Multi-component sigmoidal modeling of cumulative COVID-19 mortality in Colombia: logistic, Gompertz, and Boltzmann-type formulations

National COVID-19 mortality records combine several growth phases that cannot be represented adequately by a single sigmoid. We developed a multi-component framework for cumulative deaths in Colombia using logistic, Gompertz, and Boltzmann-type formulations. Country-level mortality data were derived from the Johns Hopkins University Center for Systems Science and Engineering COVID-19 repository and comprised 1143 daily observations from 22 January 2020 to 9 March 2023, ending at 142,339 cumulative deaths. An initial component count was estimated from peaks in a seven-day-smoothed daily series, after which orders $$m=1,\ldots ,10$$ were fitted by bounded nonlinear least squares with multiple initializations. Thus, model-order selection was automatic within a prespecified candidate range. The best in-sample representation was a ten-component Boltzmann-type model, implemented here as a logistic mixture with a global baseline. It achieved $$R^2=0.99999897$$ , RMSE $$=56.54$$ deaths, MAE $$=42.30$$ , and BIC $$=9442.23$$ . The ordinary logistic mixture was competitive ( $$\Delta \textrm{BIC}=6.97$$ ), whereas Gompertz was less accurate. On the daily-death scale, the selected derivative achieved $$R^2=0.9610$$ and RMSE $$=31.77$$ deaths/day. Residual autocorrelation remained substantial and the selected order reached the upper search boundary. The fitted terms are therefore interpreted as mathematical mortality-growth components rather than ten confirmed epidemiological waves.