters, changes in the stigmatization rate changes prevalence. The equations and solutions are given down below, for easy reference.

## Steady-state Prevalence Plot

Stigma ratio: $\mathscr{Z} =$
The parameters here are rates per year. The reciprocal of one of these rates is equal to the expected time until an event occurs. So, for example a stigmatization rate of $y = 3$ means a labelled person would be stigmatized after about 4 months.

### Theory details

In our theory, the probability that each person resides in one of our four states is governed by a continuous-time Markov chain. In matrix form, the probabilities of being in each state change according to the matrix equation \begin{gather} \begin{bmatrix} {\mathrm{d} S}/{\mathrm{d} t} \\ {\mathrm{d} C}/{\mathrm{d} t} \\ {\mathrm{d} L}/{\mathrm{d} t} \\ {\mathrm{d} Z}/{\mathrm{d} t} \end{bmatrix} = \begin{bmatrix} - m_{S} - \Lambda & 0 & 0 & 0 \\ \Lambda & - m_{C} -x & 0 & 0 \\ 0 & x & -m_{L} - y & 0 \\ 0 & 0 & y & -m_{Z} \end{bmatrix} \begin{bmatrix} S \\ C \\ L \\ Z \end{bmatrix} \end{gather}

The infection pressure $\Lambda$ can be calculated from a Ross-Lotka-McKendrick style compartmental epidemic model. Let $[S](t)$ be the density of healthy people who without signs of infection. Let $[C](t)$ be the density of infected people whose state has not yet been publicly identified. Let $[L](t)$ be the density of people who are infected and have had their infection publicly identified, and let $[Z](t)$ be the density of infected people who have been stigmatized for their state. To avoid complications associated with demographic transients, we assume a constant immigration rate of healthy people ($b$). Under a mass-action hypothesis, the infection pressure $\Lambda$ is a linear function of the density of infected individuals. Infected people become labelled at a constant rate, labelled people become stigmatized at a constant rate. This leads to a system of 4 ordinary differential equations with one additional algebraic constraint. \begin{align} \label{eq:si} \frac{d [S]}{d t} &= b - [S] m_{S} - [S] \Lambda \\ \frac{d [C]}{d t} &= [S] \Lambda - [C] m_{C} - [C] x \\ \frac{d [L]}{d t} &= [C] x - [L] m_{L} - [L] y \\ \frac{d [Z]}{d t} &= [L] y - [Z] m_{Z} \\ \Lambda &= \beta \left([C] + [L] \sigma + [Z] \eta\right) + \epsilon \end{align}

We determine steady-state community sizes and disease prevalence by setting time-derivatives in this system to zero and deriving a quadratic equation for $\Lambda$. As long as there is some external reservoir seeding infection ($\epsilon > 0$), there is a single steady-state infection pressure $\Lambda^* \in (0, \beta b/\mu_{S} + \epsilon)$ which we calculate numerically. The steady-state densities $[S]^*$, $[C]^*$, $[L]^*$, and $[Z]^*$ can then all be calculated using linear algebra. In the absence of reservoir effects ($\epsilon=0$), the basic reproduction ratio $$\mathscr{R}_0 = \frac{b \beta}{m_{S}} \left( \frac{1}{m_{C} + x} + \frac{\sigma x}{\left(m_{C} + x\right) \left(m_{L} + y\right)} + \frac{\eta x y}{m_{Z} \left(m_{C} + x\right) \left(m_{L} + y\right)} \right).$$

If we differentiate differentiate the basic reproduction ratio with respect to the stigmatization rate y, we discover that there is a special situation where it is independent of the stigmatization rate. Whether the stigmatization increases or decreases the reproduction ratio depends on a value we call the stigma ratio. From our basic parameters, we can calculate a stigma ratio $$\mathscr{Z} := \left( \frac{ \sigma \beta }{ m_{L} } \right) \left( \frac{ m_{Z} }{ \eta \beta } \right)$$ which predicts the impact of stigmatization. This stigma ratio is the average transmission by a labelled person relative to the average transmission of a stigmatized person. If $\mathscr{Z} > 1$, more stigma reduces prevalence.