| <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd"> |
| <html> |
| <head> |
| <meta http-equiv="Content-Type" content="text/html; charset=ISO-8859-1"> |
| <title>SIR Disease Model Mathematics</title> |
| </head> |
| <body> |
| <h1>SIR(S) Disease Model Mathematics</h1> |
| <img src="images/sirs.gif" align=CENTER HSPACE=10 VSPACE=10 border=0 width=80%> |
| |
| <p>The basic <em>SIR</em> (Susceptible, Infectious, Removed) |
| and <em>SIRS</em> (Susceptible, Infectious, Recovered, Susceptible) |
| disease models assume a uniform population at a single location and that |
| the population members are well "mixed", meaning that they are equally |
| likely to meet and infect each other. This model, for a normalized |
| population, is defined by the three equations below: |
| <ul> |
| <li><em>Δs = μ − βs i + σr − |
| μs</em></li> |
| <li><em>Δi = βs i − γi − μi</em></li> |
| <li><em>Δr = γi − σr − μr</em></li> |
| </ul> |
| |
| <p>Where: |
| <ul> |
| <li><em>s</em> is the proportion of the population that is |
| <em>Susceptible</em></li> |
| <li><em>i</em> is the proportion of the population |
| that is <em>Infectious</em></li> |
| <li><em>r</em> is the proportion of the population |
| that is <em>Removed</em> from the infectious and susceptible populations, |
| and therefore cannot be infected.</li> |
| <li><em>μ</em> both the rate of immigration (e.g., by birth) and emigration |
| (e.g., by death) from the population. These rates are assumed to be equal |
| over the time period of interest (this simplifies the mathematics).</li> |
| <li><em>β</em> is the disease transmission (infection) rate. |
| The rate at which infectious individuals infect susceptible individuals. Once |
| infected, susceptible individuals instantly become infectious themselves.</li> |
| <li>γ is the rate at which individuals clear infection. In this model these |
| individuals cannot be re-infected for some period of time after infection (whether through |
| immunity or removal from the population).</li> |
| <li><em>σ</em> is the immunity loss rate. This coefficient |
| determines the rate at which <em>Removed</em> population members lose |
| their immunity to the disease and become <em>Susceptible</em> again. For an SIR |
| model, this rate is 0. |
| </li> |
| </ul> |
| </p> |
| Following basically the same derivation as outlined for the |
| <a href="simath.html">SI</a> |
| model, these become: |
| |
| <p>Let |
| <ul> |
| <li><em>μ<sub>i</sub></em> be the <em>Infectious Mortality |
| Rate</em>. This is the increased rate at which infected population members die |
| specifically due to the disease.</li> |
| </ul> |
| </p> |
| <p>We modify our model to include this additional rate. |
| |
| <ul> |
| <li><em>Δs = μ − βs i + σr − |
| μ s</em></li> |
| <li><em>Δi = βs i − |
| γi − (μ+μ<sub>i</sub>)i</em></li> |
| <li><em>Δr = γi − σr |
| − μr</em></li> |
| </ul> |
| </p> |
| <h3>Spatial Adaptation</h3> |
| <p> |
| <ul> |
| <li><em>Δs P<sub>l</sub>= μP<sub>l</sub> − |
| β<sub>l</sub> s i P<sub>l</sub> + σ r P<sub>l</sub> − |
| μ s P<sub>l</sub></em></li> |
| <li><em>Δi P<sub>l</sub> = β<sub>l</sub> |
| s i P<sub>l</sub> − γ i P<sub>l</sub> − |
| (μ+μ<sub>i</sub>)i P<sub>l</sub></em></li> |
| <li><em>Δr P<sub>l</sub>= γiP<sub>l</sub> |
| − σr P<sub>l</sub>− μr P<sub>l</sub></em></li> |
| </ul> |
| |
| <p>Let <em>S<sub>l</sub> = s P<sub>l</sub></em> be the number of <em>Susceptible</em> |
| population members at location <em>l</em>. Similarly, let <em>I<sub>l</sub> |
| = i P<sub>l</sub></em> be the number of population members at location <em>l</em> |
| that are <em>Infectious</em>, and let <em>r |
| P<sub>l</sub></em> be the <em>Recovered</em> population. For readability, we |
| drop the <em>l</em> subscript and substitute. </p> |
| Substituting |
| |
| </p> |
| <ul> |
| <li><em>ΔS = μP<sub>l</sub> − β<sub>l</sub> |
| S i + σR − μ S</em></li> |
| <li><em>ΔI = β<sub>l</sub> S i |
| − γI − (μ+μ<sub>i</sub>)I </em></li> |
| <li><em>ΔR= γI − σR |
| − μR</em></li> |
| </ul> |
| |
| Continuing with |
| <em> i = I/P<sub>l</sub></em> |
| , we have: |
| <ul> |
| <li><em>ΔS = μP<sub>l</sub> − (β<sub>l</sub>/P<sub>l</sub>) |
| S I + σR − μ S</em></li> |
| <li><em>ΔI = (β<sub>l</sub>/P<sub>l</sub>) |
| S I − γI − (μ+μ<sub>i</sub>)I </em></li> |
| <li><em>ΔR= γI − σR |
| − μR</em></li> |
| </ul> |
| |
| Letting |
| <em>β<sup>*</sup> = β<sub>l</sub>/P<sub>l</sub> = β |
| (d<sub>l</sub>/(APDP<sub>l</sub>)) </em> |
| gives: |
| |
| <ul> |
| <li><em>ΔS = μP<sub>l</sub> − β<sup>*</sup> |
| S I + σR − μ S</em></li> |
| <li><em>ΔI = β<sup>*</sup> S I |
| − γI − (μ+μ<sub>i</sub>)I </em></li> |
| <li><em>ΔR= γI − σR |
| − μR</em></li> |
| </ul> |
| TSF |
| <ul> |
| <li><em>TSF<sub>l</sub> = ((S+I+R)/Area<sub>l</sub>) / |
| (P/Area(S+I+R))</em></li> |
| <li><em>TSF<sub>l</sub> = (1/Area<sub>l</sub>) / (P/Area )</em></li> |
| <li><em>TSF<sub>l</sub> = Area / (P *Area<sub>l</sub> )</em></li> |
| <li><em>TSF<sub>l</sub> = (1 / P)* (Area/Area<sub>l</sub> )</em></li> |
| </ul> |
| <h3>Neighboring Infectious Populations</h3> |
| </p> |
| <ul> |
| <li><em>ΔS = μP<sub>l</sub> − β<sup>*</sup> |
| S (I + I<sub>neighbor</sub>() ) + σR − μ S</em></li> |
| <li><em>ΔI = β<sup>*</sup> S (I |
| + I<sub>neighbor</sub>() ) − γI − (μ+μ<sub>i</sub>)I |
| </em></li> |
| <li><em>ΔR = γI − σR |
| − μR</em></li> |
| </ul> |
| |
| Specific statistics on the total number of births, deaths and deaths due |
| to the disease can be computed by adding the appropriate terms of the |
| equations above. |
| <ul> |
| <li><em>B= μ (S + I + R)</em>, is the number of <em>Births</em> |
| </li> |
| <li><em>D = μ S + (μ + μ<sub>i</sub> )I + μR</em>,is the total number of <em>Deaths</em></li> |
| <li><em>DD= μ<sub>i</sub> I</em>, is the number of |
| <em>Disease Deaths</em></li> |
| </ul> |
| </body> |
| </html> |
