A non-homogeneous Markov early epidemic growth dynamics model. Application to the SARS-CoV-2 pandemic

TL;DR

This work presents a non-homogeneous Markov Pure Birth process to model the early growth of epidemics, where the incidence rate is $\lambda_r(t) = \rho \dfrac{1 + (\gamma/\rho)\, r}{1 + \rho t}$. The ratio $\tfrac{\gamma}{\rho}$ governs the curvature of the mean infection trajectory $M(t) = \dfrac{\rho}{\gamma}\big[(1+\rho t)^{\gamma/\rho} - 1\big]$, capturing contagion versus immunization dynamics. The model yields a closed-form Case Incidence distribution $P_r(t)$ and a direct expression for the mean time between infections (MTBI), enabling practical indicators: the infection/immunization ratio, the immunization rate $\rho$, and MTBI. Applied to SARS-CoV-2 data across multiple countries, the framework achieves good fits for both incidence and deaths in the early growth phase and provides time-varying insights into outbreak control, particularly in Latin America. The approach offers an interpretable, data-driven tool for monitoring and forecasting early epidemic behavior and assessing public health interventions.

Abstract

This work introduces a new markovian stochastic model that can be described as a non-homogeneous Pure Birth process. We propose a functional form of birth rate that depends on the number of individuals in the population and on the elapsed time, allowing us to model a contagion effect. Thus, we model the early stages of an epidemic. The number of individuals then becomes the infectious cases and the birth rate becomes the incidence rate. We obtain this way a process that depends on two competitive phenomena, infection and immunization. Variations in those rates allow us to monitor how effective the actions taken by government and health organizations are. From our model, three useful indicators for the epidemic evolution over time are obtained: the immunization rate, the infection/immunization ratio and the mean time between infections (MTBI). The proposed model allows either positive or negative concavities for the mean value curve, provided the infection/immunization ratio is either greater or less than one. We apply this model to the present SARS-CoV-2 pandemic still in its early growth stage in Latin American countries. As it is shown, the model accomplishes a good fit for the real number of both positive cases and deaths. We analyze the evolution of the three indicators for several countries and perform a comparative study between them. Important conclusions are obtained from this analysis.

Figures (9)

• Figure 1: Expected shape of the MTBI indicator. Different portions show the following: A-The epidemic is spreading out, MTBI decreases since infections occur more often. B- Infections occur at a constant rate. C- MTBI increases since infections occur less often.
• Figure 2: Different behaviors of \ref{['mean-r-our']} depending on values of $\frac{\gamma}{\rho}$.
• Figure 3: Cumulative case incidence.
• Figure 4: The Uruguayan case.
• Figure 5: Cumulative number of deaths.