Session P17: Statistical Mechanics of Disease Propagation I

Live

The theoretical modeling of an epidemic poses serious challenges due to the vast complexities of the disease-host-population system. While certain qualitative behavior can be inferred from SIR-type models, they can often be unrealistic in application owing to limiting factors such as homogeneity and geospatial isolation of the modeled population. A common remedy is to spatially decompose a population into locally homogeneous metapopulations (MPs); however, without allowing a disease to spread between MPs, the true dynamics can never be recovered, and no single MP can be modeled without simulating all of them. Furthermore, if the host mobility is highly nonlocal (e.g., as with humans), the parameter space scales as O(N²) with the number of MPs, and there is typically a dearth of data available to construct reasonable estimates for these parameters. We present a reduced-order model for a single MP to act as a framework for capturing the contributions from exterior MPs through an effective “force of infection”, which encapsulates the necessary boundary conditions to accurately model a given MP while obviating a full simulation of the system. Simulation results are compared to synthetic data, and applications to modeling the current COVID-19 pandemic are discussed.   

Live

We suggest a novel mathematical framework for the inhomogeneous spatial spreading of an infectious disease in human population. SEIR-type epidemiological models assume uniform random encounters between the infectious and susceptible sub-populations, resulting in homogeneous spatial distributions. In human populations this assumption fails. Splitting the geographic region under study into areal nodes, we arrive into a continuous, ``reaction-diffusion'' model. For COVID-19, the model includes five different sub-populations. Our model accounts for the spreading evolution of infectious populations from initial epicenters, leading to different regimes of sub-exponential (e.g., power-law) growth. Importantly, we also account for the variable geographic density of the population. We show how infected ``suburban’’ areas can cause rapid migration of the infection towards a ``city’’. Predicted infection ``heat-maps" show remarkable similarity to publicly available heat-maps, e.g., from South Carolina. We further demonstrate how localized lockdown/quarantine conditions can slow down the spreading of disease from epicenters. Application of our model in different countries can provide a useful predictive tool for the authorities for planning strong lockdown measures in localized areas.   

Live

For preventing the spread of epidemics such as the coronavirus disease COVID-19, social distancing and the isolation of infected persons are crucial. However, existing reaction-diffusion equations for epidemic spreading are incapable of describing these effects. In this talk, we present an extended model for disease spread based on combining a susceptible-infected-recovered model with a dynamical density functional theory where social distancing and isolation of infected persons are explicitly taken into account [1]. We show that the model exhibits interesting transient phase separation associated with a reduction of the number of infections, and provides new insights into the control of pandemics. An extension of the model [2] allows for an investigation of adaptive containment strategies. Here, a variety of phases with different numbers of shutdowns and deaths are found, an effect that is of crucial importance for public health policy.

[1] M. te Vrugt, J. Bickmann and R. Wittkowski, Nature Communications, accepted (2020)
[2] M. te Vrugt, J. Bickmann and R. Wittkowski, arXiv:2010.00962 (2020)   

Live

The 1st wave of COVID-19 changed social distancing around the globe: severe lockdowns to stop pandemics at the cost of state
economies preceded a series of lockdown lifts. To understand social distancing dynamics it is important to combine basic epidemiology
models for viral unfold (like SIR) with game theory tools, such as a utility function that quantifies individual or government forecast
for epidemic damage and economy cost as the functions of social distancing.

We present a model that predicts a series of discontinuous transitions in social distancing during a pandemic wave. Each transition resembles Ginzburg-Landau instability and, so, maybe a general phenomenon. Data analysis of the first wave in Austria, Israel, and Germany corroborates the soundness of the model. Besides, this work presents analytical tools to analyze pandemic waves.

[1] https://arxiv.org/abs/2008.06863   

Live

The importance of a strict quarantine has been widely debated during the COVID-19 epidemic even from the purely epidemiological point of view. One argument against strict lockdown measures is that once the strict quarantine is lifted, the epidemic comes back, and so the total number of infected individuals during the entire epidemic will stay the same. We consider an SIR model on a network and follow the disease dynamics, modeling the phases of quarantine by changing the node degree distribution. We show that the system reaches different steady states based on the history despite the same final node degree distribution. The results indicate that two phases (a strict phase followed by a soft phase) are always better than one phase (the same soft one) unless all the individuals have the same number of connections (the same degree); in the latter case, the overall number of infected is indeed history-independent. The modeling also suggests that the optimal procedure of lifting the quarantine is degree-based.   

Live

We start with the SIR model (susceptible, infected, removed) on a network. See, e.g., Cooper, Mondal & Antonopoulos, in Chaos, Solitons & Fractals 2020. Since the goal is to make I = 0 a (Lyapunov) stable equilibrium, we linearize the discrete-time SIR model to obtain difference equations of the form I_new = I(1 + aS - b) at each node before including infections derived from other nodes. We assume S equal to its initial value at that node. Here a depends upon the infectivity and contact rate, b = 1/(duration of infectivity) and the traditional R_t = aS/b (R_t < 1 ↔ aS < b). This yields a vector difference equation I_new = MI. The entries of M may vary in time, even discontinuously as flows between nodes are turned on and off. The column sums of M may be interpreted as generalizations of (scalar) growth rates. Theorem. If the matrices M are column-substochastic, then the I=0 equilibrium is stable. This may yield useful design constraints for a multi-network composed of weak and strong interactions between pairs of nodes representing interactions within and among cities.   

Live

Holographic immunoassays use holographic particle characterization to measure growth in the mean diameter of thousands of colloidal beads due to the binding of antibodies to the probe bead surface. This measurement has the advantage of being label-free; binding can be detected without fluorescent labeling of the biomolecules. This reduces the cost and complexity of the technique compared with traditional assays like ELISA. Additionally, the probe beads can be batch synthesized and functionalized without the need for microfabrication. Our work on this topic measured the kinetics of irreversible binding of immunoglobulin G (IgG) and immunoglobulin M (IgM) to protein A coated probe beads. These measurements were used to determine the antibody binding rates and can be inverted to determine the concentration of antibodies in solution. This technique shows promise for detecting Antibody Deficiency Disorders and for the detection of antibodies for SARS COV-2 and other infections.

"The first two authors contributed equally to this work."   

Not Participating

In this work, we demonstrate methods for targeting and killing mammalian cancer cells with visible non-ionizing radiation. We find that photothermal efficiency is massively enhanced when cells are sensitized with synthetic melanin nanoparticles, known to be excellent absorbers of light in the ultra-violet (UV), visible (VIS) and near infrared (NIR) frequency ranges. Nanoparticle uptake is highly efficent for malignant cancer cell lines, and this uptake is further enhanced by coating the melanin nanoparticles with glucose. Death of nanoparticle-filled cells occurs primarily by heating. We show that this process of cell elimination is highly target-selective in the presence of non-sensitized cells.   

Live

The collective behavior that results from large numbers of atoms interacting in a material (mainly each atom only with its immediate neighbors for metals) determines that material's response to impulsive loading, such as a shock wave passing through a solid. Similarly, a pandemic can spread throughout a country or even worldwide as the result of a series of individual human-to-human contacts. By combining a stochastic agent-based model of disease spread among individuals at the local community level with detailed U.S. Census and Department of Transportation data on population demographics and mobility, we have extended our “SPaSM” (Scalable Parallel Short-range Molecular dynamics) code into a powerful epidemiological modeling tool for studying the spatiotemporal dynamics of regional to national-scale outbreaks. This simulation model, developed in the early 2000s, has been used to assess potential mitigation strategies – school closures, vaccination or antiviral prophylaxis campaigns, restricted air or highway travel, quarantines, ..., and various combinations thereof – in the event of an influenza pandemic in the United States. The arrival of COVID-19 presented additional challenges, in turning a simulation platform designed for “what-if” scenario exploration into one used for real-time response to an emerging pandemic outbreak.   

Live

The appearance of the coronavirus (COVID-19) in late 2019 has dominated the news in the last few months as it developed into a pandemic. In many mathematics and physics classrooms, instructors are using the time series of the number of cases to show exponential growth of the
infection. Initially we proposed to use a simple diffusion process as the mode of spreading infections to be implemented as a classroom project in computational physics courses. This model is less sophisticated than other models in the literature, but it can capture the exponential growth and it can explain it in terms of mobility (diffusion constant), population density, and probability of transmission. An account of this pedagogical application was recently published in (Am. J. Phys. 88, 604 (2020)). In this presentation we modify the initial program to include mitigation effects, such as social distancing and use of masks, as well as the effects of inter-city/neighborhood travel on the spread of the virus. Showing that one can change the caseload and explain how pockets that were initially case free can become hotspots.   

Live

We introduce a modified SIRD (Susceptible-Infected-Recovered-Deceased) model and study the impact of social distancing for interacting random walkers in two dimensional continuum using Monte Carlo methods. We use both fixed step length walkers as well as variable step length walkers drawn from a Gaussian distribution for different vulnerability factors, population density, inter-city/country traffic, and a variety of possible scenario based on COVID-19 data at various locations, and study to what extent social distancing would mitigate the spread of the disease. Our model can be used to understand how COVID-19 would have impacted at a reduced degree if social distancing were maintained more rigidly, and suggests measures to be taken to resists spread of such disease before it becomes a pandemic.