Lauro FD, Kiss IZ, Miller JC. The timing of one-shot interventions for epidemic control. Submitted. the_timing_of_one_off_interventions_for_epidemic_control.pdf
Lang J, Sterck HD, Kaiser JL, Miller JC. Random Spatial Networks: Small Worlds without Clustering, Traveling Waves, and Hop-and-Spread Disease Dynamics. Arxiv. Submitted. RandomSpatialNetworks.pdf
Miller JC, Slim AC. Analytic models of disease spread with birth, death, and concurrency. Submitted.Abstract

The existence of sexual partnerships that overlap in time (concurrent relationships) is believed by some to be a significant contributing factor to the spread of HIV, although this is controversial.  We derive an analytic model which allows us to investigate and compare disease spread in populations with and without concurrency.  We can identify regions of parameter space in which its impact is negligible, and other regions in which it plays a major role.  We also see that the impact of concurrency on the initial growth phase can be much larger than its impact on the equilibrium size.  We see that the effect of concurrency saturates, which leads to the perhaps surprising conclusion that interventions targeting concurrency may be most effective in populations with low to moderate levels of concurrency.

Miller JC. Epidemics on Networks software package. Github. 2017. Publisher's Version
Fox S, Miller JC, Meyers LA. Seasonality in risk of pandemic influenza emergence. PLoS Computational Biology. 2017.
Kiss IZ, Berthouze L, Miller JC, Simon PL. Mapping out emerging network structures in dynamic network models coupled with epidemics. In: Temporal Network Epidemiology. Springer ; 2017. Publisher's VersionAbstract


We consider the susceptible - infected - susceptible (SIS) epidemic on a dynamic network model with addition and deletion of links depending on node status. We analyse the resulting pairwise model using classical bifurcation theory to map out the spectrum of all possible epidemic behaviours. However, the major focus of the chapter is on the evolution and possible equilibria of the resulting networks. Whereas most studies are driven by determining system-level outcomes, e.g., whether the epidemic dies out or becomes endemic, with little regard for the emerging network structure, here, we want to buck this trend by augmenting the system-level results with mapping out of the structure and properties of the resulting networks. We find that depending on parameter values the network can become disconnected and show bistable-like behaviour whereas the endemic steady state sees the emergence of networks with qualitatively different degree distributions. In particular, we observe de-phased oscillations of both prevalence and network degree during which there is role reversal between the level and nature of the connectivity of susceptible and infected nodes. We conclude with an attempt at describing what a potential bifurcation theory for networks would look like.


Miller JC. Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes. Infectious Disease Modelling. 2017. Publisher's VersionAbstract

The emergence of diseases such as Zika and Ebola has highlighted the need to understand the role of sexual transmission in the spread of diseases with a primarily non-sexual transmission route.  In this paper we develop a number of low-dimensional models which are appropriate for a range of assumptions for how a disease will spread if it has sexual transmission through a sexual contact network combined with some other transmission mechanism, such as direct contact or vectors.  The equations derived provide exact predictions for the dynamics of the corresponding simulations in the large population limit.

Sherborne N, Miller JC, Blyuss KB, Kiss IZ. Mean-field models for non-Markovian epidemics on networks: from edge-based compartmental to pairwise models. Journal of Mathematical Biology. 2017. Publisher's VersionAbstract


This paper presents a novel extension of the edge-based compartmental model for epidemics with arbitrary distributions of transmission and recovery times. Using the message passing approach we also derive a new pairwise-like model for epidemics with Markovian transmission and an arbitrary recovery period. The new pairwise-like model allows one to formally prove that the message passing and edge-based compartmental models are equivalent in the case of Markovian transmission and arbitrary recovery processes. The edge-based and message passing models are conjectured to also be equivalent for arbitrary transmission processes; we show the first step of a full proof of this. The new pairwise-like model encompasses many existing well-known models that can be obtained by appropriate reductions. It is also amenable to a relatively straightforward numerical implementation. We test the theoretical results by comparing the numerical solutions of the various pairwise-like models to results based on explicit stochastic network simulations. 


Kiss IZ, Miller JC, Simon P. (Book) Mathematics of epidemics on networks: from exact to approximate models. Springer; 2017. Publisher's VersionAbstract

This book discusses many models of disease spread in networks.

Draft Table of Contents, Index, and References are in attached file.

Miller JC. EoN: Epidemics on Networks (Python module for epidemic simulations in networks.). GitHub. 2016.
Miller JC. Equivalence of several generalized percolation models on networks. Physical Review E. 2016;94 (3) :032313. Publisher's VersionAbstract

In recent years, many variants of percolation have been used to study network structure and the behavior of processes spreading on networks. These include bond percolation, site percolation, k-core percolation, bootstrap percolation, the generalized epidemic process, and the Watts threshold model (WTM). We show that—except for bond percolation—each of these processes arises as a special case of the WTM, and bond percolation arises from a small modification. In fact “heterogeneous k-core percolation,” a corresponding “heterogeneous bootstrap percolation” model, and the generalized epidemic process are completely equivalent to one another and the WTM. We further show that a natural generalization of the WTM in which individuals “transmit” or “send a message” to their neighbors with some probability less than 1 can be reformulated in terms of the WTM, and so this apparent generalization is in fact not more general. Finally, we show that in bond percolation, finding the set of nodes in the component containing a given node is equivalent to finding the set of nodes activated if that node is initially activated and the node thresholds are chosen from the appropriate distribution. A consequence of these results is that mathematical techniques developed for the WTM apply to these other models as well, and techniques that were developed for some particular case may in fact apply much more generally.

Zhang C, Zhou S, Miller JC, Cox IJ, Chain BM. Optimizing hybrid spreading in metapopulations. Scientific Reports. 2015;5 (9924). Publisher's Version zhang_et_al._-_2015_-_optimizing_hybrid_spreading_in_metapopulations.pdf
Miller JC. Complex Contagions and Hybrid Phase Transitions. Journal of Complex Networks. 2015. Publisher's VersionAbstract

A complex contagion is an infectious process in which individuals may require multiple transmissions before changing state.  These are used to model behaviors if an individual only adopts a particular behavior after perceiving a consensus among others.  We may think of individuals as beginning inactive and becoming active once they are contacted by a sufficient number of active partners.  These have been studied in a number of cases, but analytic models for the dynamic spread of complex contagions are typically complex.  Here we study the dynamics of the Watts Threshold Model (WTM) assuming that transmission occurs in continuous time as a Poisson process, or in discrete time where individuals transmit to all partners in the time step following their infection.  We adapt techniques developed for infectious disease modeling to develop an analyze analytic models for the dynamics of the WTM in Configuration Model networks and a class of random clustered (triangle-based) networks.  The resulting model is relatively simple and compact.  We use it to gain insights into the dynamics of the contagion.  Taking the infinite population limit, we derive conditions under which cascades happen with an arbitrarily small initial proportion active, confirming a hypothesis of Watts for this case.  We also observe hybrid phase transitions when cascades are not possible for small initial conditions, but occur for large enough initial conditions.  We derive sufficient conditions for this hybrid phase transition to occur.  We show that in many cases, if the hybrid phase transition occurs, then all individuals eventually become active.  Finally, we discuss the role clustering plays in facilitating or impeding the spread and find that the hypothesis of Watts that was confirmed in Configuration Model networks does not hold in general.  This approach allows us to unify many existing disparate observations and derive new results.

Miller JC, Kiss IZ. Epidemic spread in networks: Existing methods and current challenges. Mathematical Modeling of Natural Phenomena. 2014;9 :4-42. Publisher's Version equivalent_models.pdf
Miller JC. Epidemics on networks with large initial conditions or changing structure. PLoS One. 2014;9 :e101421. Publisher's Version initial_condition_fix.pdf
Rattana P, Miller JC, Kiss IZ. Pairwise and Edge-based Models of Epidemic Dynamics on Correlated Weighted Networks. Mathematical Modelling of Natural Phenomena. 2014;9 :58–81. Publisher's Version rattana_miller_kiss_weighted_epidemic.pdf
Abramson G, Kuperman MN, Morales JM, Miller JC. Space use by foragers consuming renewable resources. The European Physical Journal B. 2014;87 :1–9. Publisher's Version foragers.pdf
Slim AC, Bandi MM, Miller JC, Mahadevan L. Dissolution-driven convection in a Hele–Shaw cell. Physics of Fluids (1994-present). 2013;25 :024101. Publisher's Version convection_expt.pdf
Miller JC. Cocirculation of infectious diseases on networks. Physical Review E. 2013;87 :060801. Arxiv Version
Miller JC, Volz EM. Incorporating Disease and Population Structure into Models of SIR Disease in Contact Networks. PloS One. 2013;8 :e69162. Publisher's Version ebcm_structure.pdf