## Multistrain models

### Model types are fuzzy

Statistical

Mechanistic (mathematical v. agent-based)

### History-based models

Track individuals' histories of infection, $S_{(0)}, S_{(1)}, S_{(1,2)},...$

Cross-immunity reduces probability of infection or infectiousness

Numerically unwieldy: $2^n$ histories for $n$ strains

### Dimensional reduction

Requires cross-immunity through reduced transmission (only)

Cross-immunity must be in context of antigenic "neighborhoods"

(with $n$ strains in $m$ neighborhoods, system is $n$ x $(m+1)$)

Impossible to calculate immunity to new strains

### Status-based models

Track individuals S, I, or R to each strain

$O(n)$ equations possible

"Polarized" immunity: If $i$ and $j$ cross-react, infection with $j$ leads to perfect immunity to $i$ in some fraction of hosts

### Implicit assumption

Proportion of population not infected by second strain

Cross-immunity

### Know your assumptions

Which model promotes coexistence?

### Equations

Ecological dynamics depend on strain composition

### The question of strain space

Continuous or discrete?

How many dimensions?

### Agent-based models

Potentially most efficient with many strains

Support complex interactions

Naturally incorporate demographic stochasticity

### Analytic approaches

Equilibria: Solve for $S_{i}'=I_{i}'=...=0$

Stability analysis: Linearize at equilibria

Invasion analysis: Is $I_{2}'>0$ at $I_{1}^*$?

Integration: Calculate $I_{1}(t), I_{2}(t),...$

### Numerical integration

Approximate solution to $N'=f(N)$:

$$N(t_0+\Delta t) \approx N_0+ f(N_0) \Delta t = N_1$$

Re-iterating, we get

$$N_{z+1} \approx N_{z}+f(N_{z}) \Delta t$$

$$N_{z+1} \approx N_{z}+f(N_{z}) \Delta t$$

We could assume $f(x)$ is constant for $\Delta t$ (Euler's method)

Even better, average over multiple points!

### A second-order method

$$N_{z+1} \approx N_{z}+f(N_{z}) \Delta t$$

$$M_{z+1}=N_z+f(N_{z}) \Delta t$$

$$N_{z+1}=N_z+\frac{1}{2}\left[ f(N_{z})+f(M_{z+1})\right] \Delta t$$

### 4th-order Runge-Kutta

$$k_1=f(N_z) \Delta t$$

$$k_2=f\left(N_z+\frac{1}{2}k_1 \right) \Delta t$$

$$k_3=f\left(N_z+\frac{1}{2}k_2 \right) \Delta t$$

$$k_4=f\left(N_z+k_3 \right) \Delta t$$

$$N_{z+1}=N_z+\frac{1}{6}(k_1+2k_2+2k_3+k_4)$$

### Considerations for multistrain models

Form of immunity (duration, strength, polarity)

Effect of immunity (susceptibility, infectiousness, clearance)

Coinfections and timing of interactions

Accuracy for individuals v. populations

Dimensionality and discretization of strain space

Analytic and numeric approaches, including stochasticity