Non-Markovian dynamics and effective reproduction number in COVID-19: Evidence from Cyprus contact tracing data

Using contact tracing data provided by the Cyprus Ministry of Health, infection trees for the first four waves of the COVID-19 epidemic are constructed. In these trees, nodes represent infected individuals, while links indicate the direction of transmission between them. For each infection tree of N nodes, the hopcount distribution from the root node to all other nodes is calculated. The empirical distribution is then compared to the hopcount distribution of infection trees generated by a non-Markovian SI process on a complete graph, with Weibull infection times characterized by a shape parameter α. We compute the values of the shape parameter α that best fit the empirical distribution and find that only values of α>1 are obtained, while the Markovian case is characterized by α=1. A Weibull distributed infection time with shape parameter α>1 is characterized by a unimodal density function with a peak at finite time, consistent with previous findings in the literature. Our analysis therefore suggests that the spreading process is most likely governed by non-Markovian dynamics, and that non-Markovianity can be detected solely from the topology of the infection trees. Finally, we analyze the evolution of the empirical distribution of the number of secondary infections caused by each node in the infection trees across different time windows to estimate the effective reproduction number. In practice, the average number of secondary infections seems to often provide a lower bound of the reproduction number computed by the Cyprus Ministry of Health. When the last level of the trees, composed predominantly of terminal nodes that do not generate further infections, is excluded, the estimate reflects more accurately the dynamics of the epidemic.Author summary: We explore how diseases spread through populations by examining infection trees. Each “node” in the infection tree is a person and each “link” is a transmission that specifies who infected whom during an outbreak. Based on contact tracing data from the COVID-19 epidemic in Cyprus, we develop a method to detect non-Markovianity in disease transmission by analyzing only the structure of these trees. A Markovian process assumes that the chance of spreading a disease depends only on an individual’s current state, whereas a non-Markovian process occurs if the timing of infections is influenced by the time passed since the host was first infected. Our results demonstrate that real disease spread exhibits non-Markovian behavior, which can be measured from the structure of infection trees without employing precise chronological information. We also estimate the effective reproduction number, which indicates how many new infections are generated on average by each infected individual. The agreement between our estimates and those reported by the Cyprus Ministry of Health confirms that working with detailed infection tree data is a valid and reliable way to study disease spread. Overall, our work provides a practical framework for understanding disease transmission in real populations.