We consider the spread of infectious disease through contact networks of ConfigurationModel type. We assume that the disease spreads through contacts and infected individualsrecover into an immune state. We discuss a number of existing mathematical models used toinvestigate this system, and show relations between the underlying assumptions of themodels. In the process we offer simplifications of some of the existing models. Thedistinctions between the underlying assumptions are subtle, and in many if not most casesthis subtlety is irrelevant. Indeed, under appropriate conditions the models areequivalent. We compare the benefits and disadvantages of the different models, and discusstheir application to other populations (e.g., clustered networks).Finally we discuss ongoing challenges for network-based epidemic modeling.

We extrapolate from the exact master equations of epidemic dynamics on fully connectedgraphs to non-fully connected by keeping the size of the state space N + 1, whereN is thenumber of nodes in the graph. This gives rise to a system of approximate ODEs (ordinarydifferential equations) where the challenge is to compute/approximate analytically thetransmission rates. We show that this is possible for graphs with arbitrary degreedistributions built according to the configuration model. Numerical tests confirm that:(a) the agreement of the approximate ODEs system with simulation is excellent and (b) thatthe approach remains valid for clustered graphs with the analytical calculations of thetransmission rates still pending. The marked reduction in state space gives good results,and where the transmission rates can be analytically approximated, the model provides astrong alternative approximate model that agrees well with simulation. Given that thetransmission rates encompass information both about the dynamics and graph properties, thespecific shape of the curve, defined by the transmission rate versus the number ofinfected nodes, can provide a new and different measure of network structure, and themodel could serve as a link between inferring network structure from prevalence orincidence data.

In this paper we explore the potential of the pairwise-type modelling approach to beextended to weighted networks where nodal degree and weights are not independent. As abaseline or null model for weighted networks, we consider undirected, heterogenousnetworks where edge weights are randomly distributed. We show that the pairwise modelsuccessfully captures the extra complexity of the network, but does this at the cost oflimited analytical tractability due the high number of equations. To circumvent thisproblem, we employ the edge-based modelling approach to derive models corresponding to twodifferent cases, namely for degree-dependent and randomly distributed weights. Thesemodels are more amenable to compute important epidemic descriptors, such as early growthrate and final epidemic size, and produce similarly excellent agreement with simulation.Using a branching process approach we compute the basic reproductive ratio for both modelsand discuss the implication of random and correlated weight distributions on this as wellas on the time evolution and final outcome of epidemics. Finally, we illustrate that thetwo seemingly different modelling approaches, pairwise and edge-based, operate on similarassumptions and it is possible to formally link the two.

When modeling the spread of infectious diseases, it is important to incorporate riskbehavior of individuals in a considered population. Not only risk behavior, but also thenetwork structure created by the relationships among these individuals as well as thedynamical rules that convey the spread of the disease are the key elements in predictingand better understanding the spread. In this work we propose the weighted randomconnection model, where each individual of the population is characterized by twoparameters: its position and risk behavior. A goal is to model the effect that theprobability of transmissions among individuals increases in the individual risk factors,and decays in their Euclidean distance. Moreover, the model incorporates a combined riskbehavior function for every pair of the individuals, through which the spread can bedirectly modeled or controlled. We derive conditions for the existence of an outbreak ofinfectious diseases in this model. Our main result is the almost sure existence of aninfinite component in the weighted random connection model. We use results on the randomconnection model and site percolation in Z2.

In this paper, an SIS (susceptible-infected-susceptible)-type epidemic propagation isstudied on a special class of 3-regular graphs, called modified cycle graphs. The modifiedcycle graph is constructed from a cycle graph with N nodes by connecting nodei to thenode i +d in a way that every node has exactly three links.Monte-Carlo simulations show that the propagation process depends on the value ofd in anon-monotone way. A new theoretical model is developed to explain this phenomenon. Thisreveals a new relation between the spreading process and the average path length in thegraph.

We consider the spread of an infectious disease on a heterogeneous metapopulation definedby any (correlated or uncorrelated) network. The infection evolves under transmission,recovery and migration mechanisms. We study some spectral properties of a connectivitymatrix arising from the continuous-time equations of the model. In particular we show thatthe classical sufficient condition of instability for the disease-free equilibrium, wellknown for the particular case of uncorrelated networks, works also for the general case.We give also an alternative condition that yields a more accurate estimation of theepidemic threshold for correlated (either assortative or dissortative) networks.

The basic reproduction number, R0, is often defined as the averagenumber of infections generated by a newly infected individual in a fully susceptiblepopulation. The interpretation, meaning, and derivation of R0 arecontroversial. However, in the context of mean field models, R0 demarcatesthe epidemic threshold below which the infected population approaches zero in the limit oftime. In this manner, R0 has been proposed as a method forunderstanding the relative impact of public health interventions with respect to diseaseeliminations from a theoretical perspective. The use of R0 is made morecomplex by both the strong dependency of R0 on the model form and the stochasticnature of transmission. A common assumption in models of HIV transmission that have closedform expressions for R0 is that a single individual’sbehavior is constant over time. In this paper we derive expressions for bothR0 and probability of an epidemic in afinite population under the assumption that people periodically change their sexualbehavior over time. We illustrate the use of generating functions as a general frameworkto model the effects of potentially complex assumptions on the number of transmissionsgenerated by a newly infected person in a susceptible population. We find that therelationship between the probability of an epidemic and R0 is notstraightforward, but, that as the rate of change in sexual behavior increases bothR0 and the probability of an epidemic alsodecrease.

In epidemic modeling, the Susceptible-Alert-Infected-Susceptible (SAIS) model extends theSIS (Susceptible-Infected-Susceptible) model. In the SAIS model, “alert” individualsobserve the health status of neighbors in their contact network, and as a result, they mayadopt a set of cautious behaviors to reduce their infection rate. This alertness, whenincorporated in the mathematical model, increases the range of effective/relativeinfection rates for which initial infections die out. Built upon the SAIS model, this workinvestigates how information dissemination further increases this range. Informationdissemination is realized through an additional network (e.g., an online social network)sharing the contact network nodes (individuals) with different links. These “informationlinks” provide the health status of one individual to all the individuals she is connectedto in the information dissemination network. We propose an optimal informationdissemination strategy with an index in quadratic form relative to the informationdissemination network adjacency matrix and the dominant eigenvector of the contactnetwork. Numerical tools to exactly solve steady state infection probabilities andinfluential thresholds are developed, providing an evaluative baseline for our informationdissemination strategy. We show that monitoring the health status of a small but “central”subgroup of individuals and circulating their incidence information optimally enhances theresilience of the society against infectious diseases. Extensive numerical simulations ona survey–based contact network for a rural community in Kansas support these findings.

A feature often observed in epidemiological networks is significant heterogeneity indegree. A popular modelling approach to this has been to consider large populations withhighly heterogeneous discrete contact rates. This paper defines an individual-levelnon-Markovian stochastic process that converges on standard ODE models of such populationsin the appropriate asymptotic limit. A generalised Sellke construction is derived for thismodel, and this is then used to consider final outcomes in the case where heterogeneityfollows a truncated Zipf distribution.

We present and characterize a multi-host epidemic model of Rift Valley fever (RVF) virusin East Africa with geographic spread on a network, rule-based mitigation measures, andmosquito infection and population dynamics. Susceptible populations are depleted bydisease and vaccination and are replenished with the birth of new animals. We observe thatthe severity of the epidemics is strongly correlated with the duration of the rainy seasonand that even severe epidemics are abruptly terminated when the rain stops. Becausenaturally acquired herd immunity is established, total mortality across 25 years isrelatively insensitive to many mitigation approaches. Strong reductions in cattlemortality are expected, however, with sufficient reduction in population densities ofeither vectors or susceptible (ie. unvaccinated) hosts. A better understanding of RVFepidemiology would result from serology surveys to quantify the importance of herdimmunity in epidemic control, and sequencing of virus from representative animals toquantify the realative importance of transportation and local reservoirs in nucleatingyearly epidemics. Our results suggest that an effective multi-layered mitigation strategywould include vector control, movement control, and vaccination of young animals yearly,even in the absence of expected rainfall.