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1. Embedding properties of network realizations of dissipative reduced order models

   Druskin, Guddati ( July 2022 , HOUSEHOLDER SYMPOSIUM XXI)

   Embedding properties of network realizations of dissipative reduced order models Jörn Zimmerling, Mikhail Zaslavsky,Rob Remis, Shasri Moskow, Alexander Mamonov, Murthy Guddati, Vladimir Druskin, and Liliana Borcea Mathematical Sciences Department, Worcester Polytechnic Institute https://www.wpi.edu/people/vdruskin Abstract Realizations of reduced order models of passive SISO or MIMO LTI problems can be transformed to tridiagonal and block-tridiagonal forms, respectively, via dierent modications of the Lanczos algorithm. Generally, such realizations can be interpreted as ladder resistor-capacitor-inductor (RCL) networks. They gave rise to network syntheses in the rst half of the 20th century that was at the base of modern electronics design and consecutively to MOR that tremendously impacted many areas of engineering (electrical, mechanical, aerospace, etc.) by enabling ecient compression of the underlining dynamical systems. In his seminal 1950s works Krein realized that in addition to their compressing properties, network realizations can be used to embed the data back into the state space of the underlying continuum problems. In more recent works of the authors Krein's ideas gave rise to so-called nite-dierence Gaussian quadrature rules (FDGQR), allowing to approximately map the ROM state-space representation to its full order continuum counterpart on a judicially chosen grid. Thus, the state variables can be accessed directly from the transfer function without solving the full problem and even explicit knowledge of the PDE coecients in the interior, i.e., the FDGQR directly learns" the problem from its transfer function. This embedding property found applications in PDE solvers, inverse problems and unsupervised machine learning. Here we show a generalization of this approach to dissipative PDE problems, e.g., electromagnetic and acoustic wave propagation in lossy dispersive media. Potential applications include solution of inverse scattering problems in dispersive media, such as seismic exploration, radars and sonars. To x the idea, we consider a passive irreducible SISO ROM fn(s) = Xn j=1 yi s + σj , (62) assuming that all complex terms in (62) come in conjugate pairs. We will seek ladder realization of (62) as rjuj + vj − vj−1 = −shˆjuj , uj+1 − uj + ˆrj vj = −shj vj , (63) for j = 0, . . . , n with boundary conditions un+1 = 0, v1 = −1, and 4n real parameters hi, hˆi, ri and rˆi, i = 1, . . . , n, that can be considered, respectively, as the equivalent discrete inductances, capacitors and also primary and dual conductors. Alternatively, they can be viewed as respectively masses, spring stiness, primary and dual dampers of a mechanical string. Reordering variables would bring (63) into tridiagonal form, so from the spectral measure given by (62 ) the coecients of (63) can be obtained via a non-symmetric Lanczos algorithm written in J-symmetric form and fn(s) can be equivalently computed as fn(s) = u1. The cases considered in the original FDGQR correspond to either (i) real y, θ or (ii) real y and imaginary θ. Both cases are covered by the Stieltjes theorem, that yields in case (i) real positive h, hˆ and trivial r, rˆ, and in case (ii) real positive h,r and trivial hˆ,rˆ. This result allowed us a simple interpretation of (62) as the staggered nite-dierence approximation of the underlying PDE problem [2]. For PDEs in more than one variables (including topologically rich data-manifolds), a nite-dierence interpretation is obtained via a MIMO extensions in block form, e.g., [4, 3]. The main diculty of extending this approach to general passive problems is that the Stieltjes theory is no longer applicable. Moreover, the tridiagonal realization of a passive ROM transfer function (62) via the ladder network (63) cannot always be obtained in port-Hamiltonian form, i.e., the equivalent primary and dual conductors may change sign [1]. 100 Embedding of the Stieltjes problems, e.g., the case (i) was done by mapping h and hˆ into values of acoustic (or electromagnetic) impedance at grid cells, that required a special coordinate stretching (known as travel time coordinate transform) for continuous problems. Likewise, to circumvent possible non-positivity of conductors for the non-Stieltjes case, we introduce an additional complex s-dependent coordinate stretching, vanishing as s → ∞ [1]. This stretching applied in the discrete setting induces a diagonal factorization, removes oscillating coecients, and leads to an accurate embedding for moderate variations of the coecients of the continuum problems, i.e., it maps discrete coecients onto the values of their continuum counterparts. Not only does this embedding yields an approximate linear algebraic algorithm for the solution of the inverse problems for dissipative PDEs, it also leads to new insight into the properties of their ROM realizations. We will also discuss another approach to embedding, based on Krein-Nudelman theory [5], that results in special data-driven adaptive grids. References [1] Borcea, Liliana and Druskin, Vladimir and Zimmerling, Jörn, A reduced order model approach to inverse scattering in lossy layered media, Journal of Scientic Computing, V. 89, N1, pp. 136,2021 [2] Druskin, Vladimir and Knizhnerman, Leonid, Gaussian spectral rules for the three-point second dierences: I. A two-point positive denite problem in a semi-innite domain, SIAM Journal on Numerical Analysis, V. 37, N 2, pp.403422, 1999 [3] Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Distance preserving model order reduction of graph-Laplacians and cluster analysis, Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Journal of Scientic Computing, V. 90, N 1, pp 130, 2022 [4] Druskin, Vladimir and Moskow, Shari and Zaslavsky, Mikhail LippmannSchwingerLanczos algorithm for inverse scattering problems, Inverse Problems, V. 37, N. 7, 2021, [5] Mark Adolfovich Nudelman The Krein String and Characteristic Functions of Maximal Dissipative Operators, Journal of Mathematical Sciences, 2004, V 124, pp 49184934 Go back to Plenary Speakers Go back to Speakers Go back 

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2. The unintended consequences of inconsistent closure policies and mobility restrictions during epidemics

   https://doi.org/10.1186/s44263-023-00028-z  

   Althouse, Benjamin M. ; Wallace, Brendan ; Case, B. K. M. ; Scarpino, Samuel V. ; Allard, Antoine ; Berdahl, Andrew M. ; White, Easton R. ; Hébert-Dufresne, Laurent ( December 2023 , BMC Global and Public Health)

   Controlling the spread of infectious diseases―even when safe, transmission-blocking vaccines are available―may require the effective use of non-pharmaceutical interventions (NPIs), e.g., mask wearing, testing, limits on group sizes, venue closure. During the SARS-CoV-2 pandemic, many countries implemented NPIs inconsistently in space and time. This inconsistency was especially pronounced for policies in the United States of America (US) related to venue closure.

   Here, we investigate the impact of inconsistent policies associated with venue closure using mathematical modeling and high-resolution human mobility, Google search, and county-level SARS-CoV-2 incidence data from the USA. Specifically, we look at high-resolution location data and perform a US-county-level analysis of nearly 8 million SARS-CoV-2 cases and 150 million location visits, including 120 million church visitors across 184,677 churches, 14 million grocery visitors across 7662 grocery stores, and 13.5 million gym visitors across 5483 gyms.

   Analyzing the interaction between venue closure and changing mobility using a mathematical model shows that, across a broad range of model parameters, inconsistent or partial closure can be worse in terms of disease transmission as compared to scenarios with no closures at all. Importantly, changes in mobility patterns due to epidemic control measures can lead to increase in the future number of cases. In the most severe cases, individuals traveling to neighboring jurisdictions with different closure policies can result in an outbreak that would otherwise have been contained. To motivate our mathematical models, we turn to mobility data and find that while stay-at-home orders and closures decreased contacts in most areas of the USA, some specific activities and venues saw an increase in attendance and an increase in the distance visitors traveled to attend. We support this finding using search query data, which clearly shows a shift in information seeking behavior concurrent with the changing mobility patterns.

   While coarse-grained observations are not sufficient to validate our models, taken together, they highlight the potential unintended consequences of inconsistent epidemic control policies related to venue closure and stress the importance of balancing the societal needs of a population with the risk of an outbreak growing into a large epidemic.

    

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3. Effects of population co-location reduction on cross-county transmission risk of COVID-19 in the United States

   https://doi.org/10.1007/s41109-021-00361-y  

   Fan, Chao ; Lee, Sanghyeon ; Yang, Yang ; Oztekin, Bora ; Li, Qingchun ; Mostafavi, Ali ( December 2021 , Applied Network Science)

   null (Ed.)

   Abstract The objective of this study is to examine the transmission risk of COVID-19 based on cross-county population co-location data from Facebook. The rapid spread of COVID-19 in the United States has imposed a major threat to public health, the real economy, and human well-being. With the absence of effective vaccines, the preventive actions of social distancing, travel reduction and stay-at-home orders are recognized as essential non-pharmacologic approaches to control the infection and spatial spread of COVID-19. Prior studies demonstrated that human movement and mobility drove the spatiotemporal distribution of COVID-19 in China. Little is known, however, about the patterns and effects of co-location reduction on cross-county transmission risk of COVID-19. This study utilizes Facebook co-location data for all counties in the United States from March to early May 2020 for conducting spatial network analysis where nodes represent counties and edge weights are associated with the co-location probability of populations of the counties. The analysis examines the synchronicity and time lag between travel reduction and pandemic growth trajectory to evaluate the efficacy of social distancing in ceasing the population co-location probabilities, and subsequently the growth in weekly new cases across counties. The results show that the mitigation effects of co-location reduction appear in the growth of weekly new confirmed cases with one week of delay. The analysis categorizes counties based on the number of confirmed COVID-19 cases and examines co-location patterns within and across groups. Significant segregation is found among different county groups. The results suggest that within-group co-location probabilities (e.g., co-location probabilities among counties with high numbers of cases) remain stable, and social distancing policies primarily resulted in reduced cross-group co-location probabilities (due to travel reduction from counties with large number of cases to counties with low numbers of cases). These findings could have important practical implications for local governments to inform their intervention measures for monitoring and reducing the spread of COVID-19, as well as for adoption in future pandemics. Public policy, economic forecasting, and epidemic modeling need to account for population co-location patterns in evaluating transmission risk of COVID-19 across counties. 

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4. Intracounty modeling of COVID-19 infection with human mobility: Assessing spatial heterogeneity with business traffic, age, and race

   https://doi.org/10.1073/pnas.2020524118  

   Hou, Xiao ; Gao, Song ; Li, Qin ; Kang, Yuhao ; Chen, Nan ; Chen, Kaiping ; Rao, Jinmeng ; Ellenberg, Jordan S. ; Patz, Jonathan A. ( May 2021 , Proceedings of the National Academy of Sciences)

   The COVID-19 pandemic is a global threat presenting health, economic, and social challenges that continue to escalate. Metapopulation epidemic modeling studies in the susceptible–exposed–infectious–removed (SEIR) style have played important roles in informing public health policy making to mitigate the spread of COVID-19. These models typically rely on a key assumption on the homogeneity of the population. This assumption certainly cannot be expected to hold true in real situations; various geographic, socioeconomic, and cultural environments affect the behaviors that drive the spread of COVID-19 in different communities. What’s more, variation of intracounty environments creates spatial heterogeneity of transmission in different regions. To address this issue, we develop a human mobility flow-augmented stochastic SEIR-style epidemic modeling framework with the ability to distinguish different regions and their corresponding behaviors. This modeling framework is then combined with data assimilation and machine learning techniques to reconstruct the historical growth trajectories of COVID-19 confirmed cases in two counties in Wisconsin. The associations between the spread of COVID-19 and business foot traffic, race and ethnicity, and age structure are then investigated. The results reveal that, in a college town (Dane County), the most important heterogeneity is age structure, while, in a large city area (Milwaukee County), racial and ethnic heterogeneity becomes more apparent. Scenario studies further indicate a strong response of the spread rate to various reopening policies, which suggests that policy makers may need to take these heterogeneities into account very carefully when designing policies for mitigating the ongoing spread of COVID-19 and reopening.

    

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5. Exploring the relationship between mobility and COVID− 19 infection rates for the second peak in the United States using phase-wise association

   https://doi.org/10.1186/s12889-021-11657-0  

   Gottumukkala, Raju ; Katragadda, Satya ; Bhupatiraju, Ravi Teja ; Kamal, Azmyin Md. ; Raghavan, Vijay ; Chu, Henry ; Kolluru, Ramesh ; Ashkar, Ziad ( December 2021 , BMC Public Health)

   Abstract Human mobility plays an important role in the dynamics of infectious disease spread. Evidence from the initial nationwide lockdowns for COVID− 19 indicates that restricting human mobility is an effective strategy to contain the spread. While a direct correlation was observed early on, it is not known how mobility impacted COVID− 19 infection growth rates once lockdowns are lifted, primarily due to modulation by other factors such as face masks, social distancing, and the non-linear patterns of both mobility and infection growth. This paper introduces a piece-wise approach to better explore the phase-wise association between state-level COVID− 19 incidence data and anonymized mobile phone data for various states in the United States. Prior literature analyzed the linear correlation between mobility and the number of cases during the early stages of the pandemic. However, it is important to capture the non-linear dynamics of case growth and mobility to be usable for both tracking and forecasting COVID− 19 infections, which is accomplished by the piece-wise approach. The associations between mobility and case growth rate varied widely for various phases of the epidemic curve when the stay-at-home orders were lifted. The mobility growth patterns had a strong positive association of 0.7 with the growth in the number of cases, with a lag of 5 to 7 weeks, for the fast-growth phase of the pandemic, for only 20 states that had a peak between July 1st and September 30, 2020. Overall though, mobility cannot be used to predict the rise in the number of cases after initial lockdowns have been lifted. Our analysis explores the gradual diminishing value of mobility associations in the later stage of the outbreak. Our analysis indicates that the relationship between mobility and the increase in the number of cases, once lockdowns have been lifted, is tenuous at best and there is no strong relationship between these signals. But we identify the remnants of the last associations in specific phases of the growth curve. 

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