Critical R₀ thresholds and confidence in seeing a reduction in CRS incidence for birthrate/vaccine coverage combinations.

<p>(A) Routine vaccination only, assuming even mixing across all population age groups. (B) Routine vaccination only, assuming assortative mixing and heterogeneities in contact between age groups. (C) Routine vaccination supplemented with SIAs of 1–4 year olds with 60% coverage every 4 years (assortative mixing). (D) Routine vaccinations and SIAs supplemented with a catch-up campaign covering 1–14 year olds with 60% coverage conducted when rubella vaccine is introduced. White circle shows the cell most closely corresponding to Guinea-Bissau. The square shows the cell most closely corresponding to Guinea Bissau. The diamond indicates the cell most closely corresponding to Somalia.</p

Occupancy of Cells.

<p>The occupancy of the cells, <i>Ψ</i>^<i>Ξ</i> (m), derived from the steady-state solution (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0137482#pone.0137482.e010" target="_blank">Eq 6</a>), is shown as a function of temperature and maximum immune response, A. The same phase transition observed as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0137482#pone.0137482.g006" target="_blank">Fig 6</a> is evident here.</p

Testing for a Thermodynamic Temperature.

<p>The figure shows thermodynamic temperature <i>T</i>_{<i>thermo</i>} vs <i>T</i>_model. For <i>T</i>_model below the phase transition, the relationship is linear with slope <i>k</i>_<i>B</i>. From Eq (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0137482#pone.0137482.e027" target="_blank">15</a>) the <i>effective k</i>_<i>B</i><i>T</i>_{<i>thermo</i>} is the inverse slope derived from a plot of lnΩ(<i>E</i>) vs. <i>E</i>. Above the phase transition a negative temperature is observed as expected. In the inset we plot the slope of <i>T</i>_{<i>thermo</i>} vs <i>T</i>_model for the values of maximum immune response, A, shown in the color legend, and observe that they all fall on a single line. This suggests that the effective <i>k</i>_<i>B</i> decreases linearly with increasing immune amplitude, <i>A</i>, and shows that immune strength rescales temperature.</p

Robustness vs. Evolvability.

<p>The robustness (<i>m</i>_robust/50) as a function of “evolvability”, σ, for each quasispecies, at all studied temperatures and immunities, <i>A</i> (red points). Here the robustness is defined by the order parameter, or average , for each quasispecies distribution <i>P(m)</i>. For a symmetric distribution this corresponds to the most probable m. The curve is nearly universal, breaking apart only near the phase transition. The colored segments and the green arrows indicate the trajectories as function of increasing immunity (for temperatures represented).</p

Virus Life Cycle.

<p>The changing states of all viruses must be computed self-consistently over the entire virus life cycle. The figure shows three important stages of the model virus life cycle: (I) Infection (entering the cell), (Ξ) Immune Clearance, and (R) Reproduction and exiting the cell. Also shown are the equations for cell occupancy at each stage.</p

Order Parameter.

<p>The order parameter, <i>M</i>_<i>env</i>, as determined by sampling virus in the environment to measure the average fraction of matching codons, as a function of temperature and maximum immune response, A. The order parameter is defined in Eq (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0137482#pone.0137482.e023" target="_blank">11</a>). Sampling virus inside the cells yields almost exactly the same result (see Figure C in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0137482#pone.0137482.s001" target="_blank">S1 Appendix</a>).</p

Viral Mutation Probabilities.

<p>For our model target, the transition probability (P_mut) as a function of the number of matches, m (pre-mutation). The figure shows the probability, P_mut, of a viral mutation causing an increase, decrease, or stasis in the number of matching codons between the virus and the actual target genome (Δm = +1,0,-1).</p

The Grand Canonical Ensemble for a System of Viruses.

<p>The three thermodynamic elements of the system are shown. The “Reservoir” of all possible virus is usually referred to as the “thermal bath”.[<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0137482#pone.0137482.ref023" target="_blank">23</a>] In this case the bath of possible viral sequences is very large (effectively infinite). Free virus in the environment, at steady state, is populated from the reservoir with a distribution based on temperature and immunity. In the infection phase, virus that successfully infect cells are drawn from the environment. Virus that fails to infect are returned to the reservoir. Immunity may remove virus (from the cells back to the reservoir), and reproduction draws new offspring from the reservoir and repopulates the environment (emptying the cells). The double arrows indicate population from and return to the reservoir. The curved arrows show the virus life cycle.</p

Entropy.

<p>The figure shows the entropy of virus, <i>S/k</i>_<i>B</i><i>= ln Ω</i>, while in the cells as a function of temperature and maximum immune response, A, where <i>k</i>_<i>B</i> is the effective Boltzmann constant.</p

Three Limiting Cases of the Effect of Viral Mutation in a Single Codon.

<p>The transition probabilitiy (P_mut) as a function of the number of matches, m (pre-mutation). Curves labeled ‘+1’ represent the probability of a mutation increasing the number of matches between the virus and target genomes, ‘-1’ the probability of decreasing the number of matches, and ‘0’ the probability of no change. Given alphabet length ‘<i>a</i>,’ the figure shows the limiting behavior for: (a) Small <i>a</i>, highly degenerate target; (b) <i>a ≈</i> target+1, i.e., medium degeneracy; (c) Large <i>a</i>, i.e., low target degeneracy.</p