Matt Michalska-Smith
Kimberly L. VanderWaal, Cesar A. Corzo
Montserrat Torremorell, & Meggan E. Craft

  1. Background
  2. Model setup
  3. Small network results
  4. Implications for large networks

  1. Background
    • Multi-strain disease
    • Metapopulation networks
  2. Model setup
  3. Small network results
  4. Implications for large networks

  • Influenza:
    • Hemagglutinin
    • Neuraminidase
H1N1
H5N1
$\color{f74700}{\beta}$

$1 - \color{f74700}{\beta}$ $\color{f74700}{\sigma}$
 $\color{b6003b}{\beta}$
 $\gamma \color{b6003b}{\beta}$
  1. Background
  2. Model setup
    • One population
    • Metapopulation networks
  3. Small network results
  4. Implications for large networks

What is a strain (mathematically)?

number of loci (N) and number of alleles (M)

$$\{x_1, x_2, \ldots, x_N\} \quad\quad x_n \in y_1, y_2, \ldots, y_M $$

e.g. a disease with two loci ($a$ and $b$) and two alleles ($1$ and $2$) at each locus has four strains:

$$ \{a_1, b_1\}, \{a_1, b_2\}, \{a_2, b_1\}, \{a_2, b_2\} $$

Hosts experience cross-reactive immunity when exposed to strains similar to those they have seen before,

i.e. strains $j \ni j \cap i \neq \text{Ø}$

Compartmental modeling

  1. those that are susceptible to strain $i$
  2. those that are currently infected with strain $i$
  3. those that have previously been infected with strain $i$
  4. those that have previously been infected with a similar strain $j$
$$ \begin{aligned} \frac{\textrm{d} y_i}{\textrm{d}t} &= \color{f74700}{\beta} \left( (1 - w_i) + (1 - \color{005342}{\gamma}) (w_i - z_i) \right) y_i - \color{016394}{\sigma} y_i - \color{b6003b}{\mu} y_i\\ \frac{\textrm{d} z_i}{\textrm{d}t} &= \color{f74700}{\beta} (1 - z_i) y_i - \color{b6003b}{\mu} z_i\\ \frac{\textrm{d} w_i}{\textrm{d}t} &= \color{f74700}{\beta} (1 - w_i) \sum_{j \ni j \cap i \neq \text{Ø}} y_j - \color{b6003b}{\mu} w_i \end{aligned} $$
  • $\color{f74700}{\beta}$: force of infection
  • $\color{005342}{\gamma}$: degree of cross-reactive immunity
  • $\color{016394}{\sigma}$: recovery rate
  • $\color{b6003b}{\mu}$: death rate

Gupta, Ferguson, & Anderson 1998
Lourenço, Wikramaratna, & Gupta 2015

Lourenço, Wikramaratna, & Gupta 2015

Lourenço, Wikramaratna, & Gupta 2015

$$ \color{1b264f}{\mathbf{\Delta}} = \begin{bmatrix} -\delta & 0 & 0 & \delta & 0 & 0 & 0\\ 0 & -\delta & 0 & 0 & 0 & \delta & 0\\ \delta & 0 & -3 \delta & \delta & \delta & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \delta & \delta & 0 & 0 & -2 \delta & 0 & 0\\ \delta & 0 & \delta & 0 & \delta & -4 \delta & \delta\\ 0 & 0 & \delta & \delta & 0 & 0 & -2 \delta \end{bmatrix} $$
$$ \begin{aligned} \frac{\textrm{d} y_{i,k}}{\textrm{d}t} &= \color{f74700}{\beta} \left( (1 - w_{i,k}) + (1 - \color{005342}{\gamma}) (w_{i,k} - z_{i,k}) \right) y_{i,k} - \color{016394}{\sigma} y_{i,k} - \color{b6003b}{\mu} y_{i,k} + \sum_l \color{1b264f}{\Delta_{kl}} y_{j,l}\\ \frac{\textrm{d} z_{i,k}}{\textrm{d}t} &= \color{f74700}{\beta} (1 - z_{i,k}) y_{i,k} - \color{b6003b}{\mu} z_{i,k} + \sum_l \color{1b264f}{\Delta_{kl}} z_{j,l}\\ \frac{\textrm{d} w_{i,k}}{\textrm{d}t} &= \color{f74700}{\beta} (1 - w_{i,k}) \sum_{j \ni j \cap i \neq \text{Ø}} y_{j,k} - \color{b6003b}{\mu} w_{i,k} + \sum_l \color{1b264f}{\Delta_{kl}} w_{j,l} \end{aligned} $$
  • $\color{f74700}{\beta}$: force of infection
  • $\color{005342}{\gamma}$: degree of cross-reactive immunity
  • $\color{016394}{\sigma}$: recovery rate
  • $\color{b6003b}{\mu}$: death rate
  • $\color{1b264f}{\Delta}$: matrix of inter-population movements
  1. Background
  2. Model setup
  3. Small network results
    • Migration dampens cycles
    • Destinations inherit dynamics
    • Dynamics inherit asymmetrically
  4. Implications for large networks

  1. Background
  2. Model setup
  3. Small network results
  4. Implications for large networks
    • Effect of degree distribution
    • Pinning down why

Result (from small networks)

Prediction (in large networks)

Migration dampens cycles:

Reduced cycle amplitude and disease prevalence in networks with longer chains

Destinations inherit dynamics:

Interconnected parts of the network should have similar dynamics

Dynamics inherit asymmetrically:

Preponderance of cyclical and chaotic dynamics

  • Simple patterns emerge in small metapopulation networks
    • When linked with migration, cyclical dynamics are dampened further down chains
    • Migration tends to lead to inheritance of dynamical regimes between populations
    • When multiple source populations are considered, more complicated dynamics (cycles and chaos) dominate the inheritance
  • While there are patterns in larger networks, they are not simple
    • More skewed degree distributions lead to higher prevalence, but not necessarily greater levels of immunity
  • Further research is needed to:
    • Generalize these small-network patterns to larger networks
    • Identify relationships between more complex network structures and disease dynamics
    • Fit to empirical data, including empirical network structures

Thank you!