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Scientific and public interest in epidemiology and mathematical modelling of disease spread has increased significantly due to the current COVID-19 pandemic. Political action is influenced by forecasts and evaluations of such models and the whole society is affected by the corresponding countermeasures for containment. But how are these models structured?
Which methods can be used to apply them to the respective regions, based on real data sets? These questions are certainly not new. Mathematical modelling in epidemiology using differential equations has been researched for quite some time now and can be carried out mainly by means of numerical computer simulations. These models are constantly being refinded and adapted to corresponding diseases. However, it should be noted that the more complex a model is, the more unknown parameters are included. A meaningful data adaptation thus becomes very diffcult. The goal of this thesis is to design applicable models using the examples of COVID-19 and dengue, to adapt them adequately to real data sets and thus to perform numerical simulations. For this purpose, first the mathematical foundations are presented and a theoretical outline of ordinary differential equations and optimization is provided. The parameter estimations shall be performed by means of adjoint functions. This procedure represents a combination of static and dynamical optimization. The objective function corresponds to a least squares method with L2 norm which depends on the searched parameters. This objective function is coupled to constraints in the form of ordinary differential equations and numerically minimized, using Pontryagin's maximum (minimum) principle and optimal control theory. In the case of dengue, due to the transmission path via mosquitoes, a model reduction of an SIRUV model to an SIR model with time-dependent transmission rate is performed by means of time-scale separation. The SIRUV model includes uninfected (U) and infected (V ) mosquito compartments in addition to the susceptible (S), infected (I) and recovered (R) human compartments, known from the SIR model. The unknwon parameters of the reduced SIR model are estimated using data sets from Colombo (Sri Lanka) and Jakarta (Indonesia). Based on this parameter estimation the predictive power of the model is checked and evaluated. In the case of Jakarta, the model is additionally provided with a mobility component between the individual city districts, based on commuter data. The transmission rates of the SIR models are also dependent on meteorological data as correlations between these and dengue outbreaks have been demonstrated in previous data analyses. For the modelling of COVID-19 we use several SEIRD models which in comparison to the SIR model also take into account the latency period and the number of deaths via exposed (E) and deaths (D) compartments. Based on these models a parameter estimation with adjoint functions is performed for the location Germany. This is possible because since the beginning of the pandemic, the cumulative number of infected persons and deaths
are published daily by Johns Hopkins University and the Robert-Koch-Institute. Here, a SEIRD model with a time delay regarding the deaths proves to be particularly suitable. In the next step, this model is used to compare the parameter estimation via adjoint functions with a Metropolis algorithm. Analytical effort, accuracy and calculation speed are taken into account. In all data fittings, one parameter each is determined to assess the estimated number of unreported cases.

While reading this sentence, you probably gave (more or less deliberately) instructions to approximately 100 to 200 muscles of your body. A sceptical face or a smile, your fingers scrolling through the text or holding a printed version of this work, holding your head, sitting, and much more.
All these processes take place almost automatically, so they seem to be no real achievement. In the age of digitalization it is a defined goal to transfer human (psychological and physiological) behavior to machines (robots). However, it turns out that it is indeed laborious to obtain human facial expression or walking from robots. To optimize this transfer, a deeper understanding of a muscle's operating principle is needed (and of course an understanding of the human brain, which will, however, not be part of this thesis).
A human skeletal muscle can be shortened willingly, but not lengthened, thereto it takes an antagonist. The muscle's change in length is dependent on the incoming stimulus from the central nervous system, the current length of the muscle itself, and certain muscle--specific quantities (parameters) such as the maximum force. Hence, a muscle can be mathematically described by a differential equation (or more exactly a coupled differential--algebraic system, DAE), whose structure will be revealed in the following chapters. The theory of differential equations is well-elaborated. A multitude of applicable methods exist that may not be known by muscle modelers. The purpose of this work is to link the methods from applied mathematics to the actual application in biomechanics.
The first part of this thesis addresses stability theory. Let us remember the prominent example from middle school physics, in which the resting position of a ball was obviously less susceptible towards shoves when lying in a bowl rather than balancing at the tip of a hill. Similarly, a dynamical (musculo-skeletal) system can attain equilibrium states that react differently towards perturbations.
We are going to compute and classify these equilibria.
In the second part, we investigate the influence of individual parameters on model equations or more exactly their solutions. This method is known as sensitivity analysis.
Take for example the system "car" containing a value for the quantity "pressure on the break pedal while approaching a traffic light". A minor deviation of this quantity upward or downward may lead to an uncomfortable, abrupt stop or even to a collision, instead of a smooth stop with a sufficient gap.
The considered muscle model contains over 20 parameters that, if changed slightly, have varying effects on the model equation solutions at different instants of time. We will investigate the sensitivity of those parameters regarding different sub--models, as well as the whole model among different dynamical boundary conditions.
The third and final part addresses the \textit{optimal control} problem (OCP).
The muscle turns a nerve impulse (input or control) into a length change and therefore a force response (output). This forward process is computable by solving the respective DAE. The reverse direction is more difficult to manage. As an everyday example, the OCP is present regarding self-parking cars, where a given path is targeted and the controls are the position of the
steering wheel as well as the gas pedal.
We present two methods of solving OCPs in muscle modeling: the first is a conjunction of variational calculus and optimization in function spaces, the second is a surrogate-based optimization.

In the last years, the public interest in epidemiology and mathematical modeling of disease spread has increased - mainly caused by the COVID-19 pandemic, which has emphasized the urgent need for accurate and timely modelling of disease transmission. However, even prior to that, mathematical modelling has been used for describing the dynamics and spread of infectious diseases, which is vital for developing effective interventions and controls, e.g., for vaccination campaigns and social restrictions like lockdowns. The forecasts and evaluations provided by these models influence political actions and shape the measures implemented to contain the virus.
This research contributes to the understanding and control of disease spread, specifically for Dengue fever and COVID-19, making use of mathematical models and various data analysis techniques. The mathematical foundations of epidemiological modelling, as well as several concepts for spatio-temporal diffusion like ordinary differential equation (ODE) models, are presented, as well as an originally human-vector model for Dengue fever, and the standard (SEIR)-model (with the potential inclusion of an equation for deceased persons), which are suited for the description of COVID-19. Additionally, multi-compartment models, fractional diffusion models, partial differential equations (PDE) models, and integro-differential models are used to describe spatial propagation of the diseases.
We will make use of different optimization techniques to adapt the models to medical data and estimate the relevant parameters or finding optimal control techniques for containing diseases using both Metropolis and Lagrangian methods. Reasonable estimates for the unknown parameters are found, especially in initial stages of pandemics, when little to no information is available and the majority of the population has not got in contact with the disease. The longer a disease is present, the more complex the modelling gets and more things (vaccination, different types, etc.) appear and reduce the estimation and prediction quality of the mathematical models.
While it is possible to create highly complex models with numerous equations and parameters, such an approach presents several challenges, including difficulties in comparing and evaluating data, increased risk of overfitting, and reduced generalizability. Therefore, we will also consider criteria for model selection based on fit and complexity as well as the sensitivity of the model with respect to specific parameters. This also gives valuable information on which political interventions should be more emphasized for possible variations of parameter values.
Furthermore, the presented models, particularly the optimization using the Metropolis algorithm for parameter estimation, are compared with other established methods. The quality of model calculation, as well as computational effort and applicability, play a role in this comparison. Additionally, the spatial integro-differential model is compared with an established agent-based model. Since the macroscopic results align very well, the computationally faster integro-differential model can now be used as a proxy for the slower and non-traditionally optimizable agent-based model, e.g., in order to find an apt control strategy.