EPIDEMICS IN STRONGLY FLUCTUATING POPULATIONS

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EPIDEMICS IN STRONGLY FLUCTUATING POPULATIONS

• By
• Abdul-Aziz Yakubu
• Howard University
• ayakubu_at_howard.edu

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Epidemics In Strongly Fluctuating Populations
Constant Environments

• Barrera et al. MTBI Cornell University
  Technical Report (1999).
• Valezquez et al. MTBI Cornell University
  Technical Report (1999).
• Arreola, R. MTBI Cornell University Technical
  Report (2000).
• Gonzalez, P. A. MTBI Cornell University
  Technical Report (2000).
• Castillo-Chavez and Yakubu, Contemporary
  Mathematics, Vol 284 (2001).
• Castillo-Chavez and Yakubu, Math. Biosciences,
  Vol 173 (2001).
• Castillo-Chavez and Yakubu, Non Linear Anal
  TMA, Vol 47 (2001).
• Castillo-Chavez and Yakubu, IMA (2002).
• Yakubu and Castillo-Chavez J. Theo. Biol.
  (2002).
• K. Rios-Soto, Castillo-Chavez, E. Titi, A.
  Yakubu, AMS (In press).
• Abdul-Aziz Yakubu, JDEA (In press).

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Epidemics In Strongly Fluctuating Populations
Periodic Environments

• Franke Yakubu JDEA (2005)
• Franke Yakubu SIAM Journal of Applied
  Mathematics (2006)
• Franke Yakubu Bulletin of Mathematical
  Biology ( In press)
• Franke Yakubu Mathematical Biosciences (In
  press)

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Epidemics In Strongly Fluctuating Populations
Almost Periodic Environments

• T. Diagana, S. Elaydi and Yakubu (Preprint)

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Demographic Equation
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Examples Of Demography In Constant Environments
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Asymptotically Bounded Growth

• Demographic Equation (1) with constant rate ?
  and initial condition N(0) gives rise to the
  following
• N(t1) ?N(t)?, N(0)N0
• Since
• N(1)? N0 ?,
• N(2)?2 N0 (?1) ?,
• N(3)?3 N0 (?2 ?1) ?, ...,
• N(t)?t N0 (?t-1?t-2...?1) ?

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Asymptotically Bounded Growth(Constant
Environment)
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Geometric Growth(constant environment)

• If new recruits arrive at the positive per-capita
  rate ? per
• generation, that is, if f(N(t))?N(t) then
• N(t1)(? ? )N(t).
• That is, N(t) (? ?)t N(0).
• The demographic basic reproductive number is
• Rd?/(1-?)
• Rd, a dimensionless quantity, gives the average
  number of descendants produced by a small pioneer
  population (N(0)) over its life-time.
• Rdgt1 implies that the population invades at a
  geometric rate.
• Rdlt1 leads to extinction.

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Density-Dependent Growth Rate

• If f(N(t))N(t)g(N(t)), then
• N(t1)N(t)g(N(t))? N(t).
• That is, N(t1)N(t)(g(N(t))?).
• Demographic basic reproductive number is
• Rdg(0)/(1-?)

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The Beverton-Holt Model Compensatory Dynamics
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The Beverton-Holt ModelCompensatory Dynamics
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Beverton-Holt Model With The Allee Effect

• The Allee effect, a biological phenomenon named
  after W. C. Allee, describes a positive relation
  between population density and the per capita
  growth rate of species.

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Effects Of Allee Effects On Exploited Stocks
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The Ricker Model Overcompensatory Dynamics
g(N)exp(p-N)
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The Ricker Model Overcompensatory Dynamics
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Are population cycles globally stable?

• In constant environments, population cycles
  are not globally stable (Elaydi-Yakubu, 2002).

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Constant Recruitment In Periodic Environments
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Constant Recruitment In Periodic Environment
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Periodic Beverton-Holt Recruitment Function
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Signature Functions For Classical Population
Models In Periodic Environments

• R. May, (1974, 1975, etc)
• Franke and Yakubu Bulletin of Mathematical
  Biology (In press)
• Franke and Yakubu Periodically Forced Leslie
  Matrix Models (Mathematical Biosciences, In
  press)
• Franke and Yakubu Signature function for the
  Smith-Slatkin Model (JDEA, In press)

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Geometric Growth In Periodic Environment
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(No Transcript)
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SIS Epidemic Model
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Disease Persistence Versus Extinction
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Asymptotically Cyclic Epidemics
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Example
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Example
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Epidemics and Geometric Demographics
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Persistence and Geometric Demographics
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Cyclic Attractors and Geometric Demographics
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Multiple Attractors
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Question

• Are disease dynamics driven by demographic
  dynamics?

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S-Dynamics Versus I-Dynamics (Constant
Environment)
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SIS Models In Constant Environments

• In constant environments, the demographic
  dynamics drive both the susceptible and infective
  dynamics whenever the disease is not fatal.

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Periodic Constant Demographics Generate Chaotic
Disease Dynamics
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Periodic Beverton-Holt Demographics Generate
Chaotic Disease Dynamics
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Periodic Geometric Demographics Generate Chaotic
Disease Dynamics
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Conclusion

• We analyzed a periodically forced
  discrete-time SIS model via
• the epidemic threshold parameter R0
• We also investigated the relationship between
  pre-disease invasion
• population dynamics and disease dynamics
• Presence of the Allee effect in total
  population implies its presence in the infective
  population.
• With or without the infection of newborns, in
  constant environments
• the demographic dynamics drive the disease
  dynamics
• Periodically forced SIS models support multiple
  attractors
• Disease dynamics can be chaotic where
  demographic dynamics are
• non-chaotic

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S-E-I-S MODEL
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Other Models

1. Malaria in Mali (Bassidy Dembele Ph. D.
   Dissertation)
2. Epidemic Models With Infected Newborns (Karen
   Rios-Soto Ph. D. Dissertation)

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Dynamical Systems Theory

• Equilibrium Dynamics, Oscillatory Dynamics,
  
• Stability Concepts, etc
• Attractors and repellors (Chaotic attractors)
• Basins of Attraction
• Bifurcation Theory (Hopf, Period-doubling
  and
• saddle-node bifurcations)
• Perturbation Theory (Structural Stability)

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Animal Diseases

• Diseases in fish populations (lobster, salmon,
  etc)
• Malaria in mosquitoes
• Diseases in cows, sheep, chickens, camels,
  donkeys, horses, etc.