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

•We present a mathematical model based on a new stochastic process described by a Pure Birth process.•The proposed model matches the subexponential growth on the early stage of an epidemic.•The mathematical expression of the cumulative case incidence and cumulative death curves is obtained, with a quite accurate fit in both cases.•The model contains two parameters, the immunization and infection rates. The behavior in time of those parameters allows to assess the evolution of the outbreak.•We obtain a new indicator, the mean time between infections. This indicator allows not only to monitor the epidemic growth but also to predict the peak of cases. 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.