arXiv subjects

Thomas Götz

Publications and source records attributed to Thomas Götz.

17 recordsLinked to original sources

arXiv2026

A Mathematical Model of Dengue Transmission Incorporating Hospital Capacity and Threshold-Based Fogging Interventions

Dengue remains a major public health challenge in tropical regions, and recurring outbreaks suggest that current intervention strategies are not yet fully effective. Existing mathematical models typically assume unlimited hospital capacity and continuously applied fogging, neglecting practical constraints that strongly influence disease control. We develop a non-smooth ordinary differential equation model of dengue transmission that incorporates finite hospital capacity and a threshold-triggered fogging strategy activated when reported infections exceed a prescribed fraction of the available capacity. The model exhibits three epidemiologically relevant operating regimes, reflecting changes in hospitalization and vector-control policies as the epidemic progresses. We establish the existence and local stability of the disease-free and endemic equilibria. Numerical continuation confirms the analytical results and reveals boundary-equilibrium bifurcations at the switching thresholds, a Hopf bifurcation after hospital capacity is exceeded leading to sustained oscillatory outbreaks, and a fold bifurcation near the epidemic threshold that generates additional unstable equilibria. We further investigate periodic solutions with respect to the fogging rate and activation threshold, identifying locally optimal intervention regimes that reduce epidemic peaks while avoiding unnecessarily intensive control efforts. The results demonstrate that hospital capacity, reactive fogging, and intervention thresholds fundamentally shape dengue dynamics and provide quantitative insights for designing effective state-dependent control strategies under limited healthcare resources.

math.DS↗

arXiv2026

Age Structured Epidemic Model under Vaccination with Vector Transmission

Dengue remains a major global public health concern due to its high mortality and economic burden. Mathematical modeling is essential to understand its transmission mechanisms and for evaluating intervention strategies. In this paper, we formulate a vector host model in which the human population is structured by age, and vaccinated individuals are further described by time since vaccination. The mosquito population is coupled to the host dynamics and reduced under a quasi steady state assumption. By integrating over vaccination age, we obtain a nonlinear steady state formulation and express the endemic equilibrium as a fixed point problem for the infected mosquito population. Using Lipschitz estimates and a contraction argument, we establish existence and uniqueness of the equilibrium under a weak transmission condition. The analysis highlights the influence of age dependent vaccination on long term dengue dynamics.

q-bio.PE↗

arXiv2026

Modeling and analysis of a novel two-strain dengue epidemics model considering secondary infections with increased mortality

In this study, we develop and analyze a deterministic two-strain host-vector model for dengue transmission that incorporates key immuno-epidemiological mechanisms, including temporary cross-immunity, antibody-dependent enhancement (ADE), disease-induced mortality during secondary infections, and explicit vector co-infection. The human population is divided into compartments for primary and secondary infections, while the mosquito population includes single- and co-infected classes. ADE is modeled through distinct primary ($α$) and secondary ($σ$) transmission rates. Using the next-generation matrix method, we derive the basic reproduction number $R_0$ and establish the local stability of the disease-free equilibrium for $R_0 < 1$. Analytical results show that one-strain endemic equilibria lose stability under ADE conditions ($σ> α$), allowing invasion by a heterologous strain. Employing center-manifold theory and numerical continuation (COCO), we demonstrate the occurrence of backward bifurcation, bistability between disease-free and endemic states, and Hopf-induced oscillations. Numerical simulations confirm transitions among disease-free, endemic, and periodic regimes as key parameters vary. The model highlights how ADE, waning cross-immunity, and vector co-infection interact to generate complex dengue dynamics and provides insights useful for designing effective control and vaccination strategies in dengue-endemic regions.

math.DS↗

arXiv2025

An SIRS-model considering waning efficiency and periodic re-vaccination

In this paper, we extend the classical SIRS (Susceptible-Infectious-Recovered-Susceptible) model from mathematical epidemiology by incorporating a vaccinated compartment, V, accounting for an imperfect vaccine with waning efficacy over time. The SIRSV-model divides the population into four compartments and introduces periodic re-vaccination for waning immunity. The efficiency of the vaccine is assumed to decay with the time passed since the vaccination. Periodic re-vaccinations are applied to the population. We develop a partial differential equation (PDE) model for the continuous vaccination time and a coupled ordinary differential equation (ODE) system when discretizing the vaccination period. We analyze the equilibria of the ODE model and investigate the linear stability of the disease-free equilibrium (DFE). Furthermore, we explore an optimization framework where vaccination rate, re-vaccination time, and non-pharmaceutical interventions (NPIs) are control variables to minimize infection levels. The optimization objective is defined using different norm-based measures of infected individuals. A numerical analysis of the model's dynamic behavior under varying control parameters is conducted using path-following methods. The analysis focuses on the impacts of vaccination strategies and contact limitation measures. Bifurcation analysis reveals complex behaviors, including bistability, fold bifurcations, forward and backward bifurcations, highlighting the need for combined vaccination and contact control strategies to manage disease spread effectively.

q-bio.PE↗

arXiv2025

SIR-Model for Households

Households play an important role in disease dynamics. Many infections happening there due to the close contact, while mitigation measures mainly target the transmission between households. Therefore, one can see households as boosting the transmission depending on household size. To study the effect of household size and size distribution, we differentiated the within and between household reproduction rate. There are basically no preventive measures, and thus the close contacts can boost the spread. We explicitly incorporated that typically only a fraction of all household members are infected. Thus, viewing the infection of a household of a given size as a splitting process generating a new, small fully infected sub-household and a remaining still susceptible sub-household we derive a compartmental ODE-model for the dynamics of the sub-households. In this setting, the basic reproduction number as well as prevalence and the peak of an infection wave in a population with given households size distribution can be computed analytically. We compare numerical simulation results of this novel household-ODE model with results from an agent--based model using data for realistic household size distributions of different countries. We find good agreement of both models showing the catalytic effect of large households on the overall disease dynamics.

q-bio.PE↗

arXiv2024

Chaos in opinion-driven disease dynamics

During the COVID-19 pandemic, it became evident that the effectiveness of applying intervention measures is significantly influenced by societal acceptance, which, in turn, is affected by the processes of opinion formation. This article explores one among the many possibilities of a coupled opinion-epidemic system. The findings reveal either intricate periodic patterns or chaotic dynamics, leading to substantial fluctuations in opinion distribution and, consequently, significant variations in the total number of infections over time. Interestingly, the model is exhibiting the protective pattern.

nlin.CD↗

arXiv2023

Modelling the Spatial Spread of COVID-19 in a German District using a Diffusion Model

In this study, we present an integro-differential model to simulate the local spread of infections. The model incorporates a standard susceptible-infected-recovered (\textit{SIR}-) model enhanced by an integral kernel, allowing for non-homogeneous mixing between susceptibles and infectives. We define requirements for the kernel function and derive analytical results for both the \textit{SIR}- and a reduced susceptible-infected-susceptible (\textit{SIS}-) model, especially the uniqueness of solutions. In order to optimize the balance between disease containment and the social and political costs associated with lockdown measures, we set up requirements for the implementation of control functions, and show examples for continuous and time-dependent, continuous and space- and time-dependent, and piecewise constant space- and time-dependent controls. Latter represent reality more closely as the control cannot be updated for every time and location. We found the optimal control values for all of those setups, which are by nature best for a continuous and space-and time dependent control, yet found reasonable results for the discrete setting as well. To validate the numerical results of the integro-differential model, we compare them to an established agent-based model that incorporates social and other microscopical factors more accurately and thus acts as a benchmark for the validity of the integro-differential approach. A close match between the results of both models validates the integro-differential model as an efficient macroscopic proxy. Since computing an optimal control strategy for agent-based models is computationally very expensive, yet comparatively cheap for the integro-differential model, using the proxy model might have interesting implications for future research.

math.DS↗

arXiv2023

An integro-differential model for the spread of diseases

In this study, we present an integro-differential model to simulate the local spread of infections. The model incorporates a standard susceptible-infected-recovered (\textit{SIR}-) model enhanced by an integral kernel, allowing for non-homogeneous mixing between susceptibles and infectives. We define requirements for the kernel function and derive analytical results for both the \textit{SIR}- and a reduced susceptible-infected-susceptible (\textit{SIS}-) model, especially the uniqueness of solutions. In order to optimize the balance between disease containment and the social and political costs associated with lockdown measures, we set up requirements for the implementation of control function, and show examples for three different formulations for the control: continuous and time-dependent, continuous and space- and time-dependent, and piecewise constant space- and time-dependent. Latter represent reality more closely as the control cannot be updated for every time and location. We found the optimal control values for all of those setups, which are by nature best for a continuous and space-and time dependent control, yet found reasonable results for the discrete setting as well. To validate the numerical results of the integro-differential model, we compare them to an established agent-based model that incorporates social and other microscopical factors more accurately and thus acts as a benchmark for the validity of the integro-differential approach. A close match between the results of both models validates the integro-differential model as an efficient macroscopic proxy. Since computing an optimal control strategy for agent-based models is computationally very expensive, yet comparatively cheap for the integro-differential model, using the proxy model might have interesting implications for future research.

math.DS↗

arXiv2023

Generalizable Classification of UHF Partial Discharge Signals in Gas-Insulated HVDC Systems Using Neural Networks

Undetected partial discharges (PDs) are a safety critical issue in high voltage (HV) gas insulated systems (GIS). While the diagnosis of PDs under AC voltage is well-established, the analysis of PDs under DC voltage remains an active research field. A key focus of these investigations is the classification of different PD sources to enable subsequent sophisticated analysis. In this paper, we propose and analyze a neural network-based approach for classifying PD signals caused by metallic protrusions and conductive particles on the insulator of HVDC GIS, without relying on pulse sequence analysis features. In contrast to previous approaches, our proposed model can discriminate the studied PD signals obtained at negative and positive potentials, while also generalizing to unseen operating voltage multiples. Additionally, we compare the performance of time- and frequency-domain input signals and explore the impact of different normalization schemes to mitigate the influence of free-space path loss between the sensor and defect location.

cs.LG↗

arXiv2022

Analysis of an SIR--model with global and local infections

An epidemic model where disease transmission can occur either through global contacts or through local, nearest neighbor interactions is considered. The classical SIR--model describing the global interactions is extended by adding additional equations for the density of local pairs in different epidemic states. A locality parameter $p\in [0,1]$ characterizes the probability of global or local infections. The equilibria of the resulting model are analyzed in dependence of the locality parameter and the transmission rate of the pathogen. An explicit expression for the reproduction number in terms of the locality parameter and the disease parameters is obtained. Transient simulations confirm these findings. Neighboring pairs of one infected and one susceptible can be considered as active pairs, since local transmission of the disease can only occur in that situation. Our analysis shows, that the fraction of active pairs is minimal for intermediate values of the locality parameter.

q-bio.PE↗

arXiv2021

A two-strain SARS-COV-2 model for Germany -- Evidence from a Linearization

Currently, due to the COVID-19 pandemic the public life in most European countries stopped almost completely due to measures against the spread of the virus. Efforts to limit the number of new infections are threatened by the advent of new variants of the SARS-COV-2 virus, most prominent the B.1.1.7 strain with higher infectivity. In this article we consider a basic two-strain SIR model to explain the spread of those variants in Germany on small time scales. For a linearized version of the model we calculate relevant variables like the time of minimal infections or the dynamics of the share of variants analytically. These analytical approximations and numerical simulations are in a good agreement to data reported by the Robert Koch Institute (RKI) in Germany.

q-bio.PE↗

arXiv2020

Calculation of a local COVID-19 reproduction number for the northern Rhineland-Palatinate

Since the beginning of the corona pandemic in March 2020, various parameters for describing the spread of the disease have been specified for Germany in addition to the daily infection figures (new infections and total infections), which are also used for political decisions. In addition to excess mortality and the weekly incidence, these include the doubling time $T_2$ and the reproduction number $R_t$. For the latter, various estimates can be found on the website of the Robert-Koch-Institute, see \cite{EstR:RKI}, which are calculated from the case numbers for all of Germany; local differences are not taken into account here. In the present article, the calculations of the RKI on a local level are examined using the example of northern Rhineland-Palatinate and its districts. Here, not the reporting date but the onset of illness is used as a reference for the calculation of $R_t$. For cases where the onset of illness is not known, an adjusted generalized extreme value distribution (GEV) is first fitted to the data for which the reporting delay (difference between the onset of illness and the reporting date) is available and examined for further characteristics such as local as well as demographic differences. This GEV distribution is then used to calculate the reporting delays of incomplete data points. The calculation of the daily value of $R_t$ between the end of February and the end of October showed a similar course of the reproductive rate compared to the nationwide figures. Expectably larger statistical fluctuations were observed in the summer, mainly due to lower case numbers. The values for northern Rhineland-Palatinate have been consistently above $1$ since about mid-September. The calculations can also be transferred to other regions and administrative districts.

q-bio.PE↗

arXiv2020

An improved mathematical model for the sedimentation of microplastic particles in a lid-driven cavity with obstacle

In this paper, we developed a mathematical model for the sedimentation process of small particles in a lid-driven cavity flow with an obstacle to model the transport of microplastic particles in rivers. A stationary incompressible Navier-Stokes simulation at moderate Reynolds-numbers provides the background flow field. Spherical particles are injected into this flow field and their equation of motion is solved to determine the number of particles that sediment on the surface of the obstacle. To capture the effect of typical biological organisms and biofilms on the bottom surface of river beds, the obstacle can exert an attraction force onto the particles. Various simulations and parameter studies are carried out to determine the influence of the obstacle geometry, particle densities, and the attraction force on the sedimentation process.

physics.flu-dyn↗

arXiv2020

First attempts to model the dynamics of the Coronavirus outbreak 2020

Since the end of 2019 an outbreak of a new strain of coronavirus, called 2019--nCoV, is reported from China and later other parts of the world. Since January 21, WHO reports daily data on confirmed cases and deaths from both China and other countries. In this work we present some discrete and continuous models to discribe the disease dynamics in China and estimate the needed epidemiological parameters. Good agreement with the current dynamics has be found for both a discrete transmission model and a slightly modified SIR-model.

q-bio.PE↗

arXiv2015

Nonexistence of steady solutions for rotational slender fibre spinning with surface tension

Reduced one-dimensional equations for the stationary, isothermal rotational spinning process of slender fibers are considered for the case of large Reynolds ($δ=3/\text{Re}\ll 1$) and small Rossby numbers ($\varepsilon \ll 1$). Surface tension is included in the model using the parameter $κ=\sqrtπ/(2 \text{We})$ related to the inverse Weber number. The inviscid case $δ=0$ is discussed as a reference case. For the viscous case $δ> 0$ numerical simulations indicate, that for a certain parameter range, no physically relevant solution may exist. Transferring properties of the inviscid limit to the viscous case, analytical bounds for the initial viscous stress of the fiber are obtained. A good agreement with the numerical results is found. These bounds give strong evidence, that for $δ> 3\varepsilon^2 \left( 1- \frac{3}{2}κ+\frac{1}{2}κ^2\right)$ no physical relevant stationary solution can exist.

math-ph↗

arXiv2011

Parameter Estimation of Fiber Lay-down in Nonwoven Production - An Occupation Time Approach-

In this paper we investigate the parameter estimation of the fiber lay-down process in the production of nonwovens. The parameter estimation is based on the mass per unit area data, which is available at least on an industrial scale. We introduce a stochastic model to represent the fiber lay-down and through the model's parameters we characterize this fiber lay-down. Based on the occupation time, which is the equivalent quantity for the mass per unit area in the context of stochastic dynamical systems, an optimization procedure is formulated that estimates the parameters of the model. The optimization procedure is tested using occupation time data given by Monte-Carlo simulations. The feasibility of the optimization procedure on an industrial level is tested using the fiber paths simulated by the industrial software FYDIST.

math.PR↗

arXiv2011

Parameter Estimation from Occupation Times

We derive an equation to compute directly the expected occupation time of the centered Ornstein-Uhlenbeck process. This allows us to identify the parameters of the Ornstein-Uhlenbeck process for available occupation times via a standard least squares minimization. To test the method, we generate occupation times via Monte-Carlo simulations and recover the parameters with the above mentioned procedure.

math.NA↗