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Method

We are maximizing

<latex> \[ \ln L(x_i | \theta) = \sum_i x_i \ln(\lambda_1(t_i) + \lambda_2(t_i)) - \lambda_1(t_i) - \lambda_2(t_i) \] </latex>

where <latex>$t_i$</latex> denotes the time in weeks, <latex>$x_i$</latex> are the number of excess deaths at <latex>$t_i$</latex>.

The mean <latex>$\lambda_j(t_i)=X_j(t_i)$</latex>, that is, it equals the number of new cases in the time period <latex>$(t_{i-1},t_{i}]$</latex>.

The number of new cases <latex>$X_j(t_i)=S_j(t_{i+1})-S_j(t_{i})$</latex>, where $S_j(t)$ is the solution to the SIR model

<latex> \begin{align*} S_j' & = - \beta_j S_j I_j \\ I_j' & = \beta_j S_j I_j - \alpha I_j \end{align*} </latex>

Since the SIR model can’t be solved analytically, we will solve it numerically. R must have a subroutine for solve ODEs. You should look it up.

so, the procedure to compute the likelihood function is:

  1. Given <latex>$\theta=\{f, \beta_1,I_1(0),\beta_2,I_2(0)\}</latex>, where <latex>$f$</latex> is the case fatality proportion, compute <latex>$S_1(0)$=fN-I_1(0)</latex> and <latex>$S_2(0)$=fN-I_2(0)</latex> solve the SIR model, where <latex>$N$</latex> is the population size.
  2. Use the given parameters, <latex>$\alpha=1/7$</latex>, and the initial conditions to solve the SIR models, and obtain <latex>$S_1(t)$</latex> and <latex>$S_2(t)$</latex>.
  3. Use <latex>$S_1(t)$</latex> and <latex>$S_2(t)$</latex> to compute <latex>$\lambda_1=X_1(t)$</latex> and <latex>\lambda_2=$X_2(t)$</latex>.
  4. Substitute <latex>$\lambda_1$</latex> and <latex>$\lambda_2$</latex> into the log-likelihood function