In questa nota divulgativa presento alcuni aspetti della teoria matematica delle epidemie, con particolare attenzione al caso di una nuova infezione emergente, come nel caso del COVID-19. Parto della definizione di numero di riproduzione di base $R_0$, e presento la sua relazione col tasso di crescita esponenziale all'inizio di un'epidemia e con le possibilità di controllo in semplici modelli deterministici. Discuto poi i corrispondenti modelli stocastici, mostrando come in questo caso $R_0$ sia legato alla probabilità che esploda un'epidemia a partire dall'introduzione di pochi infetti. Infine accenno alla struttura dei modelli complessi che vengono attualmente usati come supporto previsionale alle azioni delle autorità sanitarie.
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Con il contributo del Ministero per i Beni e le Attività Culturali