key: cord-0458632-rxoz6t0m
authors: P'erez-Maldonado, Mar'ia T.; Bravo-Castillero, Juli'an; Mansilla, Ricardo; Caballero-P'erez, Rogelio O.
title: Discrete Gompertz and Generalized Logistic Models for early monitoring of the COVID-19 pandemic in Cuba
date: 2022-03-05
journal: nan
DOI: nan
sha: df92f91a5ad17745999e345290bb6a678943b69c
doc_id: 458632
cord_uid: rxoz6t0m

For the last few years there has been a resurgence in the use of phenomenological growth models for predicting the early dynamics of infectious diseases. These models assume that time is a continuous variable whereas in the present contribution, the discrete versions of Gompertz and Generalized Logistic models are used for early monitoring and short-term forecasting of the spread of an epidemic in a region. The time-continuous models are represented mathematically by first-order differential equations while their discrete versions are represented by first-order difference equations that involve parameters that should be estimated prior to forecasting. The methodology for estimating such parameters is described in detail. Real data of COVID-19 infection in Cuba is used to illustrate this methodology. The proposed methodology was implemented for the first thirty-five days, being able to predict with very good precision the data reported for the following twenty days. The codes implemented to study the Gompertz model in differences are included in an appendix with each step of the methodology identified.

Growth models are simple. They are characterized by first-order ordinary differential equations that depend on certain parameters that can be estimated using real data through optimization methods. The interrelations and generalizations of the basic growth models (Malthus, Verhulst logistic, Richards and Gompertz) can be found in Tsoularis and Wallace [2002] . A historical development of these models may be found, for instance, in Kingsland [1982] or in Bürger et al. L Logistic. GL Generalized Logistic. R Richards'. GR Generalized Richards'. G Gompertz'. GG Generalized Gompertz'.

Cumulative number of infections at continuous time t C 0 n N0 n=1

Discrete time series of the cumulative number of infections.

Discrete time series of the cumulative number of infections after averaging with a 7 day long moving window. Π Model-specific parameter vector. K Carrying capacity. γ Intrinsic Growth Rate. µ

Deceleration of Growth. α

Parameter for moving inflection point (see Tsoularis and Wallace [2002] ).

[2021]. The latter illustrates a general methodology based on phenomenological growth models, for the application of statistics to various problems in medicine and biology.

Growth models have been used to describe the early stage of several epidemics, as can be seen in the book by Brauer et al. [2019] . Zhou and Yan [2003] Jain et al. [2020] applied the L, the GL and the generalized growth models to identify the initial climbing growth period of COVID-19 outbreak in India from Apr 10, 2020, to Apr 20, 2020. Roosa et al. [2020a] use the L and the GR models to generate short-term forecasts of cumulative reported cases in Guangdong and Zhejiang, China. In Roosa et al. [2020b] , the GL and GR models were validated during earlier outbreaks to generate and evaluate short-term forecasts of the cumulative number of confirmed reported cases in Hubei province, China. Wu et al. [2021] applies the generalized growth models to quantify the evolution of COVID-19 in different countries. Simple phenomenological models (generalized growth model and logistic) were used by López et al. [2021] to characterize the two first outbreak waves of COVID-19 in Spain. Congdon [2021] considers the gain in applying phenomenological models to later stages of incomplete epidemics for the COVID-19 in UK after lockdown relaxations. Pincheira-Brown and Bentancor [2021] show that a semi-unrestricted version of the generalized growth model outperforms the traditional in several countries when predicting the number of infected people at short period. Darti et al. [2021] implement five phenomenological (R, GR, Blumberg, Tsoularis & Wallace, and Gompertz) models to predict the cumulative number of COVID-19 cases in East Java province, Indonesia.

The mathematical models used in the aforementioned works are represented by first-order differential equations on a continuous time domain:

where C is some measure of the infected population at time t and Π is a vector of parameters that is specific for each model. However, it is not common to find similar studies based on phenomenological models using first-order difference equations that can be derived from (1):

where C n represents the measure of the infected population at the n-th measurement of the (discrete) data and h represents the time difference between consecutive readings (assuming it is constant). One can imagine that, since the number of infected people is tipically reported at regular time intervals (daily, weekly, et cetera) the discrete models are suitable for describing the dynamics of the disease (see Section 2.9 of Boyce and DiPrima [2012] ). May [1976] reports an iconic review on the theoretical and applied scope of these equations. The objective of this work is to illustrate the potential of the discrete phenomenological models associated to continuous time first-order differential equations, as an alternative to estimate the early spread of epidemics. A methodology for a statistical study of the related parameters is described, the application of which is exemplified based on initial data from the COVID-19 in Cuba.

The work is structured as follows: In section 3 the methodology is described in detail. In section 4 the methodology is applied for the case of Gompertz and Generalized Logistic models. Section 5 is devoted to some concluding remarks.

The standard way of obtaining the difference equation (2) from the continuous one (1) is through the transformation:

where n represents the n-th value of the time series of N 0 length, {C n } N0 n=1 and h is the time step between recorded values (we are assuming that it is constant). The model is specified by the choice of the function f (x, t, Π) where the vector of parameters Π may have different lengths (see Table 1 ). Table 1 : Some growth models, their continuous and discrete equations and their vectors of parameters.

The parameter K is called the carrying capacity; the parameter γ, the intrinsic growth rate and the parameter α is introduced to provide flexibility in the position of the inflection point of the growth curves, see Tsoularis and Wallace [2002] . The parameter µ is called deceleration of growth by Viboud et al. [2016] when 0 < µ < 1.

A two-stage procedure was developed. In the first part: is computed through the RecurrenceTable function using the optimal values Π and C 1 = C 1 .

In the second part of the methodology, the quality of the fit is evaluated by estimating the error through bootstrap (see for instance, Efron and Tibshirani [1994] ) on M realizations (M = 10 5 in this work). Each realization corresponds to one time series {C nm } N n=1 with m = 1, 2 . . . , M . The following steps illustrate how to generate one realization: 5. For each one of the n = 1, . . . , N an error e nm is simulated. This is equal to a Poisson random number with mean λ n = C n+1 − C n . 6. The parameter vector Π m is estimated as in step 3. above from the time series {C n + e nm }. 7. For each model, the error bounds' and mean values' estimation are done over the set of parameter vectors (Π m ) M m=1 .

In this section the results for the case of the Gompertz' and Generalized Logistic models is shown. The data corresponds to the cumulative number of cases of COVID-19 during the first 61 days of the pandemic in Cuba (from March 11, 2020 to May 10, 2020). The window-averaged data consists of a time series of 55 days.

The Wolfram Mathematica code for the Gompertz' model is given in the appendix.

The optimal values Π of the parameter vector after step 3 of the methodology are reported in Table 2 . The mean values and standard deviations of the set of parameter vector after step 7 of the methodology are reported in Table 3 . 

A methodology based on discrete phenomenological models was illustrated in detail as an alternative to estimate the early evolution of epidemics. The methodology was applied to the Gompertz and Generalized Logistic models which depend on two and three parameters, respectively. The data used in the computations was the initial real data (since March 11, 2020 to May 10, 2020) on the evolution of COVID 19 in Cuba. The numerical implementation was carried out in Wolfram Mathematica in a simple and direct way. The procedure consists of two fundamental stages. In the first, the optimized parameters are obtained, while in the second, a statistical analysis is performed to evaluate the error of the optimization process. The proposed methodology was implemented for the first thirty-five days, being able to predict with very good precision the data reported for the following twenty days. The results show that the proposed methodology could be useful to guide decision-making that facilitates mitigating the early effects of the epidemic.

ROC would like to thank financing from FENOMEC, UNAM. Financial support from the PAPIIT DGAPA UNAM IN10182 project is gratefully acknowledged.

We illustrate the implementation of the methodology for Gompertz' discrete model with the initial data that corresponds to the period from March 11, 2020 until May 10, 2020 2 .

( * I n i t i a l d a t a * ) l = { 3 , 3 , 4 , 4 , 4 , 5 , 7 , 1 1 , 1 6 , 2 3 , 3 3 , 3 8 , 4 6 , 5 5 , 6 5 , 7 8 , 1 1 7 , 1 3 7 , 1 6 8 , 1 8 4 , 2 1 0 , 2 3 1 , 2 6 7 , 2 8 6 , 3 1 8 , 3 4 8 , 3 9 4 , 4 5 5 , 5 1 3 , 5 6 2 , 6 1 8 , 6 6 7 , 7 2 4 , 7 6 4 , 8 1 2 , 8 6 0 , 9 2 1 , 9 8 4 , 1 0 3 3 , 1 0 8 5 , 1 1 3 5 , 1 1 8 7 , 1 2 3 3 , 1 2 8 3 , 1 3 3 5 , 1 3 6 7 , 1 3 8 7 , 1 4 3 5 , 1 4 6 5 , 1 4 9 9 , 1 5 3 5 , 1 6 0 9 , 1 6 4 7 , 1 6 6 6 , 1 6 8 3 , 1 7 0 1 , 1 7 2 7 , 1 7 3 9 , 1 7 5 2 , 1 7 6 4 , 1 7 8 1 } ;

( * S t e p 1 . Window− a v e r a g i n g * ) 

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