COVID-19 task force of mathematical modelling and epidemiological analysis in Hungary

Research on mathematical modelling of infectious diseases has been carried out at the Bolyai Institute of the University of Szeged for already 12 years. Earlier topics studied by the research group include strategies for influenza vaccination, the impact of mandatory varicella vaccination, the risk of measles outbreaks during the 2012 European Football Championship, the spread of diseases via the global travel networks, and intervention strategies concerning the 2014 Ebola epidemic.

The research group started studying the spread of the new coronavirus in January, when a three-phase model was established to estimate the risk of outbreaks in various countries outside China due to case importations. In the first phase, using a nonautonomous system of differential equations accounting for the intervention measures of Chinese authorities, we projected the number of all cases in China (outside the quarantined Hubei province). In the second phase, we used this result as an input to an individual-based model of the global mobility network to estimate the number of infected travellers arriving to specific destinations from China. The third phase of the model is a branching process describing the early spread of the disease in the destination country which is used to estimate the probability of an outbreak. We described how the risk of an outbreak depends on the three key parameters: the number of cases in China, the connectivity between China and the destination country and the efficacy of control measures in the destination country. We also compared the risks in various countries of Europe, America and Asia. As the Figure shows, we were a bit overly pessimistic about the outcome in China, and overly optimistic about the control efforts in European countries. In any case, the model correctly identified the countries of highest risk where the first significant outbreaks occurred.

Figure is taken from . Note that the number of cases refers to all infected, not just reported cases, which was much less due to underreporting.

A COVID-19 task force of mathematical modelling and epidemiological analysis was established in March 2020, coordinated by the Ministry of Innovation and Technology. This unique, multidisciplinary group consists of 55 experts from different Hungarian universities and research institutes and includes mathematicians, biostatisticians, physicians, epidemiologists, network scientists, computer scientists, sociologists, public health and health management experts. The tasks of the research group include a continuous evaluation of the current situation and of possible interventions, collecting all relevant data and information, preparing daily and weekly reports, providing evidence base for decision making. The mathematical tools used in the research are widespread and include statistical methods, stochastic processes, compartmental, network and agent-based models. We have started an online survey to assess how the number of contacts between different age groups decreased during the lockdown which, together with phone survey and mobile phone data we could apply to estimate the contact rates in the age-structured compartmental model established to assess the possible effects of different scenarios following the release of the lockdown. This compartmental model considers multiple age groups and each age group is divided into 14 compartments depending on their infection status, namely, we consider susceptibles, three phases of exposed (with the last phase already being infectious, also called pre-symptomatic phase), three phases of symptomatically and asymptomatically infected, respectively, hospitalized, critical cases and recovered. We studied age-dependent measures like school closures and protection of the elderly and also considered seasonality and spatial heterogeneity.

Transmission diagram of the age-structured model

A working group use agent-based modelling, which means that we model the interactions among the members of the population on individual level. A prototype of the model is now being developed and tested for a virtual Szeged, where real data concerning various locations and 161,000 people endowed with properties based on real-world behaviour are incorporated.

Among others, we have also worked on modelling the effect of the exodus of large groups of susceptibles from quarantined areas just before the lockdown, a phenomenon experienced in Wuhan and Northern Italy; on the impact of symptom based testing on outbreak mitigation; and on nonlinear predictive controls with temporal logic to construct interventions for specific objectives. Currently we are working on the accurate reconstruction of our first wave, which was largely invisible due to limited testing, and also on precisely calculating the age specific infection fatality ratio to estimate future disease burden. The second wave hit Hungary with high intensity and with very different demographic profile compared to the first wave, giving us a lot of work to understand the underlying processes that are governing the present outbreak.

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