Epidemics with Pathogen Mutationon Small-World NetworksZhi-Gang Shao, Zhi-Jie Tan, Xian-Wu Zou, Zhun-Zhi JinPresenter: Daniel JacksonApril 1, 2010Tucson, AZTeam Members: Nadia FloresDaniel JacksonRobert PhillipsManuel RiveraZach RogersMentor: Toby ShearmanApril 1, 2010Introduction to the Model•Epidemiologyo Factors affecting health and illness of a populationo Viral outbreaks•What we are studyingo Socio-spatial networkso Time-dependent dynamicso Immunityo Pathogen mutationsIn 1918–1919, an estimated 40–50 million people died worldwide from the influenza pandemic.J.M. Wood and J.S. Robertson, Nat. Rev. Microbiol. 2 (2004), p. 842.April 1, 2010Why study this model?o Mathematical models can help predict the behavior of infectious agents on susceptible populationso Predictions of mathematical models can help guide efficient and effective treatment for the eradication of diseaseo Accounting for pathogen mutation and time-dependent immunity within the small-world paradigm leads to a more realistic modelExamples of infectious diseasesMeasles:• Highly infectious• Short incubation period• Short immunity duration• Control policy is long-termSmallpox:• Less infectious• Longer incubation period• Long immunity duration• “Herd Immunity” made eradication possibleApril 1, 2010Infection and Immunity• Immune response is triggered by genetically similar pathogens (antigenicity)• Measure of „how similar‟ is the cross-immunity threshold, hthr• Individuals infected when a new viral strain is sufficiently different (hmin > hthr) from any recently encountered• Individuals, represented by nodes in the network, can infect others, represented by neighboring nodes, during the infectious stage (infection duration)• Individuals, represented by nodes in the network, can become infected again after a sufficient amount of time has passed (immunity duration)April 1, 2010SIR: Differential Equation Model SIRS: Agent Based ModelDeterministic vs. ProbabilisticConsider node nIf n has a virus in the infectious stageadd to infected-population countFor each virus v in the viral historyIf v is in infectious stagesend challenge strain to neighboring nodesIf n has been immune to v longer than the immunity durationremove v from the viral historyStochastic: Data averaged to account for probabilistic factors• Many network realizations account for rewiring variability• Many runs per network account for mutation variabilitySmall-World Networks• Connected graph, highly regular but with clustering• Better depicts realistic socio-spatial networks• Purely regular and purely random networks are not realistic• Rewiring networks increases disorder• Small-World behavior emerges forD.J. Watts and S.H. Strogatz, Nature 393 (1998), p. 440.Np1April 1, 2010Collaboration graph of film actorsPower Grid of Western U.S.Neural network of worm (Caenorhabditis elegans)Examples of small world networksVariables• N Population size • 2k Average number of neighbors• p Network randomness (probability of node connections rewiring) • τIInfection time• τRImmunity duration• μ Mutation probability• hthr Cross-immunity threshold (antigenicity)• L Pathogen representation length• Bitstring model:o Abstract representation of a pathogen‟s genetic codeo Choose bitstring of length ten. e.g. 0000000000o Mutations are single random bit flips: e.g. 0000010000• Cross-immunity:o Hamming-distance: l1-norm of pathogen representationse.g. 0110001 and 0100011 have hamming-distance 2o Minimum hamming distance, hmin : smallest distance between challenge strain and all strains in the viral historyo Individual becomes infected if hmin > hthr• Mutation:o Survival mechanism of pathogeno The mutation probabilities (µ) of pathogens are generally smallHow Infections Spread In a Small World NetworkModel RealizationsImmunity DurationτR= 50, 150, 450N = 105k =2p = 0.01τI = 1µ = 0.01hthr= 2Cross-Immunity Thresholdhthr= 0, 2, 4, 10N = 105k =2p = 0.01τI = 1τR=150µ = 0.01Network SizeN = 104, 105, 106k =2p = 0.01τI = 1τR=150µ = 0.01hthr= 2(Parameters Studied)Model ImplementationApril 1, 2010MatlabC++Typical runtime (per network per run)ParameterMATLAB implementationC++ implementationτ = 50Unknown (>34 hours)15 minτ = 15010 hours3 minτ = 4501.5 hours0.35 minN = 1042 hours0.50 minN = 10510 hours3 minN = 106Unknown (not attempted)55 minFraction of infectious individuals nIas a function of time step t for a set of variations of the immunity duration τR. Cross-immunity threshold hthr= 2. Network size N=105.Epidemics with pathogen mutation on small-world networks, by Z.-G. Shao, Z.-J. Tan, X.-W. Zou, and Z.-Z. Jin, Physica A 363, 561-566 (2006).April 1, 2010Our Results Source ResultsFraction of infectious individuals nI as a function of time step t for a set of variations of the cross-immunity threshold hthr . Immunity duration TR= 150. Network size N=105.Epidemics with pathogen mutation on small-world networks, by Z.-G. Shao, Z.-J. Tan, X.-W. Zou, and Z.-Z. Jin, Physica A 363, 561-566 (2006).April 1, 2010Source ResultsOur ResultsNumber of infectious individuals NI as a function of time step t for networks of size N = 104, 105, and 106. Immunity duration τR= 150. Cross-immunity threshold hthr= 2. Epidemics with pathogen mutation on small-world networks, by Z.-G. Shao, Z.-J. Tan, X.-W. Zou, and Z.-Z. Jin, Physica A 363, 561-566 (2006).April 1, 2010Source ResultsOur ResultsSummary of results• Implementation captured behavior very similar to the original worko Long immunity durations, high cross-immunity thresholds,and small networks lead to rapid viral extinctiono Moderate immunity duration and moderate cross-immunity thresholdslead to persistent oscillatory behavioro Short immunity durations and low cross-immunity thresholdsrapidly lead to systemic infection (potential for pandemic)• Discrepancieso Implemented hmin>hthrtest for immunity, as outlined in the paper,but authors implemented hmin>= hthro Scaling factorFuture Work VaccinationsApril 1, 2010Demographic Variations in ImmunityReferencesD.J. Watts, S.H. Strogatz, Nature 393 (1998) 440.J.M. Wood, J.S. Robertson, Nat. Rev. Microbiol. 2 (2004) 842.K.L. Cooke, D.F. Calef, E.V. Level, Nonlinear Systems and its Applications, Academic Press, New York, 1977.M. Girvan, D.S. Callaway, M.E.J. Newman, S.H. Strogatz, Phys. Rev. E 65 (2001) 031915.M. Kamo, A. Sasaki, Physica D 165 (2002) 228.N.M. Ferguson, M.J. Keeling, W.J. Edmunds, R.