disease frequency and interaction patterns

[MonDie]

I realized sexual polyamory could result in lower STI frequency per partner per individual, but the decreases are only 0.0 to 0.1 even under extreme conditions and with everybody doing it. The idea is that exposing organism-A to organism-B also indirectly exposes A to all B's contacts, so it's safer for A to interact with the contacts of previous contacts, reducing the total number of contacts (direct or indirect). My equations can model other epidemiological scenarios, but I can't get a universal equation for situations involving indirect transmission within the interacting groups.

y_n = the probability of being healthy after the nth contact, hence...
y₀ = the probability of starting healthy,
x = the probability that an infected contact won't transmit the infection.

When all individuals exhibit identical behavior, and t_n is the probability that infection transmission won't occur in the nth interaction,
[math]y_{n} = y_{n-1} * t_{n}[/math]

t_nmust jointly consider the probability of contacts being infected and transmitting it.
For expediency and readability, I'll denote y_n-1 as just y from now on.

When the infection must be transmitted directly within each group, the equation is simply
[math]y_{n} = y * (y+(1-y)*x)^{w}[/math].
The 2-way (w=1) and 3-way (w=2) versions are elaborated and explained in the spoiler.

2-way (contacts are random)
[math]t_{n} = y + (1-y)*x[/math]
where y is the contact being healthy, and (1-y)*x is the contact being infected but failing to transmit.
Now we can solve for y_n with substitution.
[math]y_{n} = y * (y + (1-y)*x)[/math]

3-way direct
Interactions are triangular and transmission within a triangle must be direct.
[math]y_{n} = [/math][math] y * (y^{2} + 2(y*(1-y)*x) + ((1-y)*x)^{2})[/math]
y^2 | both contacts are healthy
y*(1-y)*x*2 is | one contact is unhealthy, no transmission
(1-y)*x)^2 | both are unhealthy, no transmission
Because these are mutually exclusive outcomes, t_n is their sum rather than their product.

To go directly from y_n to y_n+t, just substitute the equation back into itself t times. This wxMaxima code will do it for you.

y:a; x:b; w:c;

for k:1 thru t do
y:y*((y+(1-y)*x)^(w));

y;

Cell 1 says to assign to y, x, and w the values <a>, <b>, and <c>.
Cell 2 says to reassign to y the value of y*((y+(1-y)*x)^(w)), and to repeat this <t> times.
Cell 3 says to output the new value of y.

the (w+1)-way indrect equation
It gets confusing when you allow for indirect transmission. For example, when a 3-way interaction utilizes the indirect routes, A→B→T and B→A→T, in addition to the direct routes A→T and B→T.

3-way indirect
[math]y_{n} = y * (y^{2} + 2(y*(1-y)*x*(1-(1-x)^{2})) + ((1-y)*x)^{2})[/math]

(1-(1-x)^2) = failure to transmit indirectly | This is multiplied by x to calculate the failure of both direct and indirect transmission.

Since you can only be infected once, indirect transmission isn't possible if all contacts were already infected.

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One way to check these equations is to check that y_n = y when x=1. If the infection is never transmitted, then the chance of having it equals the chance of starting with it. With wxmaxima, you can check this by assigning x=1, generating a 3D graph, and checking that the plane is flat rather than curved (example code below).

x:1; kill(y); kill(w);

load(draw);

draw3d(enhanced3d=true,explicit(y*((y+(1-y)*x)^w),w,0,1,y,0,1));

Edited July 25, 2014 by MonDie