mpFitting.md

In OR_Sim_SIR_MacPan.R we have implemented a fitting mechanism with MacPan2 for a fixed time frame, i.e.

• We simulate the real data from $t=0$ and the observed real data is from a time period between $t_{min}$ and $t_{max}=t_{min}+$duration of the data.
• Then in the fitting process, we assume we know $t_{min}$ and $t_{max}$ and offset the initial fitting time to $t_{min}$ This is not ideal:
• Unless we can track to real "patient 0" in the population, we can never tell the real $t=0$, so does $t_{min}$.
• This also leads to fitting problem that we have seen, if initial $\beta, \gamma$ value for fitting is far away from the "real" value, then the mechanism will fail.

As JD suggested, $S_t$ at any time $t$ should only affect the scaling of the outbreak size since its value only scales the susceptible pool, not the tendency. Therefore, we can assume $S(t_{min})=S_0$ while $t_{min}$ remains unknown and trying to fit the $S_0$ to some level. Similarly I use notation $I(t_{min})=I_0$ We still need $I_0$ and induce $R(0)=N-S_0-I_0$ as the as start parameter for the simulation of fitting.

This leads to one more degree of freedom(now 6, previously 5 as we use $I(0)=I_0$ and naturally assume $S_0=1-I_0$) in our fitting mechanism.

• $\beta$ as the transmission rate of underlying SIR model

• $\gamma$ as the recovery rate of underlying SIR model

• $T_B$ as the baseline testing probability, i.e. probability of negative people being tested.

  • Currently assumed to be a constant.
  • Bijection with the odds $B=\frac{T_B}{1-T_B}$

• $T_Y$ as the positive testing probability, i.e. probability of positive people being tested.

  • Currently assumed to be a constant.
  • Current Odds ratio $\Phi$ between $T_B$ and $T_Y$ is also assumed to be a constant:

  $$ x = 3 $$

  $$\Phi=\frac{\frac{T_Y}{1-T_Y}}{\frac{T_B}{1-T_B}}$$

• $S_0$ as the starting point of $S$ for fitting.

• $I_0$ as the starting point of $I$ for fitting.

Deduce the $I_0$ from data and other start values

In the data we observe and fitting the model to

• Observed testing probability $OT \sim \text{Binom}(N,T)$
• Observed testing positivity $OP \sim \text{Binom}(OT,p)$
• $T(t)$ is the proportion of all tests at time $t$ s.t. $$T= (1-Y) T_B + Y T_Y $$
  • $Y(t)=I(t)/N$
• $p(t)$ the proportion of positive test at time $t$ s.t. $$p= \frac{Y T_Y}{T}= \frac{1}{1+(\frac{1}{Y}-1)\frac{T_B}{T_Y}}$$ My idea would be deduce an approximation of $I_0$ from initial data (i.e. $OT(0)$ and $OP(0)$) and other start value $T_Y$ to "reduce" the degree of freedom to help the fitting machine works better, since we need more initial value in this machinery.

The expectation of $OT$ and $OP$ is $E[OT]=N\times T$ and $E[OP]=OT \times p$ at any time $t$ respectively, and we know $OT_0$ and $OP_0$ from the data. As N and $OT$ should be relatively large, we can use: $$ \hat{T}_0=\frac{OT_0}{N}$$$$ \hat{p}=\frac{OP_0}{OT_0} $$ Then the approximation of $I_0$ could be $$\hat{I}_0=\frac{\hat{p}\hat{T}_0}{T_Y}$$for each $T_Y$ we start with for the fitting. A constraint would be $S_0+\hat{I}_0 \leq N$ but it is not hard to satisfy as $S_0$ value is basically free and easy to adjust.

A primary implementation is in OR_Sim_SIR_S0.R. The fitting has successfully converge close enough for $\beta, \gamma, T_Y, T_B$ with large disturbances to multiple initial value, which previously leads to fitting failure in previous machinery:

• $\beta = \beta_{real}+0.40$ where $\beta_{real}=0.25$
• $\gamma = \gamma_{real} + 0.1$ where $\gamma_{real}=0.1$
• $T_B=0.04$ is the "real" value
• $T_Y=T_{Y real}+0.2$ where $T_{Y real}= 0.5$
• $S_m$ is 60% of the real value
• $I_m=\hat{I}_m$ is calculated as previously indicated

Fitted parameters comparing with "real" values and 95% CI: ![[new_mech_fit.png]]

Fitted curves: ![[new_mech_curve.png]]