n
Abdul-Aziz Yakubu
Howard University
(ayakubu@howard.edu)
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• Do basic tenets in theoretical ecology and epidemiology remain true when parameters oscillate or do they need modification?
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Demographic Equations in
Constant Environments
N(t) is total population size in generation t.
N(t+1) = f(N(t))+ g
N(t), (1) where g in (0,1) is the constant
"probability" of surviving per generation, and f : R +→ R + models the birth or recruitment process.
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Examples Of Recruitment Functions In
Constant Environments
• If the recruitment rate is constant per generation , then the total population is asymptotically constant .
• If the recruitment rate is f(N(t))= m
N(t), then N(t)=( m+g) t N(0) and
Rd= m/(1-g).
• If the recruitment rate is density dependent via the Beverton-Holt model , then the total population is asymptotically constant .
• If the recruitment rate is density dependent via the Ricker model , then the total population is cyclic or chaotic .
• If the recruitment rate is density dependent via either a “modified”
BevertonHolt’s model or a “modified” Ricker’s model , then the demographic equation exhibits the Allee effect .
• References: May (1974), Hassell (1976), Castillo-Chavez-Yakubu
(2000, 2001), Franke-Yakubu (2005, 2006), Yakubu (2007).

Other important aspects of realistic demographic equations include….
• Delays and Periodic (seasonal) effects.
• Age structure and related effects.
• Genetic variations etc
• Multi-species and ecosystem effects.
• Spatial effects and diffusion (S. Levin, Amer.
Nat. 1974).
• Deterministic versus stochastic effects,
• ….
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Demographic Equation
In
Periodic Environments
If the recruitment function is p - periodically forced, then the p - periodic demographic equation is
N(t+1) = f(t,N(t))+ L N(t)
, where 0 p 5 N such that f(t,N(t)) = f(t+p,N(t)) t 5 Z
+
.
We assume throughout that f(t,N) 5 C ²( Z
+
× R
+
,
R
+
) and
L 5 (0,1).
(2)
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Examples Of Recruitment Functions In
Periodic Environments
A. Periodic constant recruitment function f(t,N(t)) = k t
(1L )
.
B. Periodic Beverton-Holt recruitment f function
( t , N ( t ))

( 1
g
)
( 1 k
-
+ g
(
) m m k
t
N
1
+
( t g
)
) N ( t )
.
1. k
2. k t t is a
= k t+p
+
.
t p
-periodic the carrying capacity.
t 5 Z
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Asymptotically Cyclic Demographics
Theorem 1 (2005): Model (2) with f(t,N(t)) = k t
(1L ) has a globally attracting positive s
- periodic cycle, where s divides p, that starts at x
0

( 1
g
)( k p
-
1
+ k p
-
2 g
1
g p
+   + k
0 g p
-
1
)
Theorem 2 (2005): Model (2) has a globally attracting positive f ( t , N ( t ))

( 1
g s
- periodic cycle when
( 1
) k t
-
+ g
(
) m k t m -
N
1
+
( t g
)
) N ( t )
.
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• Are oscillations in the carrying capacity deleterious to a population?
• Jillson, D.: Nature 1980 (Experimental results)
• Cushing, J.: Journal of Mathematical
Biology (1997).
• May, R. M.: Stability & Complexity in
Model Ecosystems (2001).
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• When the recruitment function is the period constant, then the average total biomass remains the same as the average carrying capacity (the globally attracting cycle is neither resonant nor attenuant).
• When the recruitment function is the periodic Beverton-
Holt model, then periodic environments are always disadvantageous for our population (the globally attracting cycle is attenuant, Cushing et al., JDEA 2004,
Elaydi & Sacker, JDEA 2005 ).
• When all parameters are periodically forced, then attenuance and resonance depends on the model parameters (Franke-Yakubu, Bull. Math. Biol. 2006).
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The Ricker Model in
Periodic Environments
In periodic environments, the Ricker Model exhibits multiple (coexisting) attractors via cusp bifurcation.
Reference: Franke-Yakubu JDEA 2005
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S-I-S Epidemic Models
In
Seasonal Environments
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S ( t
+
1 )

I ( t
+
1 )
 f ( t , N g
( t


 1
))
-
+



 g





I
N
I
N
( t
( t
)
( t
)
( t
)



)






S
S (
( t t
)
)
+
+ g
( 1 g
I
-
( t )

) I ( t )




, ( 3 )
Using
S(t) = N(t) - I(t) the
I
- equation becomes
I ( t
+
1 )
 g


1




I ( t )
N ( t )




( N ( t )
-
I ( t ))
+ g
I ( t )
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Let
F
N
( I )
 g


1
 
I
N

 ( N
-
I )
+ g
I .
Then
I(t+1) = F
N(t)
(I(t)), and the iterates of the nonautonomous map
F
N(t) is the set of density sequences generated by the infective equation.
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Persistence and Uniform
Persistence
Definition: The total population in Model (2) is persistent if t lim
  inf N ( t )

0 whenever
N
(0) > 0.
The total population is uniformly persistent if there exists a positive constant
R such that whenever
N
(0) > 0.
t lim
  inf N ( t )
 
Periodic constant or Beverton-Holt recruitment functions give uniformly persistent total populations.
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0
•In constant environments f(t,N(t))=f(N(t)), and
R
0
= -
JLd ′(0)/(1 – La).
Reference: Castillo Chavez and Yakubu (2001).
• In constant environments, the presence of the Allee effect in the total population implies its presence in the infective population whenever R0>1. Reference: Yakubu (2007).
• In periodic environments, if the total population is uniformly persistent then the disease goes extinct whenever R0<1.
However, the disease persists uniformly whenever R0>1.
Reference: Franke-Yakubu (2006).
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• What is the nature and structure of the basins of attraction of epidemic attractors in periodically forced discrete-time models?
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Limiting Systems Theory
(CCC, H. Thieme, and Zhao)
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Period-Doubling Bifurcations and
Chaos
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• Are disease dynamics driven by demographic dynamics? (Castillo-
Chavez & Yakubu, 2001-2002)
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Illustrative Examples: Cyclic and
Chaotic Attractors
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Illustrative Example: Multiple Attractors
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Periodic S-I-S Epidemic
Models With Delay
S(t+1)=f(t,N(t-k))+ g
S(t)G(

I(t)/N(t))+ g
I(t)(1-

),
I(t+1)= g
(1-G(

I(t)/N(t)))S(t)+ g 
I(t)
Demographic equation becomes
N(t+1)=f(t, N(t-k))+ g
N(t)
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Malaria in Seasonal Environments
……
Bassidy Dembele and
Avner Friedman
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• Malaria is one of the most life threatening tropical diseases for which no successful vaccine has been developed (UNICEF 2006
REPORT).
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• How effective is Sulfadoxine Pyrimethane
(SP) as a temporary vaccine?
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Am. J. Trop. Med. Hyg. 2002: Coulibaly et al.
Bassidy, Friedman and Yakubu, 2007
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• Protocol 2 shows no significant advantage over 1 in reducing the first malaria episode.
• Protocol 3 reduces the first episode of malaria significantly.
• Both Protocol 2 and 3 may significantly reduce the side effects of drugs because of sufficient spacing of drug administration.
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Question
• What are the effects of almost periodic environments on disease persistence and control?
• Diagana-Elaydi-Yakubu, JDEA 2007
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• In constant environments, the demographic dynamics drive the disease dynamics (CC-Y, 2001). However, in periodic environments disease dynamics are independent of the demographic dynamics.
• In constant environments, simple SIS models do not exhibit multiple attractors . However, in periodic environments the corresponding simple SIS models exhibit multiple attractors with complicated basins of attraction.
• In periodic environments, simple SIS models with no
Allee effect exhibit extreme dependence of long-term dynamics on initial population sizes . What are the implications on the persistence and control of infectious diseases?
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