Multistrain models

In multistrain models, pathogens (or immune cells) interact via resource competition


Infection by strain $i$ creates immunity to strain $i$, but also to strain $j$

Consuming antigen $i$ by clone $x$ depletes antigen $i$ for consumption by clone $y$

Simple SIR model


$dS/dt = - \beta S I$

$dI/dt = \beta S I - \gamma I$

$dR/dt = \gamma I$

Simple SIR behavior

SIR model with vital dynamics


$dS/dt = \mu - \beta S I - \mu S$

$dI/dt = \beta S I - \gamma I - \mu I$

$dR/dt = \gamma I - \mu R$

Fundamental reproductive number


Growth when 100% of population is susceptible

$$dI/dt = \beta S I - \gamma I - \mu I = \beta I - \gamma I - \mu I = (\beta - \gamma - \mu) I$$

$$R_0 = \frac{\beta}{\gamma + \mu}$$

SIR with vital dynamics behavior

Solving for endemic equilibrium


Equilibrium when $S$, $I$ and $R$ remain constant through time

$$dI/dt = 0 = \beta S^* I^* - \gamma I^* - \mu I^*$$

$$\beta S^* I^* = \gamma I^* + \mu I^*$$

$$\beta S^* = \gamma + \mu$$

$$S^* = \frac{\gamma + \mu}{\beta} = \frac{1}{R_0}$$

With this equation, we can solve for equilibrium infecteds

$$I^* = \frac{\mu}{\beta} (R_0 - 1)$$

Two-strain SIR model, independence between strains

All the following have host birth/death arrows omitted for clarity

Two-strain SIR model, cross-immunity mediated through transmission

Two-strain SIR model, cross-immunity mediated through susceptibility

Variety of ways to parameterize interactions

Kucharski et al 2015

Two-strain SIR model, cross-immunity through polarizing immunity

Exercise: invasion of second strain into an endemic population

Assuming the model of polarizing immunity, allow a single strain to reach endemic equilibrium

$$S^*_1 = \frac{1}{R_0}, I^*_1 = \frac{\mu}{\beta} (R_0 - 1)$$

After strain 1 equilibrium is reached, allow strain 2 to enter the population. Strains share $R_0, \beta, \gamma, \mu$ parameters.

Questions: What is the endemic equilibrium $S^*_2$ for strain 2? Given $S^*_2$, what is the initial growth rate of strain 2? What effect does $\sigma$ have on this growth rate? When can strain 2 invade?

Dynamical outcomes of invasion

Restif and Grenfell 2007

Dynamical outcomes of invasion

Restif and Grenfell 2007

Under some regimes, may result in continual strain turnover

Gog and Grenfell 2002

Again, looks similar to strain replacement in influenza