Analysis/Reff_derivation2.md

In cmhoove14/DDNTD: Density Dependence in Neglected Tropical Diseases

Reff Derivation

Basic Schistosomiasis Model

Beginning with a basic schistosomiasis model with S − E − I infection dynamics and logistic population growth among the intermediate host snail population and human infection modeled via the state variable W representing the mean worm burden across the human population, assumed to be negative binomially distributed with clumping parameter κ. We have four ODEs:

-%5Cmu_N%20S-%5CLambda%20S)

E)

W)

Where N = S + E + I, Λ is the man-to-snail force of infection (FOI), and λ is the snail-to-man FOI, more details on these below.

Parameter symbology, values and definitions are shown in table 1.

Value Units Description fN 10 Sd−1 Snail fecundity rate K 50 Nm−2 Snail environmental carrying capacity μN 60 Nd−1 Snail mortality rate σ 40 Ed−1 Pre-patent period μI 10 Id−1 Excess mortality of infected snails μW 3.3 Wy−1 Adult parasite mortality rate H 1.5 Hm−2 Human host population μH 60 Hy−1 Human host mortality rate λ 4820 WI−1d−1 Man-to-snail transmission parameter β 625000 ES−1d−1 Snail-to-man transmission parameter α 0.005 unitless Adult parasite density dependent fecundity parameter ξ 0.0018 unitless Density dependent parasite establishment acquired immunity parameter

Reff Derivation

Dimensionality reduction: fast snail infection dynamics

We begin with the assumption that the rate of change of the intermediate host infection dynamics is fast compared to the adult parasites and therefore reaches an equilibirum, i.e. . We then solve for the equilibirum infected snail population, I*:

Solve for I*

Solve for E*

E%5E*%3D0)

Solve for S*

First, we write the equilibrium total snail population, N*, in terms of S* by substituting for E* and I* from N* = S* + E* + I*:

%7D)

%7D%5CBig))

Then, with %5Cbig(S%5E+E%5E%5Cbig)-%5Cmu_N%20S%5E-%5CLambda%20S%5E)

we substitute for E* and N* and divide by S* to arrive at:

%7D%5Cbig)%7D%7BK%7D%5CBigg)%5CBigg(1+%5Cfrac%7B%5CLambda%20%7D%7B%5Cmu_N+%5Csigma%7D%5CBigg)-%5Cmu_N-%5CLambda%20%3D0)

Solving for S*:

%7D%5Cbig)%7D%7BK%7D%3D%5Cfrac%7B%5Cmu_N+%5CLambda%20%7D%7Bf_N%5CBig(1+%5Cfrac%7B%5CLambda%20%7D%7B%5Cmu_N+%5Csigma%7D%5CBig)%7D)

%7D%5CBig)%7D%7BK%7D%3D1-%5Cfrac%7B%5Cmu_N+%5CLambda%20%7D%7Bf_N%5CBig(1+%5Cfrac%7B%5CLambda%20%7D%7B%5Cmu_N+%5Csigma%7D%5CBig)%7D)

%7D%5CBigg)%5CBigg(%5Cfrac%7BK%7D%7B%5CBig(1+%5Cfrac%7B%5CLambda%20%7D%7B%5Cmu_N+%5Csigma%7D+%5Cfrac%7B%5Csigma%5CLambda%20%7D%7B%5Cmu_I(%5Cmu_N+%5Csigma)%7D%5CBig)%7D%5CBigg))

Now want to try and simplify this. To start, we'll assign to give:

%7D%5CBigg)%5CBigg(%5Cfrac%7BK%7D%7B%5Cbig(1+C_1+%5Cfrac%7B%5Csigma%20C_1%7D%7B%5Cmu_I%7D%5Cbig)%7D%5CBigg))

Then distribute:

%7D-%5Cfrac%7BK%5Cbig(%5Cmu_N+%5CLambda%20%5Cbig)%7D%7Bf_N%5Cbig(1+C_1%5Cbig)%5Cbig(1+C_1+%5Cfrac%7B%5Csigma%20C_1%7D%7B%5Cmu_I%7D%5Cbig)%7D)

Then multiply the LHS by %7D%7Bf_N%5Cbig(1+C_1%5Cbig)%7D)

%5Cbig)-K%5Cbig(%5Cmu_N+%5CLambda%20%5Cbig)%7D%7Bf_N%5Cbig(1+C_1%5Cbig)%5Cbig(1+C_1+%5Cfrac%7B%5Csigma%20C_1%7D%7B%5Cmu_I%7D%5Cbig)%7D)

%7D%7Bf_N%5Cbig(1+C_1%5Cbig)%5Cbig(1+C_1+%5Cfrac%7B%5Csigma%20C_1%7D%7B%5Cmu_I%7D%5Cbig)%7D)

This is basically where we left it in the Elimination Feasibility paper, but I think it might be possible to simplify a bit more by first factoring out fN from the numerator and then canceling:

%7D%7B%5Ccancel%7Bf_N%7D%5Cbig(1+C_1%5Cbig)%5Cbig(1+C_1+%5Cfrac%7B%5Csigma%20C_1%7D%7B%5Cmu_I%7D%5Cbig)%7D)

%7D%7B%5Cbig(1+C_1%5Cbig)%5Cbig(1+C_1+%5Cfrac%7B%5Csigma%20C_1%7D%7B%5Cmu_I%7D%5Cbig)%7D)

Then distribute in the denominator:

%7D%7B%5Cfrac%7B2%5Csigma%20C_1%5E2%7D%7B%5Cmu_I%7D+%5Cfrac%7B3%5Csigma%20C_1%7D%7B%5Cmu_I%7D+1%7D)

And then factor out from the denominator:

%7D%7B%5Cfrac%7B%5Csigma%20C_1%7D%7B%5Cmu_I%7D%5Cbig(2C_1+3+%5Cfrac%7B%5Cmu_I%7D%7B%5Csigma%20C_1%7D%5Cbig)%7D)

Now with the rate of change of the mean worm burden:

W)

And: %5CBig(%5Cfrac%7B%5CLambda%20%7D%7B%5Cmu_N+%5Csigma%7D%5CBig)S%5E*) and

We get:

S%5E*-(%5Cmu_W+%5Cmu_H)W)

%5CBig(%5Cfrac%7BK%5Cbig(1+C_1-%5Cfrac%7B%5Cmu_N%7D%7Bf_N%7D-%5Cfrac%7B%5CLambda%20%7D%7Bf_N%7D%5Cbig)%7D%7B%5Ccancel%7B%5Cfrac%7B%5Csigma%20C_1%7D%7B%5Cmu_I%7D%7D%5Cbig(2C_1+3+%5Cfrac%7B%5Cmu_I%7D%7B%5Csigma%20C_1%7D%5Cbig)%7D%5CBig)-(%5Cmu_W+%5Cmu_H)W)

%7D%7B%5Cbig(2C_1+3+%5Cfrac%7B%5Cmu_I%7D%7B%5Csigma%20C_1%7D%5Cbig)%7D%5CBig)-(%5Cmu_W+%5Cmu_H)W)

Now factor out (μW + μH)W to get:

W%5CBig)%5CBig(%5Cfrac%7BK%5Clambda%5Cbig(1+C_1-%5Cfrac%7B%5Cmu_N%7D%7Bf_N%7D-%5Cfrac%7B%5CLambda%20%7D%7Bf_N%7D%5Cbig)%7D%7B%5Cbig(2C_1+3+%5Cfrac%7B%5Cmu_I%7D%7B%5Csigma%20C_1%7D%5Cbig)%5Cbig((%5Cmu_W+%5Cmu_H)W%5Cbig)%7D-1%5CBig))

Given the definition of Reff as the number of adult worms produced by a single adult worm over its lifespan, we interpret (μW + μH) as the mean lifespan and therefore have:

W(R_%7Beff%7D-1))

and therefore:

%7D%7B%5Cbig(2C_1+3+%5Cfrac%7B%5Cmu_I%7D%7B%5Csigma%20C_1%7D%5Cbig)%5Cbig((%5Cmu_W+%5Cmu_H)W%5Cbig)%7D)

This gives us our first key analytic finding, the identification of the breakpoint. If Reff < 1, which implies that an Reff < 1 causes a worm population on the way to extinction (e.g. below the breakpoint)

Now plugging back in for C1:

%7D%7B%5Cbig(%5Cfrac%7B2%5CLambda%20%7D%7B%5Cmu_N+%5Csigma%7D+3+%5Cfrac%7B%5Cmu_I(%5Cmu_N+%5Csigma)%7D%7B%5Csigma%5CLambda%20%7D%5Cbig)%5Cbig((%5Cmu_W+%5Cmu_H)W%5Cbig)%7D)

Then factor out %7D) from the numerator, factor out from the denominator, and separate the pesky (μW + μH)W

%7D%5Cbig)%5Cbig(f_N(%5Cmu_N+%5Csigma)+f_N%5CLambda%20-%5Cmu_N(%5Cmu_N+%5Csigma)-%5CLambda%20(%5Cmu_N+%5Csigma)%5Cbig)%7D%7B%5Cfrac%7B1%7D%7B%5Cmu_N+%5Csigma%7D%5Cbig(2%5CLambda+3(%5Cmu_N+%5Csigma)+%5Cfrac%7B%5Cmu_I(%5Cmu_N+%5Csigma)%5E2%7D%7B%5Csigma%5CLambda%7D%5Cbig)%7D%5Ctimes%5Cfrac%7B1%7D%7B%5CBig((%5Cmu_W+%5Cmu_H)W%5CBig)%7D)

The in both the numerator and denominator cancel, the remaing can be moved to the denominator leaving:

+f_N%5CLambda%20-%5Cmu_N(%5Cmu_N+%5Csigma)-%5CLambda%20(%5Cmu_N+%5Csigma)%5Cbig)%7D%7Bf_N%5Cbig(2%5CLambda+3(%5Cmu_N+%5Csigma)+%5Cfrac%7B%5Cmu_I(%5Cmu_N+%5Csigma)%5E2%7D%7B%5Csigma%5CLambda%7D%5Cbig)%7D%5Ctimes%5Cfrac%7B1%7D%7B%5CBig((%5Cmu_W+%5Cmu_H)W%5CBig)%7D)

Some more fenagling of the numerator (distribute and then factor out (μN + σ + Λ)) gives:

%5CBig(f_N-%5Cmu_N-%5Cfrac%7B%5CLambda%5Csigma%7D%7B(%5Cmu_N+%5Csigma+%5CLambda)%7D%5CBig)%7D%7Bf_N%5Cbig(2%5CLambda+3(%5Cmu_N+%5Csigma)+%5Cfrac%7B%5Cmu_I(%5Cmu_N+%5Csigma)%5E2%7D%7B%5Csigma%5CLambda%7D%5Cbig)%7D%5Ctimes%5Cfrac%7B1%7D%7B%5CBig((%5Cmu_W+%5Cmu_H)W%5CBig)%7D)

So this tells us for Reff > 0, . I think this makes sense and is interpretable as the intermediate host birth rate, fN, has to be greater than the intermediate host baseline mortality rate, μN, and excess mortality from infection, , else the intermediate host population would go extinct. Let's keep this in mind but continue on.

Factoring out fN from the RHS of the numerator we have:

%5CBig(1-%5Cfrac%7B%5Cmu_N%7D%7Bf_N%7D-%5Cfrac%7B%5CLambda%5Csigma%7D%7Bf_N(%5Cmu_N+%5Csigma+%5CLambda)%7D%5CBig)%7D%7B%5Ccancel%7Bf_N%7D%5Cbig(2%5CLambda+3(%5Cmu_N+%5Csigma)+%5Cfrac%7B%5Cmu_I(%5Cmu_N+%5Csigma)%5E2%7D%7B%5Csigma%5CLambda%7D%5Cbig)%7D%5Ctimes%5Cfrac%7B1%7D%7B%5CBig((%5Cmu_W+%5Cmu_H)W%5CBig)%7D)

Now if we distribute and factor out in the denominator we get:

%5CBig(1-%5Cfrac%7B%5Cmu_N%7D%7Bf_N%7D-%5Cfrac%7B%5CLambda%5Csigma%7D%7Bf_N(%5Cmu_N+%5Csigma+%5CLambda)%7D%5CBig)%7D%7B%5CBig(%5Cfrac%7B1%7D%7B%5Csigma%5CLambda%7D%5CBig)%5CBig(2%5Csigma%5CLambda%5E2+3%5Csigma%5CLambda%5Cmu_N+3%5Csigma%5E2%5CLambda+%5Cmu_I%5Cmu_N%5E2+2%5Cmu_I%5Cmu_N%5Csigma+%5Cmu_I%5Csigma%5E2%5CBig)%7D%5Ctimes%5Cfrac%7B1%7D%7B%5CBig((%5Cmu_W+%5Cmu_H)W%5CBig)%7D)

Now move the Λ to the numerator and cancel out a σ in the denominator:

%5CBig(1-%5Cfrac%7B%5Cmu_N%7D%7Bf_N%7D-%5Cfrac%7B%5CLambda%5Csigma%7D%7Bf_N(%5Cmu_N+%5Csigma+%5CLambda)%7D%5CBig)%7D%7B2%5CLambda%5E2+3%5CLambda%5Cmu_N+3%5Csigma%5CLambda+%5Cfrac%7B%5Cmu_I%5Cmu_N%5E2%7D%7B%5Csigma%7D+2%5Cmu_I%5Cmu_N+%5Cmu_I%5Csigma%7D%5Ctimes%5Cfrac%7B1%7D%7B%5CBig((%5Cmu_W+%5Cmu_H)W%5CBig)%7D)

Then with some final finagling of the denominator we get:

%5CBig(1-%5Cfrac%7B%5Cmu_N%7D%7Bf_N%7D-%5Cfrac%7B%5CLambda%5Csigma%7D%7Bf_N(%5Cmu_N+%5Csigma+%5CLambda)%7D%5CBig)%7D%7B%5CBig(%5CLambda(2%5CLambda+3%5Cmu_N+3%5Csigma)+%5Cmu_I(%5Cfrac%7B%5Cmu_N%5E2%7D%7B%5Csigma%7D+2%5Cmu_N+%5Csigma)%5CBig)%5CBig((%5Cmu_W+%5Cmu_H)W%5CBig)%7D)

Which we can also write as:

%7D%7B%5CBig(%5CLambda(2%5CLambda+3%5Cmu_N+3%5Csigma)+%5Cmu_I(%5Cfrac%7B%5Cmu_N%5E2%7D%7B%5Csigma%7D+2%5Cmu_N+%5Csigma)%5CBig)%5CBig((%5Cmu_W+%5Cmu_H)W%5CBig)%7D%5Ctimes%5CBig(1-%5Cfrac%7B%5Cmu_N%7D%7Bf_N%7D-%5Cfrac%7B%5CLambda%5Csigma%7D%7Bf_N(%5Cmu_N+%5Csigma+%5CLambda)%7D%5CBig))

cmhoove14/DDNTD documentation built on Nov. 23, 2019, 7:04 p.m.