From aafc2d7677c3f6fc20fdb423b5bff015dfa508fa Mon Sep 17 00:00:00 2001
From: smitdave
In the following, we derive formulas for the force of infection (FoI) +from the model for the attack rate (AR) under the Poisson and .
In mechanistic models of malaria, the hazard rate for exposure is +generally assumed to be a linear function of the entomological +inoculation rate. In the following, we assume that the number of bites +per person over a day (or over some longer interval, \(\tau\)), is a random variable, and we +formulate approximating models for attack rates and hazard rates.
+We let \(E\) denote the EIR, the +expected number of bites per person over a day. If we assume that the +distribution of the daily EIR is Poisson, and if a fraction \(b\) of infective bites cause an infection, +then the relationship between the between EIR and the FoI is a Poisson +compounded with a binomial, which is also Poisson:
+\[ +Z \sim F_E(z) = \mbox{Poisson}(z, \mbox{mu} = bE(t)) +\]
+Over a day, the daily attack rate, \(\alpha\), is the fraction of individuals +who received at least one infection, or:
+\[ +\begin{array}{rl} +\alpha &= 1-F_E(0) \\ &= 1-\mbox{Poisson}(0, \mbox{mu} = bE(t)) +\\ +&= 1- e^{-bE(t)} \\ +\end{array} +\]
+The daily FoI, \(h\), is given by a +generic formula:
+\[ +\alpha = 1 - e^{-h} \mbox{ or equivalently } h = -\ln (1-\alpha) +\]
+In this case, the relationship between the FoI and the EIR is:
+\[ + h(t) = b E(t) +\]
+It is highly mathematically convenient that the relationship is +invariant with respect to the sampling period.
+If we assume the number of infective bites, per person, per day, has +a Gamma distribution in a population, then we could model the number of +infective bites as a Gamma - Poisson mixture process, or a negative +binomial distribution. Under this model, the counts for bites by +sporozoite positive mosquitoes over one day, \(Z\), would be a negative binomial random +variable with mean \(E\):
+\[ +Z \sim F_E(z) = \mbox{NB}(z, \mbox{mu} = bE(t), \mbox{size} = 1/\phi) +\]
+Assuming an infectious bite causes an infection with probability +\(b\), the daily attack rate is:
+\[ +\begin{array}{rl} +\alpha &= 1-F_E(0) \\ &= 1-\mbox{NB}(0, \mbox{mu} = b E(t), +\mbox{size} = 1/\phi) \\ +&= 1- \left(1+b E(t)\phi \right)^{-1/\phi} +\end{array} +\]
+This is consistent with a formula that has a continuous daily +FoI:
+\[ + h = \frac{\ln \left(1 + bE(t)\phi \right)} {\phi} +\]
+
-itn_mod <- setup_control(itn_mod)
itn_mod <- setup_control_forced(itn_mod)
+
+itn_mod <- setup_vc_control(itn_mod)
If we ran the model at this point, we would get the baseline. Instead, we set up a time-varying function to compute the coverage of ITNs at any time point. We use a sine curve with a period of 365 days @@ -259,14 +261,14 @@
t
(time) and returns a scalar
value.
-
We use the null model of human demographic dynamics, which assumes \(H\) is constant for all time.
-
-itn_mod = setup_itn_lemenach(itn_mod, phi = ITN_cov)
+itn_mod = setup_itn_lemenach(itn_mod, F_phi=ITN_cov)
xde_solve(itn_mod, 5*365) -> itn_mod
itn_mod$outputs$orbits$deout -> out
+ - +
5s&)G1B97d; F-ut*urki0vzaCk#joaoz0
zhN$D}A0f%eO1C>oD}Meo1%ZGjgH>}4HVvm%0VV{K8eA9f3QuYFgDIQ*!-GDG^O9NX
z_V1rLR?YyMvVOZ`D*`Akjk`q$kM&Q17}+yI1{t)qq-WTugPqXIZ^0eEe;k2tdM0mH
zaeEDj9`f*O1$M4Y4!k!KpDYd41M?e(*gn4F|AzNH%y}B~O@?BOZ|g?Ao#|Vy$O6eL
zjH!681zuLUJkN?*s(b?T*jkH%lY&_6PK*I4S=;Cx>$ju#IKEt5@;|9?!wJWn1l4SO
ziGfPYH?f9n8m&0Lovh>VZwPore>=XK0D!Wy*x%b-#ME*M7=i$Wch8 t$11dP{*B?3A5
z4b9PGC)EqZx`yNjNp%&?HW3tbvKK3Cu(YhiI;RNicIhzE*BL&k{6(x=S
zwNOA0b(`iFZfCv-e&`vDdg!#rxiXEE$cX0WX&0yHCCi3vCkPR%E?)t6yC#XN%YMlO?x3r|`F&1K&c@w}78lThU=9TTK)S
{k8rtP#d~11s%jM~@D|vJpa!G?DT8YdCF-vYm+nn*B%f2BtrX&>pfrHG
zV#?ZDc4CK${<(kUhx2