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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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