 Hi, I'm Jonathan, and now we're going to have a look at how we can use the ODE filter to solve a practical problem. Consider a scenario in which time series data has been observed, which we can model by an ordinary differential equation. A prominent example for this would be the COVID-19 pandemic. Here on the right, you see the number of infectious people in Germany over the course of the pandemic scaled to cases per 1,000 people. In this setting, it turns out that we can use the ODE filter in order to perform probabilistic inference over a time-varying latent process, while at the same time simultaneously solving the ODE. As we already saw in the first video, infectious diseases can be modeled by epidemiological models such as, for example, the SIRD system, which you can see here on the left as a flow chart. In this scenario, let us consider the contact rate beta, which clearly cannot be constant over the course of a pandemic. So, what we do is we put a Gauss Markov process prior over both the ODE solution and latent contact rate. We model both dynamics as independent processes by augmenting the state-space model and then use the ODE filter likelihood, which we know already from the third video, in order to couple them. So, we leverage the state-space formulation of probabilistic ODE solvers in order to feedback parts of the state back into the ODE filter likelihood. Additionally, wherever it is available, we can extract information from real data. So, we unify both probabilistic inference over a latent process with the ODE solution under a single framework, Bayesian filtering and smoothing, leveraging both data and mechanistic knowledge. So, now let us have a look at what the results would look like on COVID-19 data. What you see here is the smoothing posterior as obtained from the aforementioned framework. On top, you see the infectious case counts as obtained from the SIRD solution together with real data. When extrapolating on the right, you see the uncertainty quickly increasing due to the multitude of possible outcomes. On the bottom, you see the contact rate that led to this outcome. Remember, this is also a random process which you can sample from. The uncertainty is increased where case counts are low and wherever there is more information available, the posterior contracts. Can you spot different governmental measures in this plot? So, what we just did is we combined the probabilistic inference over a latent process together with the probabilistic ODE solution in a single linear time forward solve of the ODE filter. Thanks for watching.