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Published on March 3, 2008

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Slide1:  Carlos Castillo-Chavez Joaquin Bustoz Jr. Professor Arizona State University Tutorials 2: Epidemiological Mathematical Modeling Applications of Networks. Mathematical Modeling of Infectious Diseases: Dynamics and Control (15 Aug - 9 Oct 2005) Jointly organized by Institute for Mathematical Sciences, National University of Singapore and Regional Emerging Diseases Intervention (REDI) Centre, Singapore http://www.ims.nus.edu.sg/Programs/infectiousdiseases/index.htm Singapore, 08-24-2005 Bioterrorism:  Bioterrorism The possibility of bioterrorist acts stresses the need for the development of theoretical and practical mathematical frameworks to systemically test our efforts to anticipate, prevent and respond to acts of destabilization in a global community Epidemic Models:  Epidemic Models Basic Epidemiological Models: SIR:  Basic Epidemiological Models: SIR Susceptible - Infected - Recovered Slide5:  S(t): susceptible at time t I(t): infected assumed infectious at time t R(t): recovered, permanently immune N: Total population size (S+I+R) Slide6:  SIR - Equations Parameters SIR - Model (Invasion):  SIR - Model (Invasion) Ro:  Ro “Number of secondary infections generated by a “typical” infectious individual in a population of mostly susceptibles at a demographic steady state Ro<1 No epidemic Ro>1 Epidemic Establishment of a Critical Mass of Infectives! Ro >1 implies growth while Ro<1 extinction. :  Establishment of a Critical Mass of Infectives! Ro >1 implies growth while Ro<1 extinction. Slide10:  Phase Portraits SIR Transcritical Bifurcation:  SIR Transcritical Bifurcation unstable Deliberate Release of Biological Agents:  Deliberate Release of Biological Agents Slide13:  Effects of Behavioral Changes in a Smallpox Attack Model Impact of behavioral changes on response logistics and public policy (appeared in Mathematical Biosciences, 05) Sara Del Valle1,2 Herbert Hethcote2, Carlos Castillo-Chavez1,3, Mac Hyman1 1Los Alamos National Laboratory 2University of Iowa 3Cornell University The Model:  The Model Sn En In R V Q W Sl El Il D The subscript refers to normally active (n) or less active (l): Susceptibles (S), Exposed (E), Infectious (I), Vaccinated (V), Quarantined (Q), Isolated (W), Recovered (R), Dead (D) S E I "An Epidemic Model with Virtual Mass Transportation" :  "An Epidemic Model with Virtual Mass Transportation" Two neighborhood simulations (NYC type city):  Two neighborhood simulations (NYC type city) There are 8 million long-term and 0.2 million short-term (tourists) residents in NYC. Time span of simulation is 30 days +. Control parameters in the model are: q1 and q2 (vaccination rates) We use two ``neighborhoods”, one for NYC residents and the second for tourists. Slide18:  Curve R0 (q1, q2) =1 Slide19:  Plot R0 (q1, q2) vs q1 and q2 Slide20:  Conclusions Integrated control policies are most effective: behavioral changes and vaccination have a huge impact. Policies must include “transient” populations Delays are bad. Worst Case Scenarios?:  Worst Case Scenarios? Epidemics on Networks?:  Epidemics on Networks? Slide23:  MMUR May 9, 2003/ 52 (18); 405-411 SARS propagation network in Singapore Epidemics on Networks? Some caveats:  Epidemics on Networks? Some caveats Appeal: Networks look like the real world Typical: To study “fixed” graphs (small world, scale-free) Network/Graph structure not fixed over time A connected to B requires temporal co-habitation in the study of processes on networks Underlying Philosophy in Classical Epidemiology:  Underlying Philosophy in Classical Epidemiology Ecological view point:  Ecological view point Invasion (Networks useful at this level). Short temporal scales--single outbreaks (Networks useful at this level). Persistence Co-existence Evolution Co-evolution Processes on Networks Temporal Scales:  Processes on Networks Temporal Scales Single outbreak Long-term dynamics Evolutionary behavior Nodes? Epidemiological Units:  Nodes? Epidemiological Units Individuals Cluster of individuals (friends) Other aggregates? Farms? Reach of the Network: Level of Aggregation:  Neighborhood Cities, states, countries Reach of the Network: Level of Aggregation Highly trafficked locations in the city of Portland (EpiSims):  Highly trafficked locations in the city of Portland (EpiSims) Heterogeneity: Infection curves by routes of transmission (Ping Yang--Health Canada) :  Heterogeneity: Infection curves by routes of transmission (Ping Yang--Health Canada) Slide32:  Toronto. SARS was introduced to Toronto by a couple (Guests F and G) at Hotel M in Hong Kong. On February 23, 2003 they returned to Toronto. Two days later, the woman developed SARS, infected 5 out of her 6 adult family members and caused the first outbreak in Toronto. In mid-May, an undiagnosed case at North York General Hospital led to a second outbreak among other patients, family members and healthcare workers (from Glen Webb’s presentation). Exponential Dynamics: Hong Kong and Singapore:  Exponential Dynamics: Hong Kong and Singapore Data Model The Case of Toronto:  The Case of Toronto Data Model Slow diagnosis and effective isolation Fast diagnosis but imperfect isolation Interventions Fast diagnosis & effective isolation Predicting the Final Size of the Epidemic in Toronto:  Predicting the Final Size of the Epidemic in Toronto Model prediction = 396 cases (J. Theor. Biol 224, 1-8, 2003) Actual number as of June 23, 2003 = 377 (Health Canada website) Complexity and Networks: Population’s Characteristics:  Complexity and Networks: Population’s Characteristics Gender, ethnicity, race Social, age, economic structure Cultural and Communication structures Connectedness?:  Connectedness? Local small isolated populations Large multi-connected populations Who mixes with whom? Slide38:  Scale and topology Feb 21, ‘03 Nov 05, ‘02 Prof. Liu Mrs. Mok Mrs. Siu-Chu Johny Chen Slide39:  Modeling Challenges &Mathematical Approaches “Classical” Population Perspective Deterministic Stochastic Computational Agent Based Models Scaling Laws for the Movement of People Between Locations in a Large City Gerardo Chowell et al. Scaling Laws for the Movement of People between locations in a large city, Physical Review E, 68, 066102 (2003), Chowell, Hyman, Eubank and Castillo-Chavez LA-UR-02-6658 :  Scaling Laws for the Movement of People Between Locations in a Large City Gerardo Chowell et al. Scaling Laws for the Movement of People between locations in a large city, Physical Review E, 68, 066102 (2003), Chowell, Hyman, Eubank and Castillo-Chavez LA-UR-02-6658 Outline:  Outline Statistical properties of real world networks Network of actors in Hollywood www Internet Scientific collaboration network Power generator network of western US Outline:  Outline Analysis of a Real World Network: The city of Portland Location-based network Topological properties Traffic distribution Total traffic distribution per location Correlation between connectivity and traffic distributions Time evolution of the network Statistical properties of networks:  Statistical properties of networks Connectivity distribution (degree distribution) Clustering (C) Characteristic path length (L) 3 2 3 2 The network of actors in Hollywood:  The network of actors in Hollywood Julia Roberts Diane Lane Eric Roberts Mickey Rourke Denzel Washington Richard Gere Kevin Bacon (Watts and Strogatz, 1998) The electric power grid of western US:  The electric power grid of western US (Watts and Strogatz, 1998) The world wide web (www):  The world wide web (www) Home page web page web page web page (Barabasi et al. (1999), Kumar (2000)) Internet:  Internet (Faloutsos et al., 1998) Scientific collaboration networks:  Scientific collaboration networks M. J. Newman Statistical properties of real world networks:  Statistical properties of real world networks Small-world effect High levels of clustering Short Characteristic path length Connectivity distribution has a tail that decays as a power law of the form: P(k) ~ k -g Connectivity distribution for two real world networks:  Connectivity distribution for two real world networks Random Graph Models of Networks. M. E. J. Newman, 2002 City of Portland: A Social Network:  City of Portland: A Social Network Location-based network:  Location-based network Directed, weighted network Data set contains a detailed description of the movements of the individuals in the city of Portland. Location 1 Location 4 Location 3 Location 2 W i j Highly trafficked locations in Portland:  Highly trafficked locations in Portland Statistical properties :  Statistical properties The clustering coefficient for our location-based network is C = 0.058 (roughly 350 times larger than the expected value for an equivalent random graph). The same situation arises for the electric power grid of western US where C=0.08. Average distance between nodes = 3.38 (diameter = 9). Connectivity distribution:  Connectivity distribution Out-degree Number of locations g = 2.70 Strong or weak connections ?:  Strong or weak connections ? Very little is known about the distribution of the strength of the connections in real world networks. Only their structural properties have been analyzed. The main reason being the lack of data to quantify the strength of the connections. Traffic distribution:  Traffic distribution Out-traffic Number of connections between locations g = 3.70 Total traffic distribution:  Total traffic distribution Total out-traffic Number of locations g = 2.74 Correlation in density:  Correlation in density out-degree Número de locaciones Tráfico total de salida por nodo Semilog plot:  Semilog plot out-degree Total out-traffic Log (number of locations) Hierarchical structures at different levels of aggregation C(k) a k –b :  Hierarchical structures at different levels of aggregation C(k) a k –b a) b) c) d) Work activities School activities Social/rec activities All activities Cluster size distribution:  Cluster size distribution Gerardo Chowell t=4 hrs. t=5 hrs. t=7 hrs. t=6 hrs. Location-based network Size of the largest cluster:  Location-based network Size of the largest cluster Gerardo Chowell Slide64:  This is joint work with J.M. Hyman, S. Eubank and G Chowell. Scaling Laws for the Movement of People between locations in a large city, Physical Review E, 68, 066102 (2003), Chowell, Hyman, Eubank and Castillo-Chavez Structure and Function of Complex Networks:  Structure and Function of Complex Networks Introduction: Strogatz, Nature (2001) Comprehensive study Newman, SIAM Rev. (2003) Results:  Results Random connections Nearest neighbor, small world and random Questions:  Questions Compare disease spread on a Nearest Neighbor Network Random Network Small World Network p = 1 when random and p about 0 when nearest neighbor dominates Small-world and Scale-free networks:  Small-world and Scale-free networks Small world network of size 70 with probability of random connections p = 0.1 Scale-free network of size 70 illustrating the presence of a small number of highly nodes connected (hubs). LLYD Model:  LLYD Model Scale-Free Networks Exponentially Distributed Networks as as Barabasi-Albert (BA) Erdos-Renyi Connectivity distribution:  Connectivity distribution Out-degree Number of locations g = 2.70 Navigation in a Small-World (Kleinberg, 2000):  Navigation in a Small-World (Kleinberg, 2000) Two dimensional lattice Long-range connection between node u and v, with probability , where Greedy heuristic algorithm: each message holder forwards the message across a connection that brings it as close as possible to the target in the lattice distance T: Expected delivery time. where Highly trafficked locations in the city of Portland:  Highly trafficked locations in the city of Portland Building Epidemics:  Building Epidemics Transition Probabilities: P(S to I), P(I to R) PSI = 1 – eb I dt PIR = 1 – eg dt Epidemics on small-world networks:  Epidemics on small-world networks The rate of growth of SIR epidemics increases in a nonlinear fashion as disorder in the network increases. The rate of growth rd for the analogous deterministic homogeneous mixing model is shown. The role of the network structure in epidemics:  The role of the network structure in epidemics The dotted graph shows the rate of growth of SIR epidemics when the initial infectious source has the highest number of faraway connections (train stations, airports, etc) while the continuous line is the result of placing the source at random. Slide77:  Rate of growth of epidemics in small-world networks Growth in the number of infected in an SIR model where individuals live in a ring. Curves give the average number of infected (50 simulations) in a population of 1000 while the growth is exponential. p = 0, disorder parameter, corresponds to no long term connections and p =1 implies that everybody is connected to each other. Graph on the left from a single source (idea, virus, rumor). Top curve is when the spread begin at a pressure point. Lower spread begins at a random point. Graph on the right, three randomly placed sources of infection (ideas, whatever) versus one. SIR epidemics on Small worlds:  SIR epidemics on Small worlds For small worlds, a sharp transition occurs at small values of the disorder parameter p. 5 initial infected nodes chosen at random, =4/7, =2/7. The mean (red) of 50 realizations and the standard deviation are shown. SIR epidemics on Small worlds:  SIR epidemics on Small worlds Similar results are observed when initial infected nodes are chosen by highest degree =4/7, =2/7. The mean (red) of 50 realizations and the standard deviation are shown. SIR epidemics on Small worlds:  SIR epidemics on Small worlds Final epidemic size as a function of the transmission rate . 5 initial infected nodes chosen at random =2/7. The mean (solid) of 50 realizations and the standard deviation (bars) are shown. SIR epidemics on Scale-Free networks (Barabasi-Albert model):  SIR epidemics on Scale-Free networks (Barabasi-Albert model) Final epidemic size as a function of the transmission rate . 5 initial infected nodes chosen at random =2/7. The mean (red) of 50 realizations and the standard deviation are shown. Small worlds: Epidemic duration (five sources placed at random):  Small worlds: Epidemic duration (five sources placed at random) Slide83:  The impact of alternative agents of disease transmission and evolution—like transportation systems seems critically important. The study of epidemics on different topologies (networks) is essential (mobile individuals cause a lot of ``problems”). Worst case scenarios may occur in random networks but the focus should be on worst case plausible scenarios-one cannot ignore behavioral changes. Worst case scenarios depend on topology Bigger outbreaks are sometimes caused by releases at pressure points in the network. Conclusions Slide84:  Bioterrorism: Mathematical Modeling Applications in Homeland Security. H. T. Banks and C. Castillo-Chavez, Editors Frontiers in Applied Mathematics 28 Globalization and the possibility of bioterrorist acts have highlighted the pressing need for the development of theoretical and practical mathematical frameworks that may be useful in our systemic efforts to anticipate, prevent, and respond to acts of destabilization. Bioterrorism: Mathematical Modeling Applications in Homeland Security collects the detailed contributions of selected groups of experts from the fields of biostatistics, control theory, epidemiology, and mathematical biology who have engaged in the development of frameworks, models, and mathematical methods needed to address some of the pressing challenges posed by acts of terror. The ten chapters of this volume touch on a large range of issues in the subfields of biosurveillance, agroterrorism, bioterror response logistics, deliberate release of biological agents, impact assessment, and the spread of fanatic behaviors.

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