Szegő's problem on curves

<p>(a) Map of Kenya showing HIV prevalence distributions. The color bar from blue to red is in the order of increasing HIV prevalence. For clarity, the names of counties included are the only ones included in this study (Source of data:ArcGIS.com: shapefile-The 47 counties of Kenya (shapefile by dmuthami <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0142805#pone.0142805.s005" target="_blank">S5 Table</a>) and HIV data from [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0142805#pone.0142805.ref039" target="_blank">39</a>]. (b) Human travel networks (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0142805#pone.0142805.s006" target="_blank">S6 Table</a>) as estimated by [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0142805#pone.0142805.ref038" target="_blank">38</a>]. Monthly average number of trips per 1000 individuals between all pairs of regions over the course of the year. For clarity, only trips made per 1000 individuals that are more than 60 trips per year are shown, with arrows indicating the direction of movements from home region to a visited region. The thickness of the arrow represents the number of trips made.</p

Molecule populations <i>N</i>_<i>i</i>(<i>t</i>) for the self-replicating chemical reaction system Eq (12) at generation <i>t</i>.

<p>Molecule populations <i>N</i>_<i>i</i>(<i>t</i>) for the self-replicating chemical reaction system <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0200601#pone.0200601.e084" target="_blank">Eq (12)</a> at generation <i>t</i>.</p

The mass action population dynamics of system Eq (3) in a physical scenario.

<p>(a) Solutions of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0200601#pone.0200601.e112" target="_blank">Eq (15)</a> in log-normal scale, i.e., x-axis is in normal scale and y-axis is in logarithmic scale. (b) The normalised molecule populations for the same period as in (a).</p

Molecule populations <i>N</i>_<i>i</i>(<i>t</i>) for the self-replicating chemical reaction system Eq (3) at generation <i>t</i>.

<p>Molecule populations <i>N</i>_<i>i</i>(<i>t</i>) for the self-replicating chemical reaction system <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0200601#pone.0200601.e010" target="_blank">Eq (3)</a> at generation <i>t</i>.</p

Simulation of the model for different values of <i>α</i>, <i>β</i> ∈ [0, 1].

<p>(a) Values of <i>p</i>_<i>t</i> for all <i>α</i> and <i>β</i> combinations; (b) Corresponding values of <i>R</i>. Parameter values used are in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0172401#pone.0172401.t001" target="_blank">Table 1</a>.</p

Parameter estimates from Uganda data.

<p>Parameter estimates from Uganda data.</p

Simulation of the steady states with <i>p</i>_<i>t</i> ∈ [0, 1].

<p>The red line is for the stunted while the blue line is for the children on treatment. Note that maximum <i>T</i> (minimum <i>R</i>) is achieved at the point <i>p</i>_<i>t</i> = 0.51. This gives The parameter values used are given in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0172401#pone.0172401.t001" target="_blank">Table 1</a>.</p

Distribution of the characteristic numbers Λ for all the analysable self-replicating chemical reaction systems up to where the possibly largest molecule is .

<p>The blue lines correspond to all <i>integer self-replicating systems</i>, while the red bars correspond to all other systems.</p

Simulation of the model for different values of <i>p</i>_<i>t</i> ∈ [0, 1], and initial conditions <i>I</i> = 1,000, <i>T</i> = 0, and <i>R</i> = 0.

<p>Parameter values used are in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0172401#pone.0172401.t001" target="_blank">Table 1</a>.</p

Normalised molecule population <i>n</i>_<i>i</i>(<i>t</i>) = <i>N</i>_<i>i</i>(<i>t</i>)/∑_<i>l</i> <i>N</i>_<i>l</i>(<i>t</i>) for 60 generations for the chemically realistic self-replicating reaction system Eq (3).

<p>Specifically, lim_{<i>t</i>→∞} <i>n</i>₁(<i>t</i>) = 0.38196…, lim_{<i>t</i>→∞} <i>n</i>₂(<i>t</i>) = lim_{<i>t</i>→∞} <i>n</i>₅(<i>t</i>) = 0.23606… and lim_{<i>t</i>→∞} <i>n</i>₃(<i>t</i>) = 0.14589…</p