On Parameter Identifiability in Network-Based Epidemic Models.

Modelling epidemics on networks represents an important departure from classical compartmental models which assume random mixing. However, the resulting models are high-dimensional and their analysis is often out of reach. It turns out that mean-field models, low-dimensional systems of differential equations, whose variables are carefully chosen expected quantities from the exact model provide a good approximation and incorporate explicitly some network properties. Despite the emergence of such mean-field models, there has been limited work on investigating whether these can be used for inference purposes. In this paper, we consider network-based mean-field models and explore the problem of parameter identifiability when observations about an epidemic are available. Making use of the analytical tractability of most network-based mean-field models, e.g. explicit analytical expressions for leading eigenvalue and final epidemic size, we set up the parameter identifiability problem as finding the solution or solutions of a system of coupled equations. More precisely, subject to observing/measuring growth rate and final epidemic size, we seek to identify parameter values leading to these measurements. We are particularly concerned with disentangling transmission rate from the network density. To do this, we give a condition for practical identifiability and we find that except for the simplest model, parameters cannot be uniquely determined, that is, they are practically unidentifiable. This means that there exist multiple solutions (a manifold of infinite measure) which give rise to model output that is close to the data. Identifying, formalising and analytically describing this problem should lead to a better appreciation of the complexity involved in fitting models with many parameters to data.

The impact of spatial and social structure on an SIR epidemic on a weighted multilayer network.

A key factor in the transmission of infectious diseases is the structure of disease transmitting contacts. In the context of the current COVID-19 pandemic and with some data based on the Hungarian population we develop a theoretical epidemic model (susceptible-infected-removed, SIR) on a multilayer network. The layers include the Hungarian household structure, with population divided into children, adults and elderly, as well as schools and workplaces, some spatial embedding and community transmission due to sharing communal spaces, service and public spaces. We investigate the sensitivity of the model (via the time evolution and final size of the epidemic) to the different contact layers and we map out the relation between peak prevalence and final epidemic size. When compared to the classic compartmental model and for the same final epidemic size, we find that epidemics on multilayer network lead to higher peak prevalence meaning that the risk of overwhelming the health care system is higher. Based on our model we found that keeping cliques/bubbles in school as isolated as possible has a major effect while closing workplaces had a mild effect as long as workplaces are of relatively small size.

Epidemic threshold in pairwise models for clustered networks: closures and fast correlations.

The epidemic threshold is probably the most studied quantity in the modelling of epidemics on networks. For a large class of networks and dynamics, it is well studied and understood. However, it is less so for clustered networks where theoretical results are mostly limited to idealised networks. In this paper we focus on a class of models known as pairwise models where, to our knowledge, no analytical result for the epidemic threshold exists. We show that by exploiting the presence of fast variables and using some standard techniques from perturbation theory we are able to obtain the epidemic threshold analytically. We validate this new threshold by comparing it to the threshold based on the numerical solution of the full system. The agreement is found to be excellent over a wide range of values of the clustering coefficient, transmission rate and average degree of the network. Interestingly, we find that the analytical form of the threshold depends on the choice of closure, highlighting the importance of model selection when dealing with real-world epidemics. Nevertheless, we expect that our method will extend to other systems in which fast variables are present.

Solvability of implicit final size equations for SIR epidemic models.

Final epidemic size relations play a central role in mathematical epidemiology. These can be written in the form of an implicit equation which is not analytically solvable in most of the cases. While final size relations were derived for several complex models, including multiple infective stages and models in which the durations of stages are arbitrarily distributed, the solvability of those implicit equations have been less studied. In this paper the SIR homogeneous mean-field and pairwise models and the heterogeneous mean-field model are studied. It is proved that the implicit equation for the final epidemic size has a unique solution, and that through writing the implicit equation as a fixed point equation in a suitable form, the iteration of the fixed point equation converges to the unique solution. The Markovian SIR epidemic model on finite networks is also studied by using the generation-based approach. Explicit analytic formulas are derived for the final size distribution for line and star graphs of arbitrary size. Iterative formulas for the final size distribution enable us to study the accuracy of mean-field approximations for the complete graph.

SIS Epidemic Propagation on Hypergraphs.

Mathematical modelling of epidemic propagation on networks is extended to hypergraphs in order to account for both the community structure and the nonlinear dependence of the infection pressure on the number of infected neighbours. The exact master equations of the propagation process are derived for an arbitrary hypergraph given by its incidence matrix. Based on these, moment closure approximation and mean-field models are introduced and compared to individual-based stochastic simulations. The simulation algorithm, developed for networks, is extended to hypergraphs. The effects of hypergraph structure and the model parameters are investigated via individual-based simulation results.

Oscillating epidemics in a dynamic network model: stochastic and mean-field analysis.

An adaptive network model using SIS epidemic propagation with link-type-dependent link activation and deletion is considered. Bifurcation analysis of the pairwise ODE approximation and the network-based stochastic simulation is carried out, showing that three typical behaviours may occur; namely, oscillations can be observed besides disease-free or endemic steady states. The oscillatory behaviour in the stochastic simulations is studied using Fourier analysis, as well as through analysing the exact master equations of the stochastic model. By going beyond simply comparing simulation results to mean-field models, our approach yields deeper insights into the observed phenomena and help better understand and map out the limitations of mean-field models.

Exact deterministic representation of Markovian SIR epidemics on networks with and without loops.

In a previous paper Sharkey et al. (Bull Math Biol doi: 10.1007/s11538-013-9923-5 , 2012) proved the exactness of closures at the level of triples for Markovian [Formula: see text] (susceptible-infected-removed) dynamics on tree-like networks. This resulted in a deterministic representation of the epidemic dynamics on the network that can be numerically evaluated. In this paper, we extend this modelling framework to certain classes of networks exhibiting loops. We show that closures where the loops are kept intact are exact, and lead to a simplified and numerically solvable system of ODEs (ordinary-differential-equations). The findings of the paper lead us to a generalisation of closures that are based on partitioning the network around nodes that are cut-vertices (i.e. the removal of such a node leads to the network breaking down into at least two disjointed components or subnetworks). Exploiting this structural property of the network yields some natural closures, where the evolution of a particular state can typically be exactly given in terms of the corresponding or projected states on the subnetworks and the cut-vertex. A byproduct of this analysis is an alternative probabilistic proof of the exactness of the closures for tree-like networks presented in Sharkey et al. (Bull Math Biol doi: 10.1007/s11538-013-9923-5 , 2012). In this paper we also elaborate on how the main result can be applied to more realistic networks, for which we write down the ODEs explicitly and compare output from these to results from simulation. Furthermore, we give a general, recipe-like method of how to apply the reduction by closures technique for arbitrary networks, and give an upper bound on the maximum number of equations needed for an exact representation.

Identification of Criticality in Neuronal Avalanches: I. A Theoretical Investigation of the Non-driven Case.

In this paper, we study a simple model of a purely excitatory neural network that, by construction, operates at a critical point. This model allows us to consider various markers of criticality and illustrate how they should perform in a finite-size system. By calculating the exact distribution of avalanche sizes, we are able to show that, over a limited range of avalanche sizes which we precisely identify, the distribution has scale free properties but is not a power law. This suggests that it would be inappropriate to dismiss a system as not being critical purely based on an inability to rigorously fit a power law distribution as has been recently advocated. In assessing whether a system, especially a finite-size one, is critical it is thus important to consider other possible markers. We illustrate one of these by showing the divergence of susceptibility as the critical point of the system is approached. Finally, we provide evidence that power laws may underlie other observables of the system that may be more amenable to robust experimental assessment.

New moment closures based on a priori distributions with applications to epidemic dynamics.

Recently, research that focuses on the rigorous understanding of the relation between simulation and/or exact models on graphs and approximate counterparts has gained lots of momentum. This includes revisiting the performance of classic pairwise models with closures at the level of pairs and/or triples as well as effective-degree-type models and those based on the probability generating function formalism. In this paper, for a fully connected graph and the simple SIS (susceptible-infected-susceptible) epidemic model, a novel closure is introduced. This is done via using the equations for the moments of the distribution describing the number of infecteds at all times combined with the empirical observations that this is well described/approximated by a binomial distribution with time dependent parameters. This assumption allows us to express higher order moments in terms of lower order ones and this leads to a new closure. The significant feature of the new closure is that the difference of the exact system, given by the Kolmogorov equations, from the solution of the newly defined approximate system is of order 1/N(2). This is in contrast with the O(1/N) difference corresponding to the approximate system obtained via the classic triple closure. The fully connected nature of the graph also allows us to interpret pairwise equations in terms of the moments and thus treat closures and the two approximate models within the same framework. Finally, the applicability and limitations of the new methodology is discussed in detail.

From Markovian to pairwise epidemic models and the performance of moment closure approximations.

Many if not all models of disease transmission on networks can be linked to the exact state-based Markovian formulation. However the large number of equations for any system of realistic size limits their applicability to small populations. As a result, most modelling work relies on simulation and pairwise models. In this paper, for a simple SIS dynamics on an arbitrary network, we formalise the link between a well known pairwise model and the exact Markovian formulation. This involves the rigorous derivation of the exact ODE model at the level of pairs in terms of the expected number of pairs and triples. The exact system is then closed using two different closures, one well established and one that has been recently proposed. A new interpretation of both closures is presented, which explains several of their previously observed properties. The closed dynamical systems are solved numerically and the results are compared to output from individual-based stochastic simulations. This is done for a range of networks with the same average degree and clustering coefficient but generated using different algorithms. It is shown that the ability of the pairwise system to accurately model an epidemic is fundamentally dependent on the underlying large-scale network structure. We show that the existing pairwise models are a good fit for certain types of network but have to be used with caution as higher-order network structures may compromise their effectiveness.

Multiple sources and routes of information transmission: Implications for epidemic dynamics.

In a recent paper, we proposed and analyzed a compartmental ODE-based model describing the dynamics of an infectious disease where the presence of the pathogen also triggers the diffusion of information about the disease. In this paper, we extend this previous work by presenting results based on pairwise and simulation models that are better suited for capturing the population contact structure at a local level. We use the pairwise model to examine the potential of different information generating mechanisms and routes of information transmission to stop disease spread or to minimize the impact of an epidemic. The individual-based simulation is used to better differentiate between the networks of disease and information transmission and to investigate the impact of different basic network topologies and network overlap on epidemic dynamics. The paper concludes with an individual-based semi-analytic calculation of R(0) at the non-trivial disease free equilibrium.

Exact epidemic models on graphs using graph-automorphism driven lumping.

The dynamics of disease transmission strongly depends on the properties of the population contact network. Pair-approximation models and individual-based network simulation have been used extensively to model contact networks with non-trivial properties. In this paper, using a continuous time Markov chain, we start from the exact formulation of a simple epidemic model on an arbitrary contact network and rigorously derive and prove some known results that were previously mainly justified based on some biological hypotheses. The main result of the paper is the illustration of the link between graph automorphisms and the process of lumping whereby the number of equations in a system of linear differential equations can be significantly reduced. The main advantage of lumping is that the simplified lumped system is not an approximation of the original system but rather an exact version of this. For a special class of graphs, we show how the lumped system can be obtained by using graph automorphisms. Finally, we discuss the advantages and possible applications of exact epidemic models and lumping.

The impact of information transmission on epidemic outbreaks.

For many diseases (e.g., sexually transmitted infections, STIs), most individuals are aware of the potential risks of becoming infected, but choose not to take action ('respond') despite the information that aims to raise awareness and to increases the responsiveness or alertness of the population. We propose a simple mathematical model that accounts for the diffusion of health information disseminated as a result of the presence of a disease and an 'active' host population that can respond to it by taking measures to avoid infection or if infected by seeking treatment early. In this model, we assume that the whole population is potentially aware of the risk but only a certain proportion chooses to respond appropriately by trying to limit their probability of becoming infectious or seeking treatment early. The model also incorporates a level of responsiveness that decays over time. We show that if the dissemination of information is fast enough, infection can be eradicated. When this is not possible, information transmission has an important effect in reducing the prevalence of the infection. We derive the full characterisation of the global behaviour of the model, and we show that the parameter space can be divided into three parts according to the global attractor of the system which is one of the two disease-free steady states or the endemic equilibrium.

A contact-network-based formulation of a preferential mixing model.

Heterogeneity in the number of potentially infectious contacts and connectivity correlations ("like attaches to like", i.e., assortatively mixed or "opposites attract", i.e., disassortatively mixed) have important implications for the value of the basic reproduction ratio R(0) and final epidemic size. In this paper, we present a contact-network-based derivation of a simple differential equation model that accounts for preferential mixing based on the number of contacts. We show that results based on this model are in good qualitative agreement with results obtained from preferential mixing models used in the context of sexually transmitted diseases (STDs). This simple model can accommodate any mixing pattern ranging from completely disassortative to completely assortative and allows the derivation of a series of analytical results.

Electrolyte diodes with weak acids and bases. II. Numerical model calculations and experiments.

This is the second part of our work dealing with electrolyte diodes with weak acids and bases. In the first part an approximative analytical solution was derived for the steady-state current-voltage characteristic (CVC) of a reverse-biased diode (a quasi-one-dimensional gel connecting an acidic and an alkaline reservoir), applying either strong or weak electrolytes. An approximative analytical solution is compared here with a numerical solution free of any approximations and with CVCs measured experimentally with both strong and weak electrolytes. It is shown that the deviations between the numerical and analytical solutions are mostly due to assumptions made for the fixed charge concentration profiles. The concept of optimal analytical solution is introduced which does not use such assumptions and applies only the quasielectroneutrality and quasiequilibrium approximations. It is proven that the slope of the CVC based on the optimum analytical solution can be calculated without the complicated derivation of that solution itself. The calculation of that slope is based on the fact that in the optimum analytical solution all currents are inversely proportional to the length if the boundary conditions are held constant and realizing that in the middle part of the gel the only mobile counterions of the fixed ionized groups are hydrogen ions. In the experimental part the apparatus and the preparation of the gel are described together with the CVCs measured with strong and weak electrolytes. From these CVCs the fixed ion concentration in the middle part of the gel can be determined. That fixed ion concentration is 1.96 x 10(-4)M measured with weak electrolytes and 3.48 x 10(-4)M measured with strong electrolytes. The deviation indicates that the strong base causes some hydrolysis of the gel. Finally, possible applications of weak acid-weak base diodes are discussed.

Electrolyte diodes with weak acids and bases. I. Theory and an approximate analytical solution.

Until now acid-base diodes and transistors applied strong mineral acids and bases exclusively. In this work properties of electrolyte diodes with weak electrolytes are studied and compared with those of diodes with strong ones to show the advantages of weak acids and bases in these applications. The theoretical model is a one dimensional piece of gel containing fixed ionizable groups and connecting reservoirs of an acid and a base. The electric current flowing through the gel is measured as a function of the applied voltage. The steady-state current-voltage characteristic (CVC) of such a gel looks like that of a diode under these conditions. Results of our theoretical, numerical, and experimental investigations are reported in two parts. In this first, theoretical part governing equations necessary to calculate the steady-state CVC of a reverse-biased electrolyte diode are presented together with an approximate analytical solution of this reaction-diffusion-ionic migration problem. The applied approximations are quasielectroneutrality and quasiequilibrium. It is shown that the gel can be divided into an alkaline and an acidic zone separated by a middle weakly acidic region. As a further approximation it is assumed that the ionization of the fixed acidic groups is complete in the alkaline zone and that it is completely suppressed in the acidic one. The general solution given here describes the CVC and the potential and ionic concentration profiles of diodes applying either strong or weak electrolytes. It is proven that previous formulas valid for a strong acid-strong base diode can be regarded as a special case of the more general formulas presented here.

Electrolyte diodes and hydrogels: determination of concentration and pK value of fixed acidic groups in a weakly charged hydrogel.

Current-voltage (CV) characteristics of polyvinyl alcohol (PVA)-glutardialdehyde hydrogel cylinders were measured in aqueous KCl solutions. To this end a new special apparatus was constructed where the gel cylinder connects two electrolyte reservoirs. The measured quantities are the electric current flowing through the gel and the potential difference between the two reservoirs. Concentration polarization near the gel-liquid interfaces is decreased considerably by applying an intense mechanical stirring in both reservoirs. Under these conditions below 1 V concentration polarization is negligible, and the CV curves are nearly straight lines. It was found that the gel applied here is a weakly charged anionic hydrogel. Concentration of fixed anions was determined from the slope of these lines measured in 0.001 and 0.01 molar KCl solutions. Fixed anion concentration of the same piece of gel was measured also with a different method, when the gel was used in an acid-base diode. In this case one reservoir contained 0.1 molar HCl, and the other 0.1 molar KOH. From the results of the two measurements, the concentration (4.45 x 10(-3) M) and the pK value (4.03) of the fixed acid groups responsible for the anionic character of the gel was calculated. The pK value is compatible with fixed carboxylic acid groups contaminating the PVA gel. Furthermore, concentration polarization phenomena in the boundary layers nearby the gel were studied in 0.001 M KCl solutions, measuring the diodelike CV characteristic of a gel cylinder, when stirring was applied only at one side of the gel. Boundary layers facing the cathode or the anode responded in a different way to stirring. The difference cannot be explained completely with the hypothesis of electroconvection suggested previously.

Isothermal flame balls: effect of autocatalyst decay.

The steady, spherically symmetric solutions to the reaction-diffusion equations based on a simple autocatalytic reaction followed by the decay of the autocatalyst are considered. Three parameters-the orders with respect to the autocatalyst in the autocatalysis p and in the decay q and the rate of decay of the autocatalyst relative to its autocatalytic production K-determine the steady concentration profiles. Numerical integrations for a fixed value of the order of the autocatalyst show that the concentration profiles have different forms depending on whether q

/=p. In the former case, there is a critical decay rate K(crit) for solutions to exist, with multiple solutions for K

Isothermal flame balls.

The existence of steady, spherically symmetric wave fronts ("isothermal flame balls") in chemical reaction systems exhibiting autocatalysis is demonstrated. Such solutions require relatively high kinetic orders p with respect to the autocatalytic species, with p>5, but occur even with equal diffusion coefficients. The flame balls are unstable, but have relevance as they indicate the minimum size for a perturbation to initiate a propagating front. A flame ball radius R(b) is identified and the dependence of this quantity on the autocatalytic order is determined. This shows R(b) tending to infinity as p-->5(+) and as p--> infinity, with a minimum for p approximately 6.71. Numerical computations are confirmed by asymptotic analysis appropriate for p-->5(+) and for systems with p large.