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A history-dependent approach for accurate initial condition estimation in epidemic models
Fig 2
Schematic figure for deriving the loss function to estimate the initial condition.
(a) To address the limitation of the history-independent method (left), we developed a novel history-dependent method (right). (b) (i) We established the connection between the known data, 
, and the unknown 
by treating the 
as a convolutional output of 
and the probability density function of the latent period, 
. (ii) By discretizing this relationship and (iii) assuming 
remains consistent before 
, we can express the known 
as a linear combination of unknown 
and unknown 
with known coefficients 
and 
. 
represents the probability of an individual having a latent period of exactly 
days, while 
represents the probability of the latent period being longer or equal to 
days. 
and 
can be obtained by integrating the convolution of 
and 
, where 
represents the characteristic function supported on [0,1] (See Methods for more details). (c) Extending the linear combination expression to the whole data (i.e., 
for 
), we can construct a matrix that describes the relationship between known data and unknown parameters. (d) We utilized this matrix equation that 
must satisfy to establish the data loss function, then sought to minimize this data loss by finding optimal values for unknown parameters, including 
. However, as the number of unknown parameters (
) exceeds the number of equations (
), the parameters cannot be determined solely from the data loss. This leads us to incorporate the regularization loss for the 
parameters, which aims to smooth the 
parameters by minimizing their second order derivatives. Consequently, by finding the parameters that minimizes the total loss function (
), which includes both the data loss and the regularization loss, we can estimate 
. By summing up the difference between daily incidence of exposure (
) and daily incidence of becoming infectious at 
(
), we finally get the initial condition of E.
doi: https://doi.org/10.1371/journal.pcbi.1013438.g002