Suppose that in times of epidemic we have at our disposal data ((t_i,n_i,d_i,h_i)), (i=1,points ,m)on the number of new confirmed cases ((or))newly deceased ((d_i))and the total number of people hospitalized ((Salvation)) the day (t_{i})during a certain observed period ([t_1,t_m]).
Assuming that the number of deceased on the day you depends on the number of new cases on a previous day (t-tau), (tau >0)and that the number of new hospitalized on the day you depends on the number of new cases on a previous day (t-delta), (delta >0)we will try to estimate the number of deceased in the coming period ([t_m,t_m+tau ])and hospitalized in the coming period ([t_m,t_m+delta ])thus providing important indicators for monitoring and combating the pandemic.
In order to operationalize these assumptions using the available data, we will first determine the corresponding growth model function (Phi) for newly reported cases.
Due to different methodologies and daily testing dynamics throughout the week, daily new case data is unreliable (i.e. contains random errors). Therefore, instead of using the daily data, we will estimate the parameter values a, band vs growth model function (Phi) based on cumulative Numbers (N_i=sum nolimits _{s=1}^i n_s), (i=1,points ,m)new cases during the same period ([t_1,t_m]). The derivative (tmapsto Phi ‘
(1)
which is the solution of the differential equation
$$begin{aligned} y’=c,y,(ay),quad a,c>0, end{aligned}$$
(2)
describing the following law: The growth rate of the number of cases at the moment it is T proportional to the total number of cases at that time, there(you), and the number of potentially new cases, ((ay
(3)
The parameters obtained will be noted (a^star), (b^star)and (c^star)and the model function (Phi) will be the function (tmapsto y(t;a^star ,b^star ,c^star )).
Simple global optimization problems (G.O.P.) (3) can be solved using the Mathematical module Fitting the nonlinear model27.
Prediction of the number of deceased
As we have already said, the dependence of the number of deceased on the number of new cases is based on the following assumptions:
-
(I)
there is a delay (tau) between infection and death;
-
(ii)
The number of deceased depends linearly on the derivative (Phi ‘(t-tau ;a^star ,b^star ,c^star ))or (a^star), (b^star), (c^star) are optimal parameters obtained by solving the nonlinear least squares problem (3) on the basis of cumulative data ((t_i,N_i)), (i=1,points ,m).
Know the function of the model (Phi) of the cumulative number of new cases, i.e. knowing the parameters (a^star ,,b^star ,,c^star)the function of the daily deceased can be expressed by (see also similar models in refs.24,28,29)
$$begin{aligned} varphi (t;u,alpha ,tau )=u+alpha , Phi ‘(t-tau ;a^star ,b^star ,c^star ) . end{aligned}$$
(4)
The settings you, (alpha) and (tau) of the model function (varphi) will be determined using data ((t_i,d_i)), (i=kappa ,dots ,m)on the numbers of deceased as follows G.O.P. (the data of the deceased concern (kappa)say 15 days later since we assume that deaths before that belong to the previous pandemic wave):
$$begin{aligned} smash
(5)
The solution to this problem will be noted ((u^star , tau ^star , alpha ^star )): (tau ^star) is the average time (in days) between new cases and corresponding deaths, and (alpha ^star) represents the proportion of people who died among the new cases reported (tau ^star) days earlier.
Since in general the number (tau ^star) will not be an integer, we will denote by (hat{tau }=lfloor tau ^star rceil) the nearest integer. Thus, the number of people who died during the period ([t_{m+1},t_m+hat{tau }:]) of (hat{tau}) days can be calculated using the formula
$$begin{aligned} sum limits _{t=t_m+1}^{t_m+hat{tau }} lfloor varphi (t;u^star ,alpha ^star ,tau ^ star )rceil . end{aligned}$$
(6)
This procedure should be repeated, adding new data, every day.
Note 1
Due to the digital sensitivity of the G.O.P. (5), we will solve this problem in the same way as it was done in refs.30.31: we first determine the initial approximation using the global optimization algorithm DIRECT32,33,34then we apply Newton’s optimization method25.26 using Mathematical module Fitting the nonlinear model. The same will be done to solve the G.O.P. (9).
Forecast of the number of hospitalizations
The link between the number of hospitalized and newly infected is more complicated than in the case of deceased persons. That is, the total number (Salvation) day hospitalization (t_i) can be defined as
$$begin{aligned} h_i=h_{i-1}+nu _i-rho _i-d_i, end{aligned}$$
(7)
or (nu _i) is the number of people admitted to the hospital on the day (t_i), (rho _i) is the number of hospital discharges during the day (t_i)and (d_i) is the number of patients who died on the day (t_i). Note that (nu _i-rho _i-d_i) means the change (growth or decrease) in the number of patients hospitalized on the day (t_i). We will assume the following:
-
(I)
there is a delay (delta) between new cases and new hospitalizations;
-
(ii)
The number of hospitalizations depends linearly on the derivative (Phi ‘(t-delta ;a^star ,b^star ,c^star ))or (a^star), (b^star), (c^star) are optimal parameters obtained by solving the nonlinear least squares problem (3) on the basis of cumulative data ((t_i,N_i)), (i=1,points ,m).
Based on these assumptions and knowing the function of the growth model (tmapsto Phi (t;a^star ,b^star ,c^star )) for the cumulative number of new cases, the estimated number of hospitalizations can be expressed as follows (see also similar models in refs.28.29)
$$begin{aligned} psi (t;u,v,delta )=u+ v,Phi ‘(t-delta ;a^star ,b^star ,c^star ), end{aligned}$$
(8)
where the parameters you, vand (delta) of the model function (psi) will be determined using data ((t_i,h_i)), (i=kappa ,dots ,m)on the number of people hospitalized, as follows G.O.P.:
$$begin{aligned} mathop {mathrm {argmin}}nolimits limits _{u,v,delta >0}F(u,v,delta ),qquad F(u,v, delta )=sum limits _{i=kappa }^m (h_i-psi (t_i;u,v,delta ))^2, end{aligned}$$
(9)
where data for hospitalized persons are for (kappa), say 15 days later since we assume that previous hospitalizations belong to the previous wave. The solution to this problem will be noted ((u^star ,v^star ,delta ^star )). The number (delta ^star) is the average time (in days) between new cases and corresponding hospitalizations.
The number (Salvation) day hospitalization (t_i), (I>m)can now be estimated as
$$begin{aligned} h_iapprox u^star +v^star ,Phi ‘(t_i-delta ^star ;a^star ,b^star ,c^star ). end{aligned}$$
(ten)
By comparing (7) and (10), we have
$$begin{aligned} nu _i-rho _i-d_i approx v^star Phi ‘(t_i-delta ^star )-v^star Phi ‘(t_{i-1}- delta ^star ) approx v^star ,Phi ”(t_i-delta ^star )= psi ‘(t_i,u^star ,v^star ,delta ^star ) . end{aligned}$$
(11)
Therefore, the change (growth or decrease) (nu _i-rho _i-d_i) the number of patients hospitalized during the day (t_i), (I>m)can be expressed as the value (psi ‘(t_i,u^star ,v^star ,delta ^star )).
Important points of the function (psi ‘) are its maximum M which is reached at
$$begin{aligned} t_M=delta ^star +tfrac{ln (b^star (2-sqrt{3}))}{c^star }, end{aligned}$$
(12)
and its zero point (T_0=(t_0,0)). We can say that the number of hospitalized has a gradual growth until now (t_M)and one declining growth after that, and the number of hospitalizations decreases after the time (t_0).
Since in general the number (delta ^star) is not an integer, we note (hat{delta }=lfloor delta ^star rceil) the nearest integer. Therefore, the projected number of hospitalizations over the next (hat{delta }) the days are
$$begin{aligned} lfloor psi (t_m+1;u^star ,v^star ,delta ^star )rceil ,;dots ,;lfloor psi (t_m+hat {delta };u^star ,v^star ,delta ^star )rceil . end{aligned}$$
(13)
This procedure should be repeated, adding new data, every day.
We have also developed software support for the proposed method in the form of Mathematical modules, see “Availability of data and computer code”.
