MTBI Mathematical and Theoretical Biology Institute
Modeling Influenza Epidemics
and
Related Issues
Carlos Castillo-Chavez
Department of Biological Statistics and
Computational Biology
Department of Theoretical and Applied Mechanics
Current Collaborators (Direct or Indirect)
Miriam Nuno, Cornell University
Zhilan Feng, Purdue University
Maia Martcheva, Polytechnic University
Laura Jones, Cornell University
Influenza virus Picture
Historical Overview
History suggests that influenza pandemics have probably happened during at least the last
four centuries. During the 20th century, three pandemics and several "pandemic scares"
occurred.
1918: Spanish Flu
The Spanish Influenza pandemic is the catastrophe against which all modern pandemics are
measured. It is estimated that approximately 20 to 40 percent of the worldwide population became
ill and that over 20 million people died. Between September 1918 and April 1919, approximately
500,000 deaths from the flu occurred in the U.S. alone. Many people died from this very quickly.
Some people who felt well in the morning became sick by noon, and were dead by nightfall. Those
who did not succumb to the disease within the first few days often died of complications from the
flu (such as pneumonia) caused by bacteria.
One of the most unusual aspects of the Spanish flu was its ability to kill young adults. The reasons
for this remain uncertain. With the Spanish flu, mortality rates were high among healthy adults as
well as the usual high-risk groups. The attack rate and mortality was highest among adults 20 to 50
years old. The severity of that virus has not been seen again.
Historical Overview
(Taken from Web Site)
History suggests that influenza pandemics have probably happened during at least the last
four centuries. 20th century pandemics and "pandemic scares."
1918: Spanish Flu
It is estimated that approximately 20 to 40 percent of the worldwide population
became ill and that over 20 million people died. Between September 1918 and
April 1919, approximately 500,000 deaths from the flu occurred in the U.S. alone.
Some people who felt well in the morning became sick by noon, and were dead by
nightfall. Those who did not succumb to the disease within the first few days often
died of complications from the flu (such as pneumonia) caused by bacteria. With
the Spanish flu, mortality rates were high among healthy adults as well as the usual
high-risk groups. The attack rate and mortality was highest among adults 20 to 50
years old. The severity of that virus has not been seen again.
1957: Asian Flu
In February 1957, the Asian influenza pandemic was first identified in the Far East. Immunity to
this strain was rare in people less than 65 years of age, and a pandemic was predicted.
The 1957 pandemic virus was quickly identified, due to advances in scientific technology. Vaccine
was available in limited supply by August 1957. The virus came to the U.S. quietly, with a series of
small outbreaks over the summer of 1957. When U.S. children went back to school in the fall, they
spread the disease in classrooms and brought it home to their families. Infection rates were highest
among school children, young adults, and pregnant women in October 1957. Most influenza-and
pneumonia-related deaths occurred between September 1957 and March 1958. The elderly had the
highest rates of death.
During January and February 1958, there was another wave of illness among the elderly. This is an
example of the potential "second wave" of infections that can develop during a pandemic. The
disease infects one group of people first, infections appear to decrease and then infections increase
in a different part of the population. Although the Asian flu pandemic was not as devastating as the
Spanish flu, about 69,800 people in the U.S. died.
1968: Hong Kong Flu
In early 1968, the Hong Kong influenza pandemic was first detected in Hong Kong. The
first cases in the U.S. were detected as early as September of that year, but illness did not
become widespread in the U.S. until December. Deaths from this virus peaked in
December 1968 and January 1969. Those over the age of 65 were most likely to die. The
same virus returned in 1970 and 1972. The number of deaths between September 1968
and March 1969 for this pandemic was 33,800, making it the mildest pandemic in the
20th century.
1976: Swine Flu Scare
When a novel virus was first identified at Fort Dix, it was
labeled the "killer flu." Experts were extremely concerned
because the virus was thought to be related to the Spanish flu
virus of 1918. The concern that a major pandemic could sweep
across the world led to a mass vaccination campaign in the
United States. In fact, the virus--later named "swine flu"--never
moved outside the Fort Dix area. Research on the virus later
showed that if it had spread, it would probably have been much
less deadly than the Spanish flu.
1977: Russian Flu Scare
In May 1977, influenza A/H1N1 viruses isolated in northern China, spread rapidly, and
caused epidemic disease in children and young adults (< 23 years) worldwide. The 1977
virus was similar to other A/H1N1 viruses that had circulated prior to 1957. (In 1957, the
A/H1N1 virus was replaced by the new A/H2N2 viruses). Because of the timing of the
appearance of these viruses, persons born before 1957 were likely to have been exposed
to A/H1N1 viruses and to have developed immunity against A/H1N1 viruses. Therefore,
when the A/H1N1 reappeared in 1977, many people over the age of 23 had some
protection against the virus and it was primarily younger people who became ill from
A/H1N1 infections. By January 1978, the virus had spread around the world, including
the United States. Because illness occurred primarily in children, this event was not
considered a true pandemic. Vaccine containing this virus was not produced in time for
the 1977-78 season, but the virus was included in the 1978-79 vaccine.
1997: Avian Flu Scare
The most recent pandemic "scares" occurred in 1997 and 1999. In 1997, at least a few
hundred people became infected with the avian A/H5N1 flu virus in Hong Kong and 18
people were hospitalized. Six of the hospitalized persons died. This virus was different
because it moved directly from chickens to people, rather than having been altered by
infecting pigs as an intermediate host. In addition, many of the most severe illnesses
occurred in young adults similar to illnesses caused by the 1918 Spanish flu virus. To
prevent the spread of this virus, all chickens (approximately 1.5 million) in Hong Kong
were slaughtered. The avian flu did not easily spread from one person to another, and
after the poultry slaughter, no new human infections were found.
In 1999, another novel avian flu virus – A/H9N2 – was found that caused illnesses in two
children in Hong Kong. Although both of these viruses have not gone on to start
pandemics, their continued presence in birds, their ability to infect humans, and the
ability of influenza viruses to change and become more transmissible among people is an
ongoing concern.
Influenza and Cross-Immunity
OUTLINE
 Epidemiology of Influenza








Cross-Immunity
Two strain models (CHALL)
Age structure
Results
Limitations
Example
Conclusions
Future Work
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.
It is believed that interactions between human
populations with host (animal) populations such as
swine, duck, equine may be responsible for the observed
epidemic and pandemic outbreaks.
Evidence of Cross-Immunity
Study in 1979 shows that only 3% of individuals
previously exposed to strain A/Hong Kong/68 or
A/England/72 were infected with a similar subtype H3N2,
while 23% were infected without a previous exposure.
Frequency of detection of antibody-positive sera between
1977 and 1978 changed from 0% to38% for young people.
(reappearance of H1N1 subtype).
Houston study indicates that no cross-reactive immunity
exist between subtypes H1N1 and H3N2.
What is Cross-Immunity?
There are three subtypes of influenza A:
H1N1, H2N2, and H3N2
Infection with an influenza subtype A strain may provide
cross protection against other antigenically similar circulating
strains.
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
EXPERIMENTAL EVIDENCE
(1) 1974 Study < 3% With Prior
A/HONG KONG/68 (H3N2) OR
A PRIOR A/ENGLAND/72 (H3N2)
GOT
A/PORT CHALMERS/73
Vs. 23% With NO Prior Experience.
(2) 1976 Appearance of
A/VICTORIA/75 (H3N2)
Relative Frequency of First Infected/Previously Infected (By
Another Strain of H3N2( Was Approx. 41%).
EXPERIMENTAL EVIDENCE
(3) 1977 Co-Circulating H1N2 Strains
Individuals bon before 1952 “GOT” a strain of H1N1
DETECTION OF ANTIBODY-POSITIVE SERA
YOUNG: Changed From 0% to 9%.
OLDER: Did not change (remained at 9%).
(4) 1982 (Glezen) No Cross-Immunity Between Subtypes
(H1N1 & H3N2)
Facts
Experimental results indicate that Cross-immunity shares the
following features (Couch and Kasel, 1983):
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
reinfection with H1N1 may last 20 years
First Approaches
In 1975 epidemiological interference of virus populations
was introduced [Dietz].
In 1989 age-structure, proportionate mixing and crossimmunity 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.
Models with age structure
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.
a2
# of susceptible individuals with ages in (a1 , a2)
s
(
t
,
a
)
da
:

a1
at time t
a2
 i(t, a)da : # of infectious individuals with ages
a1
in (a1 , a2) at time t
a2
 r(t, a)da : # of recovered individuals with ages in (a1 , a2)
a1
at time t
Parameters





: 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.
Equations
    s(t , a)    (a)c(a) B(t ) s(t , a)   (a)s(t , a),
dt da 
 
  i(t , a)   (a)c(a) B(t ) s(t , a)  ( (a)   (a))i(t , a),

 dt da 
 
  r (t , a)   (a)i(t , a) -  (a)r (t , a).

 dt da 




B(t)   i(t,a') p(t,a,a')da'
0 n(t, a')
p(t,a,a')  c(a')n(t,a')

 c(a)n(t,a)da
0
Initial and Boundary Conditions
Initial conditions ;
s(t,0)  , i(t,0)  r(t,0)  0.
Boundary conditions ;
s(0,a)  s0 (a),
i(0,a)  i0 (a),
r(0,a)  r0 (a),
Total; populatuio n density :
n(t,a)  s(t,a)  i(t,a)  r(t,a)
Demographic Steady State
n(t,a): density of individual with age a at time t
n(t,a) satisfies the Mackendrick Equation



   n(t , a)    (a)n(t , a),
dt da 
n(t, a)  e
 0a (a)da
, as t  
We assume that the total population density has reached
this demographic steady state.
Mixing Rules
1.
p(t,a,a`)  0

2.
 p(t,a,a')da'1
0
3. c(a) p(t,a,a')n(t,a)  c(a') p(t,a',a)n(t,a')
Proportionate mixing:
p(t,a,a')  p(t,a')   c(a')n(t,a')
 c(u)n(t,u)du
0
Stability of Disease-free Steady State
Disease - free steady state : (n (a),0,0)
Variable separated solutions :
s(t, a)  e t s(a)
i(t, a)  et i(a)
r (t, a)  et r (a)
 satisfies the characteristic equation
Characteristic Equation
The characteristic equation has a unique real solution which
is dominant, that is, the real part of rest solutions is less than
This dominant solution.
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;
 Whenever R0<1, the disease-free steady state is
locally asymptotically stable.
Endemic Steady States
ds* (a)    (a)c(a) B*s* (a)   (a)s* (a)
da
di* (a)   (a)c(a) B* s* (a)  ( (a)   (a))i* (a)
da
dr * (a)   (a)i* (a) -  (a)r * (a)
da
 i * (a )
p (a)da
B*  
0 n (a)
p (a) 
c (a ) n  a )

0
 c(a)n (a)da
Endemic Steady States
Moving forward, 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.
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
Transfer Diagram:
X

Y1

Z1


Y2
V2


Z2

V1

W
NOTATION:
X
Susceptible Individuals
Yi
Individuals Infected Strain i.
Zi
Individuals Recovered Strain i.
But Susceptible to Other Strain.
Vi
Individuals Infected by Strain I but
Recovered From the Other Strain.
W
Recovered Individuals.
Hopf-Bifurcation point
 > 0 a <aC
1

C
(
)
a

<
a
>
a
w2 1  0 1
1
 
Hopf - Bifurcation


I1
a1
Hopf -Bifurcation
2
Strain 2
E
Strain 1
2
Strain 2 is present
E
0
E1
1
Strain 1 is present
*We have oscillations for a range of the quarantine period
Results
Age-structure is sufficient to drive sustained oscillations in a
multiple strain model[Castillo-Chavez, Hethcote, Andreasen,
Liu and Levin].
For a heterogeneous population and age-dependent mortality,
cross-immunity provides an explanation to the observed
recurrence of strains [Castillo-Chavez, et.al].
Under the assumption of cross-immunity and multiple strains,
quarantine periods support sustained oscillations
[Castillo-Chavez, et.al].
A model on the role of
cross-immunity and quarantine
In the following model we explore crossimmunity and quarantine periods as the
mechanisms driving the observed epidemic
outbreaks.
Multiple outbreaks appear and grow in
number as we increase the levels of cross
protection for the circulating type A influenza
strains.
I
S
 S(I  V
1

1
)
1
dI
1 1
I
Q
1
I
1
aQ
1
1
1
R
1
1
 S(I  V )
2
1
A
S
2
Q
R
1
 2 R 1 (I 2  V 2 )
2
A
A
I
Q
dI
2
2
2
2
2
V
2
I
2
2
I
2
Q a Q
1
R
V
2
 V
2
2
V
2 1R 2 (I1  V 1 )
1
V 1
A
R 2
V
1
W
1
2
W
2
The Model
 S(I  V )  S(I  V
dS


dt
A
A
dI   S( I  V )  (    d )
Ii
dt
A
i
i
i
i
i
i
j
j
j
) 
i
i
i
dQi
 d i Ii  (   a i ) Q i
dt
(I  V
dR
  I  a Q  R    R
dt
A
(I  V ) (
dV
  R
    )V
dt
A
i
j
i
i
i
i
i
i
i
i
j
i
j
i
dW
  V   V  W
dt
i
i
i
j
j
i
j
)
S
Hopf-Bifurcation point
 > 0 a <aC
1

C
(
)
a

<
a
>
a
w2 1  0 1
1
 
Hopf - Bifurcation


I1
a1
Hopf -Bifurcation
2
Strain 2
E
Strain 1
2
Strain 2 is present
E
0
E1
1
Strain 1 is present
*We have oscillations for a range of the quarantine period
Conclusions




Cross-immunity allows for the incorporation
of immune system level phenomena into
population models.
Population structure is critical in the study of
communicable diseases.
The impact of reservoir of infection needs
further study.
Identification of mechanisms capable of
generating sustained oscillations is important.
Future Work


The study of influenza and other
communicable diseases at global scales is
critical (several Russian researchers and
Longini have done some important past
work).
The impact of continued population growth
and environmental changes also must be
considered in the study of disease spread.