April 1 2010 Tucson AZ Epidemics with Pathogen Mutation on Small World Networks Zhi Gang Shao Zhi Jie Tan Xian Wu Zou Zhun Zhi Jin Presenter Daniel Jackson April 1 2010 Team Members Nadia Flores Daniel Jackson Robert Phillips Manuel Rivera Zach Rogers Mentor Toby Shearman Introduction to the Model Epidemiology o Factors affecting health and illness of a population o Viral outbreaks What we are studying In 1918 1919 an estimated 40 50 million people died worldwide from the influenza pandemic o o o o Socio spatial networks Time dependent dynamics Immunity Pathogen mutations J M Wood and J S Robertson Nat Rev Microbiol 2 2004 p 842 April 1 2010 Why study this model o Mathematical models can help predict the behavior of infectious agents on susceptible populations o Predictions of mathematical models can help guide efficient and effective treatment for the eradication of disease o Accounting for pathogen mutation and timedependent immunity within the small world paradigm leads to a more realistic model April 1 2010 Examples of infectious diseases Measles Highly infectious Short incubation period Short immunity duration Control policy is long term Smallpox Less infectious Longer incubation period Long immunity duration Herd Immunity made eradication possible April 1 2010 Infection 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 Deterministic vs Probabilistic SIR Differential Equation Model SIRS Agent Based Model Consider node n If n has a virus in the infectious stage add to infected population count For each virus v in the viral history If v is in infectious stage send challenge strain to neighboring nodes If n has been immune to v longer than the immunity duration remove v from the viral history Stochastic Data averaged to account for probabilistic factors Many network realizations account for rewiring variability Many runs per network account for mutation variability Small 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 1 Small World behavior emerges for p N D J Watts and S H Strogatz Nature 393 1998 p 440 April 1 2010 Examples of small world networks Collaboration graph of film actors Power Grid of Western U S Neural network of worm Caenorhabditis elegans Variables N Population size 2k Average number of neighbors p Network randomness probability of node connections rewiring I Infection time Mutation probability hthr Cross immunity threshold antigenicity L Pathogen representation length R Immunity duration How Infections Spread In a Small World Network Bitstring model o Abstract representation of a pathogen s genetic code o Choose bitstring of length ten e g 0000000000 o Mutations are single random bit flips e g 0000010000 Cross immunity o Hamming distance l1 norm of pathogen representations e g 0110001 and 0100011 have hamming distance 2 o Minimum hamming distance hmin smallest distance between challenge strain and all strains in the viral history o Individual becomes infected if hmin hthr Mutation o Survival mechanism of pathogen o The mutation probabilities of pathogens are generally small Model Realizations Parameters Studied Immunity Duration R 50 150 450 Cross Immunity Threshold hthr 0 2 4 10 Network Size N 104 105 106 N 105 k 2 p 0 01 I 1 0 01 hthr 2 N 105 k 2 p 0 01 I 1 R 150 0 01 k 2 p 0 01 I 1 R 150 0 01 hthr 2 April 1 2010 Model Implementation Typical runtime per network per run Parameter MATLAB implementation C implementation Unknown 34 hours 15 min 150 10 hours 3 min 450 1 5 hours 0 35 min N 104 2 hours 0 50 min N 105 10 hours 3 min N 106 Unknown not attempted 55 min 50 C Matlab April 1 2010 Our Results Source Results Fraction of infectious individuals nI as a function of time step t for a set of variations of the immunity duration R 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 2010 Our Results Source Results Fraction 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 2010 Our Results Source Results Number of infectious individuals NI as a function of time step t for networks of size N 104 105 and 106 Immunity duration R 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 Summary of results Implementation captured behavior very similar to the original work o Long immunity durations high cross immunity thresholds and small networks lead to rapid viral extinction o Moderate immunity duration and moderate cross immunity thresholds lead to persistent oscillatory behavior o Short immunity durations and low cross immunity thresholds rapidly lead to systemic infection potential for pandemic Discrepancies o Implemented hmin hthr test for immunity as outlined in the paper but authors implemented hmin hthr o Scaling factor April 1 2010 Future Work Vaccinations Demographic Variations in Immunity References 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 D 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 Gani B T Grenfell R M Anderson S Leach Nature 425 2003 681 Thank you for your interest Acknowledgements o o Ildar Gabitov Toby Shearman 20