10.1 Summary
In the present thesis, the author’s motivation for researching the epidemiological topic treated was revealed at the beginning. Subsequently, the fundamental theory for the investigation and solution of ordinary differential equations, the use of these for epidemi-ological mathematical modelling and fundamentals for static and dynamical optimization were presented. The combination of static and dynamical optimization to solve a L2 norm based least squares problem for parameter fitting of models to real data sets was introduced and performed using adjoint functions.
In the case of dengue, a model reduction of an SIRU V to an SIR model with time–
dependent transmission rate via time–scale separation was initially necessary in order to work practically with real data sets from Colombo and Jakarta. This enabled us to perform a useful and realistic parameter estimation with the adjoint approach. Furthermore, this could be used to study the direct impact of seasonal meteorological conditions on the disease. In addition, the data was used to test the extent to which the dengue model can be utilized to predict future peaks and their intensity and duration. The results show that both short and long–term forecasts are possible with a certain quality of the data sets. Especially in the case of Jakarta this could be shown impressively which was also examined via multipatch model with daily commuter movements. In this case, realistic and useful results could be obtained regarding the parameter fitting and also regarding the predictive power of this model, forecasts matching the real data sets could be made.
Regarding the spread of the neurogenic Coronavirus with the disease COVID–19 a SEIRD model with and without time delay was used. This model was applied to the initial spread in Germany. The results show that the model with time delay and the presented parameter estimation provided very realistic values, especially with respect to the detection rate and lethality rate which were hardly valid at that time. Subsequently, the time delaySEIRD model was applied to a more advanced data set in Germany and this time the adjoint approach was compared with a Metropolis algorithm regarding the parameter estimation. It became clear that the former requires a higher analytical effort but converges much faster with suitable initial values and is therefore less computationally demanding. This refers only to local minima since the Metropolis algorithm has proven to be more effective on a global level.
Regarding vector–borne diseases, it is worth using the reducedSIR system applied in the present thesis to simulate other diseases of this type and make possible predictions.
Especially in the case of dengue, a refinement of the approach to a multistrain model should be considered. Also a differentiated modelling with regard to external influences like the entire meteorology is desirable.
Regarding the COVID–19 models, we are currently still in a learning process as the disease itself still needs further research. Accordingly, the models can be adapted and refined in future, of course as practical as possible. So far, our research has been limited mainly to Germany. Here it is desirable to use the presented models also for data sets of different countries and regions. Similar to the dengue model, the mobility component should also be included in this study and its significance for the dynamics of disease should be better investigated with the models.
In principle, the mathematical models should be examined with regard to their pre-dictive power and, in the case of a positive evaluation, with the help of optimal control theory, the use of possible control variables should be optimized for disease containment.
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