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| <h1>SI(S) Disease Model Mathematics</h1> | |
| <img src="images/sis.gif" align=CENTER HSPACE=10 VSPACE=10 border=0> | |
| <p>The basic <em>SI</em> (Susceptible, Infectious) | |
| and <em>SIS</em> (Susceptible, Infectious 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, can be defined by the two difference equations below: | |
| <ul> | |
| <li><em>Δs = μ − βs i + γi − | |
| μs</em></li> | |
| <li><em>Δi = βs i − γi − μi</em></li> | |
| </ul> | |
| <p>Where: | |
| <ul> | |
| <li><em>s</em> is the proportion of the population <em>Susceptible</em> to the disease. </li> | |
| <li><em>i</em> is the proportion of the population that is <em>Infectious</em></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. In the STEM | |
| new disease wizard, <em>β</em> is specified using the "Transmission Rate" property.</li> | |
| <li>γ is the rate at which individuals clear infection. In this model these | |
| individuals become susceptible again after clearing infection. In a pure <em>SI</em> model, | |
| this rate is 0. In the STEM new disease wizard, <em>γ</em> is specified using | |
| the "Infectious Recovery Rate" property.</li> | |
| </ul> | |
| </p> | |
| <p>In the first equation, the <em>Susceptible</em> population | |
| increases when new members are born. This value is the birth rate μ | |
| multiplied by the total population which, since the values are normalized, is 1. It also increases | |
| due to <em>Infectious</em> population members recovering. The <em>Susceptible</em> population | |
| decreases by members who die. That value is μ, the mortality rate, | |
| multiplied by the <em>Susceptible</em> population, <em>s</em>. The <em>Susceptible</em> | |
| population also decreases by having members become <em>infected</em>. | |
| The product of <em>β</em> and <em>i</em> gives the normalized | |
| number of <em>Susceptible</em> population members that would become | |
| infected for each <em>Infectious</em> population member assuming all | |
| population members are in the <em>Susceptible</em> state. Multiplying | |
| that by <em>s</em>, the fraction that actually are <em>Susceptible</em>, | |
| gives the normalized amount that become <em>Infectious</em>.</p> | |
| <p>In the second equation, the <em>Infectious</em> population | |
| increases by the number of <em>Susceptible</em> population members that | |
| become <em>Infectious</em> (the first term). It also decreases by the | |
| proportion that <em>Recover</em> from the disease (middle term) and by | |
| the proportion that die (last term).</p> | |
| <p>Frequently, being infected by a disease will increase the | |
| likelihood that a population member will die. The model above needs to | |
| be enhanced to include the likelihood of a fatal infection and a | |
| potentially different rate at which infected members die.</p> | |
| <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. The actual mortality rate at which infected population | |
| members die is <em>μ+μ<sub>i</sub></em> (the background mortality rate plus | |
| the infectious mortality rate) </li> | |
| </ul> | |
| </p> | |
| <p>We modify our model to include this additional rate | |
| <ul> | |
| <li><em>Δs = μP − βs i + γi | |
| − μ s</em></li> | |
| <li><em>Δi = βs i − | |
| γi − (μ+μ<sub>i</sub>)i</em></li> | |
| </ul> | |
| </p> | |
| <h3>Spatial Adaptation</h3> | |
| <p>The "SI" disease model computations in STEM enhance these | |
| equations by adapting them to populations that are spatially | |
| distributed. This relaxes the assumption that the populations are at a | |
| single location and opens up the possibility that different locations | |
| could have different areas and numbers of population members (i.e., | |
| different population densities). To accommodate this situation STEM | |
| maintains separate disease state values for each location and uses | |
| unnormalized versions of the equations presented above. We develop those | |
| below. | |
| <p>To account for population differences at different locations, we | |
| define a new parameter <em>P<sub>l</sub></em> which is the number of | |
| population members at location <em>l</em> (Note: <em>P<sub>l</sub> | |
| = S<sub>l</sub> + I<sub>l</sub></em>). We also need to account for variability in the disease | |
| transmission (infection) rate, β, due to potentially different | |
| population densities. This modification is based upon the assumption | |
| that locations with greater population densities will have a higher | |
| effective transmission rates than locations with lower densities (i.e., | |
| one value can't be used for all locations). Thus, we need to replace the | |
| <em>β</em> in the non-spatial versions of the equations with a <em>β | |
| <sub>l</sub></em> that is specific to the location.</p> | |
| Making the substitution for | |
| <em>β</em> | |
| and multiply both sides of the equations by | |
| <em>P<sub>l</sub></em> | |
| , we obtain: | |
| </p> | |
| <ul> | |
| <li><em>Δs P<sub>l</sub>= μP<sub>l</sub> − | |
| β<sub>l</sub> s i P<sub>l</sub> + γ i 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> | |
| </ul> | |
| We choose the value for | |
| <em>β<sub>l</sub></em> | |
| to be original | |
| <em>β</em> | |
| scaled by the ratio between the population density at the location and | |
| the average population density of all locations. | |
| <ul> | |
| <li><em> β<sub>l</sub> = β d<sub>l</sub>/APD</em></li> | |
| </ul> | |
| Where | |
| <em>d<sub>l</sub></em> | |
| is the population density at location | |
| <em>l</em> | |
| , and | |
| <em>APD</em> | |
| is the average population density for all locations. | |
| <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>. , 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> (both states combined). For readability, we | |
| drop the <em>l</em> subscript and substitute.</p> | |
| <ul> | |
| <li><em>ΔS = μP<sub>l</sub> − β<sub>l</sub> | |
| S i + γI − μ S</em></li> | |
| <li><em>ΔI = β<sub>l</sub> S i | |
| − γI − (μ+μ<sub>i</sub>)I </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 + γI − 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> | |
| </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 + γI − μ S</em></li> | |
| <li><em>ΔI = β<sup>*</sup> S I | |
| − γI − (μ+μ<sub>i</sub>)I </em></li> | |
| </ul> | |
| <p>Computing <em>β<sup>*</sup></em> is straightforward. Let <em>TSF<sub>l</sub> | |
| = (d<sub>l</sub>/(APDP<sub>l</sub>)</em> be the <em>transmission scale | |
| factor</em> at location <em>l</em>.</p> | |
| <p>Thus | |
| <ul> | |
| <li><em>β<sup>*</sup> = β TSF<sub>l</sub></em></li> | |
| </ul> | |
| </p> | |
| <p>Substituting <em>P<sub>l</sub> = S + I</em>, <em>APD = | |
| P/Area</em> and <em>d<sub>l</sub> = (S+I)/Area<sub>l</sub></em>, where <em>P</em> | |
| is the total population for all locations, <em>Area<sub>l</sub></em> is | |
| the area of location <em>l</em>, and <em>Area</em> is the total area of | |
| all locations, we get: | |
| <ul> | |
| <li><em>TSF<sub>l</sub> = ((S+I)/Area<sub>l</sub>) / (P/Area | |
| (S+I))</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> | |
| So the | |
| <em>TSF<sub>l</sub></em> | |
| is the product of the reciprocal of the total population of all | |
| locations and the constant ratio between the area of the location and | |
| the total area of all locations. The former can be computed by | |
| accumulating the population of all locations as they are generated and | |
| the later ratio can be computed once at the start of the simulation. | |
| </p> | |
| <h3>Neighboring Infectious Populations</h3> | |
| <p>The extension of the non-spatial model into a spatial one in STEM | |
| also needs to account for infectious population members that reside in a | |
| location's "neighbors". Consider a location with no infections that is | |
| physically adjacent to several locations that have large infectious | |
| populations. This physical adjacency would naturally lead to | |
| population-to-population contact and eventually to disease transmission. | |
| We need to further extend the equations we are here to incorporate this | |
| aspect of a spatially distributed population.</p> | |
| <p>In STEM, a location has another location as a neighbor <em>Relationship</em> | |
| that links it to that location. If the <em>Relationship</em> represents | |
| the exchange of of population members (i.e., some kind of transportation | |
| relationship like pathways, roads or air travel) then it would be | |
| possible for <em>Infectious</em> population members from a neighbor to | |
| "visit" a location. We need to account for this potential by increasing | |
| the "effective" <em>Infectious</em> population at a location when doing | |
| our computations. Each Relationship has a rate at which population | |
| members travel from one location to another. It is assumed that the | |
| visitors would have the same level of "infectious contact" as an | |
| infectious member of the population at the current location (i.e., that | |
| they could could be counted as an infectious member of the population at | |
| the current location).</p> | |
| <p>The equations become:</p> | |
| <ul> | |
| <li><em>ΔS = μP<sub>l</sub> − β<sup>*</sup> | |
| S (I + I<sub>neighbor</sub>() ) + γI − 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> | |
| </ul> | |
| </p> | |
| <p>Where | |
| <ul> | |
| <li><em>I<sub>neighbor</sub>() = (∑ <sub>m</sub> I <sub>m</sub>mixingfactor)/(∑<sub>m</sub>P<sub>m</sub>mixingfactor)</em> | |
| is the number of <em>Infectious</em> visitors from neighbor <em>m</em>.</li> | |
| <li><em>mixingfactor</em> is a constant that varies depending on the type of relationship. For neighboring locations | |
| (i.e. people migrating across border) STEM uses the "Phys.Adj.Inf.Proportion" property on the new disease | |
| wizard to specify the mixing factor. For road transportation, STEM uses "Road.Net.Inf.Proportion" to | |
| specify the mixing factor for roads between locations.</li> | |
| <li>P<sub>m</sub> is the number of population members at neighbor | |
| <em>m</em></li> | |
| <li>I<sub>m</sub> is the number of <em>Infectious</em> population | |
| members at neighbor <em>m</em></li> | |
| </ul> | |
| </p> | |
| <p>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)</em>, is the number of <em>Births</em></li> | |
| <li><em>D = μ S + (μ + μ<sub>i</sub> )</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> | |
| </p> | |
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