# Borrowed from Mac Hyman Work of Mac Hyman in addition to Tara La Force SIR Model with Loss of Immunity End Work of Mac Hyman in addition to Tara La Force Motivation

## Borrowed from Mac Hyman Work of Mac Hyman in addition to Tara La Force SIR Model with Loss of Immunity End Work of Mac Hyman in addition to Tara La Force Motivation

Bryan, Kim, Local News Reporter (City of Homewood) has reference to this Academic Journal, PHwiki organized this Journal Carlos Castillo-Chavez Joaquin Bustoz Jr. Professor Arizona State University August 22, 2005 The role of crossimmunity on influenza dynamics Mathematical Modeling of Infectious Diseases: Dynamics in addition to Control (15 Aug – 9 Oct 2005) Jointly organized by Institute as long as Mathematical Sciences, National University of Singapore in addition to Regional Emerging Diseases Intervention (REDI) Centre, Singapore http://www.ims.nus.edu.sg/Programs/infectiousdiseases/index.htm Recent work: Joint with Miriam Nuno, Harvard School of Public Health Zhilan Feng, Purdue University Maia Martcheva, University of Florida Impact of Influenza Epidemics/P in addition to emics 1918 Spanish Flu (H1N1): 20% – 40% illness, 20 million deaths. 1957 Asian Flu (H2N2): 70,000 deaths in US. 1968 Hong Kong Flu (H3N2): 34,000 deaths in US. 1976 Swine Flu Scare (H1N1 related) 1977 Russian Flu Scare (H1N1 related) 1997 Avian Flu Scare (H5N1, human human)

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Borrowed from Mac Hyman THE DIFFUSION OF INFLENZA, Patterns in addition to Paradigms, by Gerald F. Pyle The p in addition to emic of 1781-82 originated in the Orient. St = -lS + .k S It = lS + .kI l = r b(x,t,t) I/(S + I) k = mobility of the population THE DIFFUSION OF INFLENZA, Patterns in addition to Paradigms, by Gerald F. Pyle

The p in addition to emic of 1847-48 started in the Orient. In this epidemic, the diffusion pathways within western Europe changed after the railroads began running. THE DIFFUSION OF INFLENZA, Patterns in addition to Paradigms, by Gerald F. Pyle Diffusion Pathways as long as Primary Outbreaks of influenza P in addition to emic of 1918-19 THE DIFFUSION OF INFLENZA, Patterns in addition to Paradigms, by Gerald F. Pyle Diffusion Pathways as long as Primary Outbreaks of influenza Core Areas in addition to During the Beginning of the 1967-68 Season THE DIFFUSION OF INFLENZA, Patterns in addition to Paradigms, by Gerald F. Pyle

Core Areas in addition to Diffusion Pathways as long as Primary Outbreaks of Influenza During the Beginning of the 1968-69 Season THE DIFFUSION OF INFLENZA, Patterns in addition to Paradigms, by Gerald F. Pyle Susceptibility of the population is different as long as the second flu season of the same virus. Weighted Network Nodes are cities weighted by their population. Edges are weighted by the mobility of people between the cities THE DIFFUSION OF INFLENZA, Patterns in addition to Paradigms, by Gerald F. Pyle Work of Mac Hyman in addition to Tara La Force

SIR Model with Loss of Immunity Flow of people through a simple SIR model Flow of people through a SIRP model with return to susceptibility Partially Immune Recovered Immune Infected Susceptible Joint research with Tara LaForce City 1 City 2 City 3 City 4 m13 m31 Mobility

The URT Data is rough compared with the smooth model predictions Y axis is in 100s of people/week infected. Comparison of model in addition to data as long as upper respiratory track illness Comparison of model in addition to data as long as upper respiratory track illness

End Work of Mac Hyman in addition to Tara La Force Motivation Researchers have explored the possible mechanism(s) underlying the recurrence of epidemics in addition to persistence of co-circulating virus strains of influenza types between p in addition to emics. We (CHALL I in addition to II) began to explore role of cross-immunity in 1988 with the aid of mathematical models JMB Paper 1988: Castillo-Chavez, Hethcote, Andreasen, Levin in addition to Liu Influenza A reemerges year after year, despite the fact that infection leads to lifetime immunity to a strain

Figure: Modified w/permission from H.N. Eisen in addition to Lippincott-Raven, Microbiology, Fourth Ed., J.B. Lippincott Company, Philadelphia, 1990 Schematic Model as long as Influenza Virus Particles The 8 influenza A viral RNA segments encode 10 recognized gene products (PB1,PB2, in addition to PA polymerases, HA, NP, NA, M1 in addition to M2 proteins, in addition to NS1 in addition to NS2 proteins. Surface proteins HA (hemagglutinin) in addition to NA (neuraminidase) are the principal targets of the humoral immune response (i.e. response involving antibodies). Influenza Strains in addition to Subtypes in addition to the role of Cross-immunity Little evidence support the existence of cross-immunity between influenza A subtypes Houston in addition to Seattle studies show that cross-immunity exists between strains within the same subtype. Influenza Epidemiology Antigenic drift (resulting in minor yearly epidemics) Antigenic shift (resulting in major epidemics with periods of ~ 27 years) Seasonal occurrence Low transmission rates out-of-season Explosive onset of epidemics Rapid termination of epidemics despite the continued abundance of susceptibles (Tacker) Highest attack rates observed among children Highest risk group observed in the elderly

Two-Strain Model with Seasonality The effects of seasonal variation in the transmission coefficient leads to changes in the qualitative behavior of the system. (3-D trajectories reconstructed using time-delay embedding). Results Multiple in addition to sub-threshold coexistence is possible Conditions that guarantee a winning strain type or coexistence have been established Cross-immunity in addition to isolation can lead to periodic outbreaks (sustained oscillations) Oscillatory coexistence is established via Hopf-bifurcation. Numerical simulations using realistic parameter values show that periods are consistent with observations Approximation have been provided as long as the period between oscillations () region of strain coexistence: results show that coexistence is more likely to as long as weak immunity levels whereas competitive exclusion occurs as long as strong immunity levels () Probability of having long periods between oscillations is low approximately 0.0055 as long as a somewhat typical case.

## Bryan, Kim Local News Reporter (City of Homewood)

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