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.

Dengue remains a serious public health problem in urban areas; Jakarta and Bandung, Indonesia, are no exception. Our work aims to advance the understanding of dengue transmission dynamics in Bandung by integrating vehicle mobility data from tollgate records into the modeling framework. Most traditional models often overlook the impact of human mobility on the spread of disease. However, this study recognizes the critical role of vehicular movement as a contributing factor to dengue transmission. The research uses a comprehensive approach, combining epidemiological data with high-resolution tollgate records to map population movements. By integrating these mobility data into the infectious disease model, the study aims to refine predictions of dengue hotspots and transmission risk. Hence, incorporating mobility dynamics could significantly enhance the model’s accuracy and provide specific insights into patterns of disease spread, helping design more precise and impactful prevention strategies. The research explores the interconnectedness between human mobility and dengue transmission, sheds light on potential routes of disease spread, and identifies areas of increased risk. The findings of this study have implications for public health interventions, highlighting the need to consider mobility in developing proactive measures against dengue outbreaks. The integration of tollgate data provides a foundation for informed decision-making in public health strategies tailored to the unique challenges posed by urban environments. Our findings show that the infection rate and the effective reproduction ratio have strong correlations with the forward weekly infection, which could be used to detect the trend of the weekly infection early on.

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.

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.

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.

Robots are increasingly being used across various sectors, from industry and healthcare to household applications. In practice, a pivotal challenge is the reaction to unexpected external disturbances, whose real-time feedback often relies on (noisy) sensor measurements. Subsequent inverse-dynamics calculations demand noise-amplifying numerical differentiation, leading to impracticable results. Although much effort has been spent on establishing direct measurement approaches, their measurement uncertainty quantification has not or yet insufficiently been tackled in the literature. Here, we propose a multi-method framework to develop an angular acceleration reference and provide evidence that it can serve as a measurement standard to calibrate various kinematic sensors. Within the framework, we use Monte-Carlo simulations to quantify the uncertainty of a direct measurement sensor recently developed by our team; the inertial measurement cluster (IMC). For angular accelerations up to 21 rad/s 2 , the standard deviation of the IMC was on average only 0.3 rad/s 2 (95% CI: [0.28,0.31] rad/s 2 ), which constitutes a reliable data-sheet record. Further, using least-squares optimization, we show that the deviation of IMC with respect to the reference was not only less on the level of angular acceleration but also on the level of angular velocity and angle, when compared to other direct and indirect measurement methods.

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 coupled opinion–epidemic systems. 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 exhibits a protective pattern.

In this study, we focus on modeling the local spread of COVID-19 infections. As the pandemic continues and new variants or future pandemics can emerge, modelling the early stages of infection spread becomes crucial, especially as limited medical data might be available initially. Therefore, our aim is to gain a better understanding of the diffusion dynamics on smaller scales using partial differential equation (PDE) models. Previous works have already presented various methods to model the spatial spread of diseases, but, due to a lack of data on regional or even local scale, few actually applied their models on real disease courses in order to describe the behaviour of the disease or estimate parameters. We use medical data from both the Robert-Koch-Institute (RKI) and the Birkenfeld district government for parameter estimation within a single German district, Birkenfeld in Rhineland-Palatinate, during the second wave of the pandemic in autumn 2020 and winter 2020-21. This district can be seen as a typical middle-European region, characterized by its (mainly) rural nature and daily commuter movements towards metropolitan areas. A basic reaction-diffusion model used for spatial COVID spread, which includes compartments for susceptibles, exposed, infected, recovered, and the total population, is used to describe the spatio-temporal spread of infections. The transmission rate, recovery rate, initial infected values, detection rate, and diffusivity rate are considered as parameters to be estimated using the reported daily data and least square fit. This work also features an emphasis on numerical methods which will be used to describe the diffusion on arbitrary two-dimensional domains. Two numerical optimization techniques for parameter fitting are used: the Metropolis algorithm and the adjoint method. Two different methods, the Crank-Nicholson method and a finite element method, which are used according to the requirements of the respective optimization method are used to solve the PDE system. This way, the two methods are compared and validated and provide similar results with good approximation of the infected in both the district and the respective sub-districts.

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.

The current pandemic situation due to COVID-19 demonstrates the need for epidemiologic models to represent infection events as accurately as possible. An important factor is the mobility of the affected individuals which can be investigated with discrete or continuous spatial models. In this contribution, parameter estimation via adjoint functions is presented to fit a reaction-diffusion PDE system with epidemiological SIS model to data sets. For this purpose static and dynamic optimization methods are used to solve an L-2-norm based least squares problem. An artificial data set is generated to test the accuracy of the procedure. Subsequently, the PDE system is adapted to this data set using methods of optimal control theory. Unknown parameters like diffusivity and transmission rate can be determined. The noise in the data set is also taken into account by fitting the initial conditions. The results show that the method is well suited for this purpose and should be further used with real data sets.

Abstract Background COVID-19 continues to disrupt social lives and the economy of many countries and challenges their healthcare capacities. Looking back at the situation in Germany in 2020, the number of cases increased exponentially in early March. Social restrictions were imposed by closing e.g. schools, shops, cafés and restaurants, as well as borders for travellers. This reaped success as the infection rate descended significantly in early April. In mid July, however, the numbers started to rise again. Of particular reasons was that from mid June onwards, the travel ban has widely been cancelled or at least loosened. We aim to measure the impact of travellers on the overall infection dynamics for the case of (relatively) few infectives and no vaccinations available. We also want to analyse under which conditions political travelling measures are relevant, in particular in comparison to local measures. By travel restrictions in our model we mean all possible measures that equally reduce the possibility of infected returnees to further spread the disease in Germany, e.g. travel bans, lockdown, post-arrival tests and quarantines. Methods To analyse the impact of travellers, we present three variants of an susceptible–exposed–infected–recovered–deceased model to describe disease dynamics in Germany. Epidemiological parameters such as transmission rate, lethality, and detection rate of infected individuals are incorporated. We compare a model without inclusion of travellers and two models with a rate measuring the impact of travellers incorporating incidence data from the Johns Hopkins University. Parameter estimation was performed with the aid of the Monte–Carlo-based Metropolis algorithm. All models are compared in terms of validity and simplicity. Further, we perform sensitivity analyses of the model to observe on which of the model parameters show the largest influence the results. In particular, we compare local and international travelling measures and identify regions in which one of these shows larger relevance than the other. Results In the comparison of the three models, both models with the traveller impact rate yield significantly better results than the model without this rate. The model including a piecewise constant travel impact rate yields the best results in the sense of maximal likelihood and minimal Bayesian Information Criterion. We synthesize from model simulations and analyses that travellers had a strong impact on the overall infection cases in the considered time interval. By a comparison of the reproductive ratios of the models under traveller/no-traveller scenarios, we found that higher traveller numbers likely induce higher transmission rates and infection cases even in the further course, which is one possible explanation to the start of the second wave in Germany as of autumn 2020. The sensitivity analyses show that the travelling parameter, among others, shows a larger impact on the results. We also found that the relevance of travel measures depends on the value of the transmission parameter: In domains with a lower transmission parameter, caused either by the current variant or local measures, it is found that handling the travel parameters is more relevant than those with lower value of the transmission. Conclusions We conclude that travellers is an important factor in controlling infection cases during pandemics. Depending on the current situation, travel restrictions can be part of a policy to reduce infection numbers, especially when case numbers and transmission rate are low. The results of the sensitivity analyses also show that travel measures are more effective when the local transmission is already reduced, so a combination of those two appears to be optimal. In any case, supervision of the influence of travellers should always be undertaken, as another pandemic or wave can happen in the upcoming years and vaccinations and basic hygiene rules alone might not be able to prevent further infection waves.

Challenges set by COVID-19 in terms of deadliness, propagation speed, and gained immunity remain unprecedented. As the vaccine production races against virus mutations, international communities are settling toward the new normal guided by local government and health authorities. This study asks how well such transition has been applied in Germany by assessing the functionality of typical measures against the disease. We propose a mathematical model to govern mechanistic processes behind the epidemics. The analysis focuses on to what extent face mask, its efficacy, and community awareness in the likelihood of enhancing physical distancing may significantly suppress the incidence in the long run. Focusing on reemerging peaks of outbreaks during winter season, our sensitivity analysis and optimal decision framework recommend that guaranteeing the locals to uphold physical distancing during the festive season (beginning of November to the end of December) is urgent and effective in reducing the inflow of new cases.

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 ∈ [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.

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.

This paper stresses its base contribution on a new SIR-type model including direct and fomite transmission as well as the effect of distinct household structures. The model derivation is modulated by several mechanistic processes inherent from typical airborne diseases. The notion of minimum contact radius is included in the direct transmission, facilitating the arguments on physical distancing. As fomite transmission heavily relates to former-trace of sneezes, the vector field of the system naturally contains an integral kernel with time delay indicating the contribution of undetected and non-quarantined asymptomatic cases in accumulating the historical contamination of surfaces. We then increase the complexity by including the different transmission routines within and between households. For airborne diseases, within-household interactions play a significant role in the propagation of the disease rendering countrywide effect. Two steps were taken to include the effect of household structure. The first step subdivides the entire compartments (susceptible, exposed, asymptomatic, symptomatic, recovered, death) into the household level and different infection rates for the direct transmission within and between households were distinguished. Under predefined conditions and assumptions, the governing system on household level can be raised to the community level. The second step then raises the governing system to the country level, where the final state variables estimate the total individuals from all compartments in the country. Two key attributes related to the household structure (number of local households and number of household members) effectively classify countries to be of low or high risk in terms of effective disease propagation. The basic reproductive number is calculated and its biological meaning is invoked properly. The numerical methods for solving the DIDE-system and the parameter estimation problem were mentioned. Our optimal model solutions are in quite good agreement with datasets of COVID-19 active cases and related deaths from Germany and Sri Lanka in early infection, allowing us to hypothesize several unobservable situations in the two countries. Focusing on extending minimum contact radius and reducing the intensity of individual activities, we were able to synthesize the key parameters telling what to practice.

Microplastic in freshwater has been known to absorb, adsorb, and later desorb persistent organic pollutants as well as in its tiny size acts as an infiltrator to vital tissues; it may therefore corrupt physiological processes of organic lives. The fate of microplastic particles can be understood by revealing to what extent certain material properties (e.g., size and density) determine local behavior such as sedimentation and interaction with biofilm. This work seeks to gain an understanding of the short-range transport of microplastic particles in freshwater through devising a lid-driven cavity with a biofilm-covering obstacle as the medium. A stationary Navier-Stokes equation for an incompressible fluid at a moderate Reynolds number provides the background flow field. Microbeads are injected into the flow field, where their motion is governed by a Lagrangian system of equations. Advanced features such as dry particle-particle and particle-wall collisions as well as adhesion between particles and biofilm portraying particle entrapment are presented. Various simulations and parameterization studies are carried out to determine the impact of material properties, obstacle geometry, and adhesion force on the deposition profiles. In most cases, particles are trapped in the biofilm and in regions around the cavity with negative Okubo-Weiss numbers whereby the relative vorticity is dominating against the local strains. (c) 2021 Elsevier B.V. All rights reserved.

Abstract At the end of 2019, an outbreak of a new coronavirus, called SARS–CoV–2, was reported in China and later in other parts of the world. First infection reported in Germany by the end of January 2020 and on March 16th, 2020 the federal government announced a partial lockdown in order to mitigate the spread. Since the dynamics of new infections started to slow down, German states started to relax the confinement measures as to May 6th, 2020. As a fall back option, a limit of 50 new infections per 100,000 inhabitants within seven days was introduced for each district in Germany. If a district exceeds this limit, measures to control the spread of the virus should be taken. Based on a multi–patch SEAIRD–type model, we will simulate the effect of choosing a specific upper limit for new infections. We investigate, whether the politically motivated bound is low enough to detect new outbreaks at an early stage. Subsequently, we introduce an optimal control problem to tackle the multi–criteria problem of finding a bound for new infections that is low enough to avoid new outbreaks, which might lead to an overload of the health care system, but is large enough to curb the expected economic losses.

Since the end of 2019 an outbreak of a new strain of coronavirus, called SARS-CoV-2, is reported from China and later other parts of the world. Since January 21, World Health Organization (WHO) reports daily data on confirmed cases and deaths from both China and other countries (www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-reports). The Johns Hopkins University (github.com/CSSEGISandData/COVID-19/blob/master/csse_COVID_19_data/csse_COVID_19_time_series/time_series_COVID19_confirmed_global.csv) collects those data from various sources worldwide on a daily basis. For Germany, the Robert-Koch-Institute (RKI) also issues daily reports on the current number of infections and infection related fatal cases (www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Gesamt.html). However, due to delays in the data collection, the data from RKI always lags behind those reported by Johns Hopkins. In this work we present an extended SEIRD-model to describe the disease dynamics in Germany. The parameter values are identified by matching the model output to the officially reported cases. An additional parameter to capture the influence of unidentified cases is also included in the model.

This article thematizes the qualitative estimation of transmission dynamics of Dengue fever. At first, a single-compartment vector-host model for the total infective cases in one homogeneous area has been set up and simplified using a steady-state approximation. Seasonality has been considered in the transmission parameter which is modelled by a Fourier sum. The equilibria of the model and their stability as well as the computation of the basic reproductive number are presented. As a modification, models considering a segmenting of the area in separate districts and their inter-district mobility have been set up, both with and without dependence of the disease transmission parameters on the district. Those have also been analysed in terms of equilibria and stability. Parameter estimation on available Dengue data from Jakarta in the time interval of 2008–2016 using the Metropolis algorithm has been done. $${\mathscr {L}}_1$$ and $${\mathscr {L}}_2$$ comparisons show that using the multi-patch model with district-dependent parameters a decent approximation to the infection data is possible.

On the basis of a semi-realistic SIR microsimulation for Germany and Poland, we show that the R0 parameter interval for which the COVID-19 epidemic stays overcritical but below the capacity limit of the health care system to reach herd immunity is so narrow that a successful implementation of this strategy is likely to fail. Our microsimulation is based on official census data and involves household composition and age distribution as the main population structure variables. Outside household contacts are characterised by an out-reproduction number R* which is the only free parameter of the model. For a subcritical domain we compute the time till extinction and prevalence as a function of the initial number of infected individuals and R*. For the Polish city of Wroclaw we also discuss the combined impact of testing coverage and contact reduction. For both countries we estimate R* for disease progression until 20th of March 2020.