cccarlos

Slide1: 

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 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

Recent work: Joint with: 

Recent work: Joint with Miriam Nuno, Harvard School of Public Health Zhilan Feng, Purdue University Maia Martcheva, University of Florida

Slide3: 

Impact of Influenza Epidemics/Pandemics 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)

Borrowed from Mac Hyman: 

Borrowed from Mac Hyman

Slide5: 

THE DIFFUSION OF INFLENZA, Patterns and Paradigms, by Gerald F. Pyle

Slide6: 

The pandemic of 1781-82 originated in the Orient. St = -lS + .k S It = lS + .kI l = r b(x,t,t) I/(S + I) k = mobility of the population THE DIFFUSION OF INFLENZA, Patterns and Paradigms, by Gerald F. Pyle

Slide7: 

The pandemic 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 and Paradigms, by Gerald F. Pyle

Slide8: 

Diffusion Pathways for Primary Outbreaks of influenza Pandemic of 1918-19 THE DIFFUSION OF INFLENZA, Patterns and Paradigms, by Gerald F. Pyle

Slide9: 

Diffusion Pathways for Primary Outbreaks of influenza Core Areas and During the Beginning of the 1967-68 Season THE DIFFUSION OF INFLENZA, Patterns and Paradigms, by Gerald F. Pyle

Slide10: 

Core Areas and Diffusion Pathways for Primary Outbreaks of Influenza During the Beginning of the 1968-69 Season THE DIFFUSION OF INFLENZA, Patterns and Paradigms, by Gerald F. Pyle Susceptibility of the population is different for the second flu season of the same virus.

Slide11: 

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 and Paradigms, by Gerald F. Pyle

Work of Mac Hyman and Tara La Force: 

Work of Mac Hyman and Tara La Force

SIR Model with Loss of Immunity: 

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

Slide14: 

City 1 City 2 City 3 City 4 m13 m31 Mobility

Slide17: 

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

Slide18: 

Comparison of model and data for upper respiratory track illness

End Work of Mac Hyman and Tara La Force: 

End Work of Mac Hyman and Tara La Force

Motivation: 

Motivation Researchers have explored the possible mechanism(s) underlying the recurrence of epidemics and persistence of co-circulating virus strains of influenza types between pandemics. We (CHALL I and II) began to explore role of cross-immunity in 1988 with the aid of mathematical models JMB Paper 1988: Castillo-Chavez, Hethcote, Andreasen, Levin and Liu

Influenza A reemerges year after year, despite the fact that infection leads to lifetime immunity to a strain: 

Influenza A reemerges year after year, despite the fact that infection leads to lifetime immunity to a strain

Slide25: 

Modeling the Dynamics of Two-Strain Influenza Strains with Isolation and Partial Cross-Immunity Previous Results (CHAL I and II, plus): Herd-immunity, cross-immunity and age-structure are possible factors supporting influenza strain coexistence and/or disease oscillations Set up: We put two-influenza strains under various levels of (interference) competition with isolation periods and cross-immunity Some New Results (SIAM 2005 (Vol. 65: 3, 962-982) and …) We establish that cross-immunity and host isolation lead to period epidemic outbreaks (sustained oscillations) where the periods of oscillations mimic those in real data Multiple coexistence of strains even under sub-threshold conditions Oscillatory coexistence is established via Hopf-bifurcation theory and numerical simulations using realistic parameter values

Slide26: 

Figure modified from : Microbiological Reviews, March ,1992, pp 152-179

Slide27: 

Emergence and Reemergence of “New” Influenza A Virus in Humans The emergence of H5N1 influenza in Hong Kong H5N1 (nonpathogenic) flu could have spread from migrating shorebirds to ducks by fecal contamination of water. The virus was transmitted to chickens and became established in live bird markets in Hong Kong. During transmission between different species, the virus became highly pathogenic for chickens and occasionally was transmitted to humans from chickens in the markets. Despite high pathogenicity for chickens (and humans), H5N1 were nonpathogenic for ducks and geese. Molecular changes associated with emergence of a highly pathogenic H5N2 influenza virus in chicken in Mexico In 1994 H5N2 (pathogenic) in Mexican chickens related to H5N2 isolated in shorebirds (Delaware Bay, US, These H5N2 isolates replicated, spread rapidly and were not highly pathogenic. However, in 1995 virus became highly pathogenic and HA acquired an insert of 2 basic amino acids (Arg-Lys) possibly due to recombination and a mutation. Pathogenic: Capable of causing disease

Slide28: 

Figure: Modified w/permission from H.N. Eisen and Lippincott-Raven, Microbiology, Fourth Ed., J.B. Lippincott Company, Philadelphia, 1990 Schematic Model for Influenza Virus Particles The 8 influenza A viral RNA segments encode 10 recognized gene products (PB1,PB2, and PA polymerases, HA, NP, NA, M1 and M2 proteins, and NS1 and NS2 proteins. Surface proteins HA (hemagglutinin) and NA (neuraminidase) are the principal targets of the humoral immune response (i.e. response involving antibodies).

Slide29: 

Influenza Strains and Subtypes and the role of Cross-immunity Little evidence support the existence of cross-immunity between influenza A subtypes Houston and Seattle studies show that cross-immunity exists between strains within the same subtype.

Slide30: 

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

What is Cross-Immunity?: 

What is Cross-Immunity? Infection with an influenza subtype A strain may provide cross protection against other antigenically similar circulating strains.

Slide32: 

Experimental Evidence of Cross-immunity (1) 1974: Study shows <3% with prior exposure to A/Hong Kong/68 (H3N2) OR A PRIOR A/ENGLAND/72 (H3N2) GOT A/Port Chalmers/73 VS 23% with NO prior experience got infected 1976: Appearance of A/Victoria/75 (H3N2) Relative Frequency of First Infected/Previously Infected (By another strain of H3N2 subtype was approximately 41%) 1977: Co-circulating H1N2 strains Individuals born before 1952 “GOT” a strain of H1N1 Detection of antibody-positive sera YOUNG: Changed from 0% to 9% OLDER: Did not changed (remained at 9%)

Slide33: 

Experimental Evidence of Cross-immunity (2) 1979: Christ’s Hospital study shows that past infection with H1N1 protected 55%. Protection (%): (Rate in ‘susceptibles’-Rate in ‘immunes)X100 Rate in ‘susceptibles’ 1982: (Glezen) No cross-immunity between subtypes H1N1 & H3N2

Couch and Kasel (1983) Cross-immunity: 

Couch and Kasel (1983) Cross-immunity Experimental results indicate that cross-immunity shares the following features: Exhibits subtype specificity Exhibits cross-reactivity to variants within a subtype, but with reduced cross-reactivity for variants that are antigenically distant from the initial variant. Exhibits a duration of at least five to eight years Be able to account for the observation that resistance to re-infection with H1N1 may last 20 years

Modeling Cross Immunity: 

Modeling Cross Immunity -coefficient of cross-immunity Relative reduction on susceptibility due to prior exposure to a related strain. =0, represents total cross-immunity =1, represents no cross-immunity 0<<1, represents partial cross-immunity >1, represents immune deficiency

Early Modeling Approaches: 

Early Modeling Approaches In 1975 epidemiological interference of virus populations was introduced [Dietz]. In 1989 age-structure, proportionate mixing and cross- immunity are studied [Castillo-Chavez, et.al]. In 1989 interactions between human and animal host populations are studied as a source of recombinants in strains and cross-immunity.

Basic Epidemiological Models: SIR: 

Basic Epidemiological Models: SIR Susceptible - Infected - Recovered

Slide38: 

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)

Slide39: 

SIR - Equations Parameters

SIR - Model (Invasion): 

SIR - Model (Invasion)

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.

Slide42: 

Phase Portraits

SIR Transcritical Bifurcation: 

SIR Transcritical Bifurcation unstable

Models without population structure: 

Models without population structure

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

Ro = 2: 

Ro = 2

Ro = 2: 

Ro = 2

Ro = 2: 

Ro = 2 ( End )

Ro < 1: 

Ro < 1

Ro < 1: 

Ro < 1

Ro < 1: 

Ro < 1

Ro < 1: 

Ro < 1 ( End )

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.

Slide54: 

Phase Portraits

SIR Transcritical Bifurcation: 

SIR Transcritical Bifurcation unstable

Models with age structure: 

Models with age structure

Slide57: 

SIR Model with Age Structure s(t,a) : Density of susceptible individuals with age a at time t. i(t,a) : Density of infectious individuals with age a at time t. r(t,a) : Density of recovered individuals with age a at time t. # of recovered individuals with ages in (a1 , a2) at time t # of infectious individuals with ages in (a1 , a2) at time t # of susceptible individuals with ages in (a1 , a2) at time t

Slide58: 

: recruitment/birth rate. (a): age-specific probability of becoming infected. c(a): age-specific per-capita contact rate. (a): age-specific per-capita mortality rate. (a): age-specific per-capita recovery rate. Parameters

Slide59: 

Equations

Slide60: 

Initial and Boundary Conditions

Slide61: 

n(t,a) satisfies the Mackendrick Equation We assume that the total population density has reached this demographic steady state. Demographic Steady State n(t,a): density of individual with age a at time t

Slide62: 

p(t,a,a`): probability that an individual of age a has contact with an individual of age a` given that it has a contact with a member of the population . Mixing

Slide63: 

p(t,a,a`)  0 Proportionate mixing: Mixing Rules

Slide64: 

Stability of Disease-free Steady State

Ro: 

Ro “Number of secondary infections generated by a “typical” infectious individual in a population of mostly susceptibles Ro<1 No epidemic; Role of vaccination to reduce Ro and eliminate the disease. Ro>1 Epidemic (often leading to and endemic state) Role of vaccination to reduce Ro but disease still endemic

Slide66: 

Characteristic Equation The characteristic equation has a unique real dominant solution, that is, its real part is larger than the real part of all other solutions separable solutions.

Slide67: 

R0<1, Disease-free State Is Stable The characteristic equation has a unique dominant real solution. That is, the real part of all other solutions is less than this dominant solution; The dominant solution is negative iff R0<1; The dominant solution is positive iff R0>1; Whenever R0<1, the disease-free steady state is locally asymptotically stable.

Slide68: 

If R0 < = 1, the disease-free equilibrium (1,0) is globally asymptotically stable while if R0 > 1, the unique endemic equilibrium is globally asymptotically stable. Qualitative Analysis SIR model undergoes a global forward (transcritical) bifurcation.

Slide69: 

Endemic Steady States

Slide70: 

Endemic Steady States One can formally solve for the steady states. The existence of endemic steady states is determined by the roots of the following equation: f(B*) is a decreasing function of B* with f()=0. R0>1, there exists a unique endemic (e.g. non trivial) steady states; R0<1, an endemic steady state does not exist.

Slide72: 

If R0 < = 1, the disease-free equilibrium (1,0) is globally asymptotically stable while if R0 > 1, the unique endemic equilibrium is globally asymptotically stable. Qualitative Analysis SIR model undergoes a global forward (transcritical) bifurcation.

Two-strain Models: 

Two-strain Models

Modeling Cross Immunity: 

Modeling Cross Immunity -coefficient of cross-immunity relative reduction on susceptibility due to prior exposure to a related strain. =0, represents total cross-immunity =1, represents no cross-immunity 0<<1, represents partial cross-immunity >1, represents immune deficiency

Slide75: 

Two-Strain Influenza Model without Quarantine

Slide76: 

If R0 < = 1, the disease-free equilibrium (1,0) is globally asymptotically stable while if R0 > 1, the unique endemic equilibrium is globally asymptotically stable. Qualitative Analysis-no age structure SIR model undergoes a global forward (transcritical) bifurcation.

Slide77: 

Probability of Survival

Hopf -Bifurcation: 

Hopf -Bifurcation We “saw” oscillations on a simulation model. The analysis of particular cases (characteristic equation) supported this.

Slide85: 

I (t+T2) I (t+T1) I (t)

Early Results: 

Early Results Age-structure is sufficient to drive sustained oscillations in a multi- strain model [Castillo-Chavez, Hethcote, Andreasen, Liu and Levin, 1988 and 1989]. For a heterogeneous population with age-dependent mortality, cross-immunity provides an explanation to the observed recurrence of strains [Castillo-Chavez, et.al]. Cross-immunity without age structure not enough to support sustained oscillations Extensions by Andreasen, Lin, Levin and others to more than two strains.

Slide87: 

Two-Strain Influenza Model with Quarantine

Slide88: 

Invasion Reproductive Numbers Basic Reproductive Number The average number of secondary infections generated by the simultaneous introduction of both strains in a fully susceptible population Invasion reproductive number of strain 2 given that strain 1 is at equilibrium where Invasion reproductive number of strain 1 given that strain 2 is at equilibrium where

Slide89: 

Stability Regions for symmetric strains Bifurcation diagram in the ( , ) plane. The curves divide the regions into sub-regions I, II, III. In region I (II) only strain 1 (2) will be maintained (stable boundary equilibrium or sustained oscillations of a single strain). In Region III, both strains will be maintained (a stable boundary equilibrium or sustained oscillations). Further Observations: (a) As cross-immunity increases the region of stability of each individual strain increases significantly (I and II) (b) As cross-immunity decreases we observe an increase in coexistence region (III) but a decrease in the stability regions of each individual strain ( I and II)

Slide90: 

Stability Regions for asymmetric strains ( )

Slide91: 

Multiple and Sub-threshold Coexistence

Slide92: 

Approximation of long/short period oscillations and coexistence Regions A1 and A2 are used to approximate the likelihood of having coexistence of both strains and oscillations with long periods.

Slide93: 

Numerical Simulations (1) Fraction of infective individuals with strain 1 versus time. Cross-immunity is chosen such that strains are completely uncoupled ( no shared cross-immunity between strains).

Slide94: 

Sustained Oscillations: The role of Quarantine and Cross-Immunity

Slide95: 

Numerical Simulations (2) Fraction of infective individuals with strain 1 (solid) and strain 2 (dashed) versus time. Differences in cross-immunity levels between strains 1 and 2 increase (from above to below) 0.01, 0.02 and 0.03.

Slide96: 

Seasonal forcing of the infectious process Observation: The introduction of seasonality in the infection transmission process yields epidemic outbreaks that range from period to quasi-period to possibly chaotic.

Slide97: 

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).

Slide98: 

Results Multiple and sub-threshold coexistence is possible Conditions that guarantee a winning strain type or coexistence have been established Cross-immunity and 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 for the period between oscillations (*) region of strain coexistence: results show that coexistence is more likely to for weak immunity levels whereas competitive exclusion occurs for strong immunity levels (**) Probability of having long periods between oscillations is low “approximately” 0.0055 for a somewhat “typical” case.