Epidemic Models with Immunization and Mutations on a Finite Population

We propose two variants of a stochastic epidemic model in which the disease is spread by mobile particles performing random walks on the complete graph. For the first model, we study the effect on the epidemic size of an immunization mechanism that depends on the activity of the disease. In the second model, the transmission agents can gain lives at random during their existences. We prove limit theorems for the final outcome of these processes. The epidemic model with mutations exhibits phase transition, meaning that if the mutation parameter is sufficiently large, then asymptotically all the individuals in the population are infected

Epidemic spreading with immunization and mutations

The spreading of infectious diseases with and without immunization of individuals can be modeled by stochastic processes that exhibit a transition between an active phase of epidemic spreading and an absorbing phase, where the disease dies out. In nature, however, the transmitted pathogen may also mutate, weakening the effect of immunization. In order to study the influence of mutations, we introduce a model that mimics epidemic spreading with immunization and mutations. The model exhibits a line of continuous phase transitions and includes the general epidemic process (GEP) and directed percolation (DP) as special cases. Restricting to perfect immunization in two spatial dimensions we analyze the phase diagram and study the scaling behavior along the phase transition line as well as in the vicinity of the GEP point. We show that mutations lead generically to a crossover from the GEP to DP. Using standard scaling arguments we also predict the form of the phase transition line close to the GEP point. It turns out that the protection gained by immunization is vitally decreased by the occurrence of mutations.Comment: 9 pages, 13 figure

Graph Theory and Networks in Biology

In this paper, we present a survey of the use of graph theoretical techniques in Biology. In particular, we discuss recent work on identifying and modelling the structure of bio-molecular networks, as well as the application of centrality measures to interaction networks and research on the hierarchical structure of such networks and network motifs. Work on the link between structural network properties and dynamics is also described, with emphasis on synchronization and disease propagation.Comment: 52 pages, 5 figures, Survey Pape

Localization, epidemic transitions, and unpredictability of multistrain epidemics with an underlying genotype network

Mathematical disease modelling has long operated under the assumption that any one infectious disease is caused by one transmissible pathogen spreading among a population. This paradigm has been useful in simplifying the biological reality of epidemics and has allowed the modelling community to focus on the complexity of other factors such as population structure and interventions. However, there is an increasing amount of evidence that the strain diversity of pathogens, and their interplay with the host immune system, can play a large role in shaping the dynamics of epidemics. Here, we introduce a disease model with an underlying genotype network to account for two important mechanisms. One, the disease can mutate along network pathways as it spreads in a host population. Two, the genotype network allows us to define a genetic distance across strains and therefore to model the transcendence of immunity often observed in real world pathogens. We study the emergence of epidemics in this model, through its epidemic phase transitions, and highlight the role of the genotype network in driving cyclicity of diseases, large scale fluctuations, sequential epidemic transitions, as well as localization around specific strains of the associated pathogen. More generally, our model illustrates the richness of behaviours that are possible even in well-mixed host populations once we consider strain diversity and go beyond the "one disease equals one pathogen" paradigm

Deconvolving mutational patterns of poliovirus outbreaks reveals its intrinsic fitness landscape.

Vaccination has essentially eradicated poliovirus. Yet, its mutation rate is higher than that of viruses like HIV, for which no effective vaccine exists. To investigate this, we infer a fitness model for the poliovirus viral protein 1 (vp1), which successfully predicts in vitro fitness measurements. This is achieved by first developing a probabilistic model for the prevalence of vp1 sequences that enables us to isolate and remove data that are subject to strong vaccine-derived biases. The intrinsic fitness constraints derived for vp1, a capsid protein subject to antibody responses, are compared with those of analogous HIV proteins. We find that vp1 evolution is subject to tighter constraints, limiting its ability to evade vaccine-induced immune responses. Our analysis also indicates that circulating poliovirus strains in unimmunized populations serve as a reservoir that can seed outbreaks in spatio-temporally localized sub-optimally immunized populations

A minimal stochastic model for influenza evolution

We introduce and discuss a minimal individual-based model for influenza dynamics. The model takes into account the effects of specific immunization against viral strains, but also infectivity randomness and the presence of a short-lived strain transcending immunity recently suggested in the literature. We show by simulations that the resulting model exhibits substitution of viral strains along the years, but that their divergence remains bounded. We also show that dropping any of these features results in a drastically different behavior, leading either to the extinction of the disease, to the proliferation of the viral strains, or to their divergence

Laws of large numbers for the frog model on the complete graph

The frog model is a stochastic model for the spreading of an epidemic on a graph, in which a dormant particle starts to perform a simple random walk on the graph and to awake other particles, once it becomes active. We study two versions of the frog model on the complete graph with $N + 1$ vertices. In the first version we consider, active particles have geometrically distributed lifetimes. In the second version, the displacement of each awakened particle lasts until it hits a vertex already visited by the process. For each model, we prove that as $N \to \infty$, the trajectory of the process is well approximated by a three-dimensional discrete-time dynamical system. We also study the long-term behavior of the corresponding deterministic systems

The Timing and Targeting of Treatment in Influenza Pandemics Influences the Emergence of Resistance in Structured Populations

abstract: Antiviral resistance in influenza is rampant and has the possibility of causing major morbidity and mortality. Previous models have identified treatment regimes to minimize total infections and keep resistance low. However, the bulk of these studies have ignored stochasticity and heterogeneous contact structures. Here we develop a network model of influenza transmission with treatment and resistance, and present both standard mean-field approximations as well as simulated dynamics. We find differences in the final epidemic sizes for identical transmission parameters (bistability) leading to different optimal treatment timing depending on the number initially infected. We also find, contrary to previous results, that treatment targeted by number of contacts per individual (node degree) gives rise to more resistance at lower levels of treatment than non-targeted treatment. Finally we highlight important differences between the two methods of analysis (mean-field versus stochastic simulations), and show where traditional mean-field approximations fail. Our results have important implications not only for the timing and distribution of influenza chemotherapy, but also for mathematical epidemiological modeling in general. Antiviral resistance in influenza may carry large consequences for pandemic mitigation efforts, and models ignoring contact heterogeneity and stochasticity may provide misleading policy recommendations.The article is published at http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.100291

A quantitative narrative on movement, disease and patch exploitation in nesting agent groups

Abstract Animal relocation data has recently become considerably more ubiquitous, finely structured (collection frequencies measured in minutes) and co-variate rich (physiology of individuals, environmental and landscape information, and accelerometer data). To better understand the impacts of ecological interactions, individual movement and disease on global change ecology, including wildlife management and conservation, it is important to have simulators that will provide demographic, movement, and epidemiology null models against which to compare patterns observed in empirical systems. Such models may then be used to develop quantitative narratives that enhance our intuition and understanding of the relationship between population structure and generative processes: in essence, along with empirical and experimental narratives, quantitative narratives are used to advance ecological epistemology. Here we describe a simulator that accounts for the influence of consumer-resource interactions, existence of social groups anchored around a central location, territoriality, group-switching behavior, and disease dynamics on population size. We use this simulator to develop new and reinforce existing quantitative narratives and point out areas for future study. Author summary The health and viability of species are of considerable concern to all nature lovers. Population models are central to our efforts to assess the numerical and ecological status of species and threats posed by climate change. Models, however, are crude caricatures of complex ecological systems. So how do we construct reliable assessment models able to capture processes essential to predicating the impacts of global change on population viability without getting tied up in their vast complexities? We broach this question and demonstrate how models focusing at the level of the individual (i.e., agent-based models) are tools for developing robust, narratives to augment narratives arising purely from empirical data sources and experimental outcomes. We do this in the context of nesting social groups, foraging for food, while exhibiting territoriality and group-switching behavior; and, we evaluate the impact of disease on the viability of such populations