One of the earliest dreams of humanity, reflected in dozens of legends and fantastical stories, was to see the future. Such a power would grant its wielder not only a considerable advantage over others, but also a chance to shape the course of coming events. Whether it is in an economical setting, or choice of treatment for a patient, or even – as we all have experienced first-hand in the past couple of years – managing a pandemic, the ability to predict the consequences of our actions cannot be overstated. Using the computational power of modern computers, we are now closer than ever to achieving this goal, replacing the mystical oracles with mathematical models.
Since the beginning of the COVID-19 pandemic, countless attempts have been made to accurately model the spread of the virus. Government policies and restrictions, tightened or relaxed in the hope of mitigating the pandemic, can also be included in the simulation – if their effectiveness can be correctly assessed. A well parametrized model would allow us to test different scenarios, adjusting the control strategy to minimize the infections in as non-burdening way as possible.
There is, however, one catch, painfully familiar to all scientists working with biomedical systems. For a model to be well parametrized, abundant data for fitting are needed. Meanwhile, nearly all available real-life datasets are scarce, noisy, and internally correlated. For example, as seen in Fig 1A, the timelines of two policies, debt contract relief and contact tracing, are the same for a single country. Effectively, they are indistinguishable in terms of modelling and the corresponding parameters are non-estimable as increase in one can be compensated with a decrease in the other. This problem can be countered by using data from multiple countries (Fig 1B) – the pandemic is a global crisis.
A common, ‘one-fits-all' model, while tempting, is intrinsically unrealistic – countries (or patients, or any other entity) differ from one another in many ways. Taking this into account we developed an individualized modelling approach, suitable for every scenario involving a heterogeneous cohort of entities, such as a set of 42 European countries suffering from SARS-CoV-2 pandemic. We used a classic SEIR (susceptible-exposed-infected-removed) epidemiological model , in which we assumed that the virus transmission parameter β varies in time and is a function of imposed policies (Fig 1C). Incorporating 13 different non-pharmaceutical interventions, belonging to three categories (containment and closure, economic measures, and health system) means that together with the bias, there are 14 parameters to estimate. In the individualized approach, we divide the parameters into two subsets – ai, representing policy efficiencies, are common for all countries, and bias b is country-specific. This formulation reduces the number of parameters to estimate from 14×42=588 in case of modelling each country independently to only 13+42=55 while retaining individual characteristics.
We estimated the parameters using a two-stage procedure depicted in Fig 2. In the first stage, a time-dependent function 𝛽opt(t) is estimated for each country using a gradient method based on adjoint sensitivity analysis . In the second stage, data from all the countries are integrated and a function of restrictions is fitted to the 𝛽opt using a non-linear least squares method.
We compared the performance of our framework to the independent and common approaches, estimating the parameters on the period between 1 February 2020 and 31 November 2020 and using two months (December 2020 and January 2021) for validation. As a general measure of predictive ability, we adopted NRMSE (normalized root mean square error). The individualized modelling outperformed the two other approaches, achieving the lowest average error and much lower standard deviation of errors. Moreover, it displayed a distinct numerical advantage over the independent modelling, as the estimated parameter values were within realistic, interpretable ranges, without any prior assumptions. This confirms the applicability of individualized modelling for poorly described systems.
Coming back to the initial question – how can the pandemic be controlled using non-pharmaceutical interventions? Estimating weights ai as common for all the countries lets us construct a ranking of policies according to their effectiveness. This ranking leads to an interesting observation – the most effective policies are straightforward and well-defined. Public information campaigns, income support, school closing, cancellation of public events, workplace closing, and open testing policy are all top-down, independent of the individual and easily traceable, which makes them powerful tools in the fight against the virus. On the other hand, less tangible and harder to enforce policies like restrictions on internal movement, debt/contract relief, contact tracing, restrictions on gatherings, stay at home requirements, closing public transport and, surprisingly, facial coverings, proved less effective.
No study is perfect - our estimations were based on confirmed cases constituting only a fraction of the actual number of infections, many of which were asymptomatic. Furthermore, the SEIR model used to model the pandemic is relatively simple as it does not consider repeated infections, vaccinations, or population structure. Still, our methodology is not limited by the model formulation and can be applied to other models or scenarios.
Biomedical systems differ from each other because each cell, tissue, person, and society is unique. Capturing this uniqueness in a mathematical model basing on limited and noisy data is difficult, but, fortunately, not impossible. Our modelling approach provides a way to construct individualized models able to retain entity specific characteristics while avoiding poor numerical conditioning by using an entire cohort for estimation.
 Hethcote, H. W. The Mathematics of Infectious Diseases. SIAM Rev. 42, 599–653, DOI: 10.1137/S0036144500371907 (2000).
 Łakomiec, K., Wilk, A., Psiuk-Maksymowicz, K., Fujarewicz, K. (2022). Finding the Time-Dependent Virus Transmission Intensity via Gradient Method and Adjoint Sensitivity Analysis. In: Pietka, E., Badura, P., Kawa, J., Wieclawek, W. (eds) Information Technology in Biomedicine. ITIB 2022. Advances in Intelligent Systems and Computing, vol 1429. Springer, Cham. https://doi.org/10.1007/978-3-031-09135-3_41