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Mathematical Biology
J. Math. Biol.
DOI 10.1007/s00285-013-0645-y
Seasonal dynamics in an SIR epidemic system
E. Augeraud-Véron · N. Sari
Received: 2 July 2012 / Revised: 16 January 2013
© Springer-Verlag Berlin Heidelberg 2013
Abstract We consider a seasonally forced SIR epidemic model where periodicity
occurs in the contact rate. This periodical forcing represents successions of school
terms and holidays. The epidemic dynamics are described by a switched system.
Numerical studies in such a model have shown the existence of periodic solutions.
First, we analytically prove the existence of an invariant domain D containing all
periodic (harmonic and subharmonic) orbits. Then, using different scales in time and
variables, we rewrite the SIR model as a slow-fast dynamical system and we establish
the existence of a macroscopic attractor domain K , included in D, for the switched
dynamics. The existence of a unique harmonic solution is also proved for any value
of the magnitude of the seasonal forcing term which can be interpreted as an annual
infection. Subharmonic solutions can be seen as epidemic outbreaks. Our theoretical
results allow us to exhibit quantitative characteristics about epidemics, such as the
maximal period between major outbreaks and maximal prevalence.
Keywords Periodic SIR epidemic model · Slow-fast system · Canard solution ·
Averaging · Periodic motion
Mathematics Subject Classification (2000)
34E17
92D30 · 34C15 · 34C25 · 34E15 ·
E. Augeraud-Véron · N. Sari (B)
Laboratoire de Mathématiques, Image et Applications (MIA),
Université de La Rochelle, 17042 La Rochelle, France
e-mail: [email protected]
E. Augeraud-Véron
e-mail: [email protected]
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E. Augeraud-Véron, N. Sari
1 Introduction
Epidemics are sensitive to seasonal forcing (Grassly and Fraser 2006). The cause
of seasonal patterns may vary from natural causes such as true seasons phenomena
that generate periodic temperatures, humidity profiles or periodic birth rates (He and
Earn 2007), to human forced phenomena such as the succession of school terms and
holidays. More specifically, childhood diseases such as measles, rubella and whooping
cough are highly sensitive to seasonal forcing due to the alternation of school terms
and holidays (Hethcote 2000). One of the first studies about seasonality due to school
periods is by Soper (1929), who, studying weekly periods of cases of measles in
England and Wales, demonstrated a decline of transmission during school holidays.
This schooling seasonality has also been studied by London and York (1973) and
Altizer et al. (2006).
Many epidemiological models have been studied using numerical simulations to
examine the effect of a seasonally varying contact rate on the behavior of the disease.
Dietz (1976); Grossman et al. (1977); Aron and Schwartz (1984); Schwartz (1985,
1992); Dowell (2001); Rohani et al. (2002) use continuous models of seasonality. In
these different studies, a particular interest has been taken in the computation of solutions, period doubling bifurcations and the description of basins of attraction of stable
periodic solutions. Theoretical studies of periodic continuous models were performed
by several authors (Smith (1983); Schwartz and Smith (1983)). These studies are based
on a method of continuation in bifurcation theory by Hale and Taboas (1978). They
show the existence of harmonic and subharmonic solutions near the endemic steady
state for a low mortality rate and a seasonality parameter.
According to data of Fine and Clarkson (1982), another approach has been proposed
by Schenzle (1984), describing seasonality due to school terms and holidays by termtime (switch) forcing. Many numerical studies use this kind of forcing (Earn et al.
2000; Keeling et al. 2001; Moneim 2007; Black and McKane 2010). Keeling et al.
(2001) show numerical results, detailing how seasonal forcing can be explained by
switching between attractive focuses, leading to a unique harmonic solution. When
the degree of seasonality is small, this previous study proves the existence of such a
solution using linearization and studying local dynamics around the two close focuses.
However, their method is not valid for a larger degree of seasonality.
In epidemiological studies, disease and population dynamics have different
timescales (Anderson and May 1991) due to small parameters in the models. Using
different scales in time and variables may help us to study the dynamics. These are
used to reduce the dimensionality of the model as, for example, in Song et al. (2002),
or to perform asymptotic studies.
When the epidemiological dynamics take seasonality into account, they exhibit
periodic solutions. Small parameters lead to relaxation oscillations which can be analyzed by redimensioning and rescaling (Schwartz and Smith 1983; Schwartz 1985).
Olinky et al. (2008) use different timescales to study the outcome of epidemic phenomena, defined as skips, between annual, biennial or longer periods. They provide
some results for the study of the number of skips as a function of the initial number
of susceptibles. The main tool of their study uses some variables, which may be small
in relation to some parameters, to enable them to build slow dynamics.
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Seasonal dynamics in an SIR epidemic system
Oscillations can also be generated by delay due to temporary immunity (Taylor
and Carr 2009). In this work the basic SIR model for low mortality can be nondimensionalized and rescaled as a perturbed conservative system. This allows us to
build a Taylor approximation of the Poincaré map to study oscillatory relaxations.
We consider the SIR seasonal model as a switched system with high and low contact
rates corresponding to school periods and holidays. We analytically study the existence
of an invariant domain which contains the two stable focuses and all the periodic
(harmonic and subharmonic) solutions of the seasonally switched system. Then in the
second part of our work, using different timescales and variables, we rewrite the SIR
model as a slow-fast dynamical system. Due to the switching, the system cannot be
written as a perturbation of a conservative system, as in Taylor and Carr (2009) and
in Schwartz and Erneux (1994) since the change in time and in variables needed to do
this rescaling would depend on periodic parameters.
Studying properties of slow and fast dynamics, we state the existence of a macroscopically invariant domain for the seasonally switched dynamics and the existence
and uniqueness of an annual cycle.
The article is organized as follows. In the next section we introduce the problem, the
hypothesis and the main results of our work. In Sect. 3, we prove Theorems 2 and 3.
After scaling the time and variables, in Sect. 4 we study the system in the framework of
a singular perturbation theory, especially the theory of slow-fast systems. In particular,
using a method of averaging, we give an approximation of the trajectory of the switched
system in the slow motion phase. Then we prove Theorems 4 and 5. In Sect. 5, we
extend our results to the case of an SIR model with disease-induced mortality, proving
that it does not modify qualitative results. In the final section, we conclude our article
by giving biological insights that are yielded by the theoretical results and then present
a numerical illustration of our results.
2 The problem and main results
We consider a classical SIR epidemic model, where, at time τ , the population is
composed of a number S (τ ) of susceptibles, I (τ ) of infectives and R (τ ) of removed
individuals.
Vital dynamics are taken into account by considering a natural mortality rate per
capita μ > 0, and a number μN (τ ) of births per time unit, where N (τ ) is the total host
population size. We assume, in Sect. 2, 3 and 4, that the disease is not lethal. According
to these hypotheses, the size of the population is constant N = S (τ ) + I (τ ) + R (τ ).
Let β > 0 denote the rate of transmission per infective, so that a number βNI S
of individuals leave the susceptible compartment to pass into the infective one. We
assume that β (τ ) is periodic, with period Tτ and infected individuals may recover at
the rate γ > 0.
Dynamics are therefore described by the following system
⎧ dS
SI
⎪
⎨ dτ = μ (N − S) − β(τ ) N
SI
dI
dτ = β(τ ) N − (γ + μ) I
⎪
⎩ dR
dτ = γ I − μR
(1)
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E. Augeraud-Véron, N. Sari
Let x = NS , y = NI be respectively the fraction of the population which is susceptible and infectious. According to the triangular nature of the system, the R variable
is irrelevant in this study. In the sequel, we consider the following system
dx
dτ
dy
dτ
= μ (1 − x) − β(τ )x y
(2)
= β(τ )x y − (γ + μ) y
Transmission functions, which mimic school opening and closing times with terms
and holidays were first proposed by Schenzle (1984) and are used by several authors
(Bolker and Grenfell 1993; Keeling et al. 2001; Moneim 2007; Black and McKane 2010). The transition from one period to another may be considered quasiinstantaneous in the childhood diseases model since it is governed by periods of
school terms and holidays. Hence, this binary nature of the seasonal forcing generates switched dynamics between two systems. Moreover, the amplitude of seasonality
that can be estimated from data (Fine and Clarkson 1982; Finkenstädt and Grenfell
2000) fits term-time forcing better than continuous sinusoidal forcing, so the termtime forced SIR model is well adapted when compared to the data (Earn et al. 2000).
However, continuous models are more adapted to describe changes in transmissibility
due to natural phenomena. Following this approach, that seems natural in this context,
we assume that the dynamics forced by switching, where β(τ ) is the switch signal and
defined as follows:
δ+ = 1 + δ,
δ− = 1 − δ,
β+ = β0 δ+ ,
β− = β0 δ− ,
β(τ ) is given by
β (τ ) =
τ ∈ [nTτ , nTτ + θ Tτ )
β+ for
n∈N
β− for τ ∈ [nTτ + θ Tτ , (n + 1) Tτ )
(3)
where β0 is the mean contact rate, δ ∈ (0, 1) the seasonal forcing and θ Tτ ,with
θ ∈ (0, 1), the high epidemic period length (i.e. the proportion of time per period
spent at school).
Hence a trajectory of system (2) is a concatenation of arcs of trajectories of the
following systems
dx
dτ
dy
dτ
= μ (1 − x) − β+ x y
= β+ x y − (γ + μ) y
(4)
and
dx
dτ
dy
dτ
= μ (1 − x) − β− x y
= β− x y − (γ + μ) y
integrated alternatively during the time θ Tτ and (1 − θ )Tτ .
123
(5)
Seasonal dynamics in an SIR epidemic system
β
Remember that for constant β, the Reproduction Ratio is defined by R0 = γ +μ
and
the
unstable
disease-free
equilibrium
if R0 > 1, system (2) admits two steady states,
(DFE) (1, 0) and an endemic steady state x F , y F which is a stable focus
x ,y
F
F
=
1 μ
, (R0 − 1)
R0 β
(6)
If R0 < 1, the only steady state of the dynamics is the DFE, which is stable.
This parametrization may be found in Keeling et al. (2001), but also in Black and
McKane (2010), where β (τ ) = β0 (1 + δ ter m (τ )). The effective transmission rate
is defined by
βm = β0 (θ (1 + δ) + (1 − θ ) (1 − δ))
and the reproduction ratio is defined by R0 = βγm .
Parameter μ is the inverse of the average human life time; so, if we take 75 years
as this life time, it leads to μ = 0.0133 per year, that is μ = 3.6510−5 per day.
Values of the other parameters of the model can be found in Keeling et al. (2001) for
different childhood diseases and are used in our numerical simulation to illustrate our
results. In accordance with these different data we will assume the following hypothesis
in the sequel, which guarantees the existence of asymptotically stable endemic states
for the two systems (5) and (5).
Hypothesis 1
γ + μ < β−
and that μ is small enough so that the endemic states are stable focuses.
Thus system (5) and system (5) have the unstable DFE (1, 0) and the following
stable focuses respectively
x+F , y+F =
γ + μ 1 − x F+
,μ
β+
β+ x F+
x−F , y−F =
γ + μ 1 − x−F
,μ
β−
β− x−F
(7)
We will now state the main results of this work.
Theorem 2 Under hypothesis 1, there is a positively invariant domain D ⊂ R+ ×R+
for seasonally switched system (2), delimited by an arc of trajectory of system (5) and
an arc of trajectory of system (5), and it is surrounding the two focuses (7).
Let D be the domain of the above theorem.
Theorem 3 Under hypothesis 1, for all k ∈ N∗ , seasonally switched system (2) possesses at least one kTτ -periodic solution in D.
Proof of the above theorems will be given in Sect. 3.
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E. Augeraud-Véron, N. Sari
According to the diseases data cited above, we can also consider μ as a small
parameter and make the following change of state variable y = μY . Hence system
(2) becomes
dx
dτ
dY
dτ
= μ (1 − x − β(τ )xY )
(8)
= (β(τ )x − γ − μ) Y
System (8) appears as a slow-fast system and in Sect. 4 we shall study the dynamics of
system (8) in the framework of the singular perturbation theory (the reader can find the
main tools of the theory in Kokotović and Khalil, 1986 or O’Malley 1991). Here the
fundamental tools are Thikonov’s theorem for convergence of solutions of slow-fast
systems (Kokotović et al. (1986); Lobry et al. (1998); O’Malley (1991); Tikhonov
(1952)) and an averaging method for perturbed systems (Sanders and Verhulst 1985).
We can see in Fig. 1 that the solutions of the seasonally switched system seem to
be attracted by a smaller domain K that is included in domain D.
This result is stated in Theorem 4, which is proved in Sect. 4.4 using the notion
of macroscopical invariance defined as follows: K is macroscopically invariant if any
solution of system (2), starting on the frontier ∂ K of K , remains in a neighborhood V
of K such that ∀X ∈ V, d(X, K ) = O(μ).
Theorem 4 For small μ << 1, and under hypothesis 1, there is a domain K ⊂ D
containing focuses (7); it is macroscopically and positively invariant for seasonally
switched system (2) and delimited by an arc of trajectory of system (5) and an arc of
trajectory of the averaged system
dx
dt
dz
dt
= 1−x
= β0 δm x − γ
where δm = θ δ+ + (1 − θ ) δ− and
t = μτ
z = μ ln( μy )
(9)
Figure 2 shows a harmonic solution existing in the neighborhood of point (xm , ym )
given by
(xm , ym ) =
μ
γ
,
(1 − xm )
β0 δm γ + μ
(10)
The existence and uniqueness of this harmonic solution is stated in Theorem 5, which
is proved in Sect. 4.5.
Theorem 5 For μ << 1, and under hypothesis 1, seasonally switched system (2) has
a unique Tτ -periodic solution in a neighborhood V of point (10) with diameter δ(V )
√
of order O( μ).
123
Fig. 1 Domain K (dotted line), domain D (dotted line), periodic solutions (thick line), transitory solutions entering domains D and K (thin line)
Seasonal dynamics in an SIR epidemic system
123
E. Augeraud-Véron, N. Sari
Fig. 2 Periodic solution of
period Tτ in the neighborhood of
the point (xm , ym ). Curved lines
represent isoclines x = 0 for
systems (5) and (5), the straight
line joining the focuses is the set
of points where systems (5) and
(5) have collinear vector fields
3 Proofs of theorems 2 and 3
3.1 Proof of theorem 2
Firstly, straightforward calculations show that the set of points of the phase plan,
where the vector fields associated to systems (5) and (5) are collinear, is the straight
line defined by
y=
μ
(1 − x)
γ +μ
(11)
this line contains the common DFE equilibrium and the two focuses of systems (5)
and (5).
Denoted by Δ+ and Δ− , the segments of the line given by equation (11) are delimμ
) and (x+F , y+F ), and by points (x−F , y−F ) and (1, 0).
ited respectively by points (0, γ +μ
Now we define a first return continuous map on segment Δ+ :
→ Δ+
P : Δ+
(x0 , y0 ) → (x2 , y2 )
(12)
where (x2 , y2 ) ∈ Δ+ is defined as follows.
We consider the arc of trajectory of system (5) starting at (x0 , y0 ) and ending at
the first intersection point (x1 , y1 ) with Δ− . Then we consider the arc of trajectory of
system (5) starting at (x1 , y1 ) and ending at the first intersection point (x2 , y2 ) with
Δ+ . Figure 3 shows an illustration of map P : Δ+ → Δ+
μ
It is easy to see that P(0, γ +μ
) ∈ Δ+ and P(x+F , y+F ) ∈ Δ+ , hence P(Δ+ ) ⊂ Δ+
and, by the uniqueness of the trajectories of systems (5) and (5), P has a unique fixed
point (x+ , y+ ) ∈ Δ+ . Then, the concatenation of the arc of trajectory of system (5),
starting at (x+ , y+ ) and ending at the point (x− , y− ) ∈ Δ− , and the arc of trajectory of system (5), starting at (x− , y− ) and ending at the point (x+ , y+ ), delimitate
domain D.
Domain D is positively invariant by system (2). Indeed, on frontier ∂ D of D the
vector field defined by system (2) is directed towards the interior of D or is tangent to
∂ D since this frontier consists of arcs of trajectories.
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Seasonal dynamics in an SIR epidemic system
Fig. 3 Illustration of the map
P : Δ+ → Δ+
3.2 Proof of theorem 3
Using Brouwer’s fixed point theorem, we shall prove the existence of at least one
periodic solution for system (2) in domain D. We apply this theorem to the Poincaré
map associated with the Tτ -periodic seasonal system (2)
Πk : R+ × R+ → R+ × R+
(x0 , y0 ) → (x(kTτ ), y(kTτ ))
(13)
where k ∈ N∗ and (x(τ ), y(τ )) is the solution of the periodic system (2) starting at
point (x0 , y0 ) and at time τ0 = 0. This map is obviously a continuous application.
We have already proved that D is a positively invariant compact domain for system
(2). Hence for all k ∈ N∗ we have Πk (D) ⊂ D, then, by using Brouwer’s fixed point
theorem, Πk has at least one fixed point (xk∗ , yk∗ ) in D for all k ∈ N∗ which is an initial
condition of a kTτ -periodic solution for the seasonal switched system (2).
4 The case: µ is a small parameter
In this section we consider μ as a small parameter. Before giving a description of
the dynamics of the seasonally switched system (8), we shall study, using Thikonov’s
theory of slow-fast systems, system (5) and system (5), where y = μY . Hence, we
obtain
dx
dτ
dY
dτ
= μ (1 − x − β± xY )
= (β± x − γ − μ) Y
(14)
Here we give a brief description of Tikhonov’s result: we consider two timescales, fast
time τ and slow time t = μτ , which leads us to system
dx
dt
μ dY
dt
= 1 − x − β± xY
= (β± x − γ − μ) Y
(15)
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E. Augeraud-Véron, N. Sari
We see that system (14) has two limit systems:
At fast time τ , this system is a regular perturbation of the unperturbed system
dx
dτ
dY
dτ
= 0,
= (β± x − γ ) Y
(16)
Hence Y varies quickly and is approximated by the solution of the boundary layer
equation
dY
= (β± x0 − γ ) Y
(17)
dτ
The manifold L = (x, Y ) ∈ R2 , Y = 0 is called the slow manifold (or slow curve
in our case) and it is the set of the stationary points of equation (17).
This slow curve L is attractive for x < x±F and repulsive for x > x±F where
F
x± = βγ± .
Letting (x0 , Y0 ) be an initial condition in the basin of attraction of the attractive
part of L, the solution of system (14) quickly jumps close to the slow manifold where
slow motion takes place and it is approximated by the solution of the reduced problem
dx
=1−x
dt
(18)
which is the limit system at slow time t:
dx
dt = 1 − x − β± xY
0 = (β± x − γ ) Y
(19)
Hence, by Tikhonov’s theory of slow-fast systems (see Theorem 1 of Lobry et al.
(1998), Theorem 3.1 of Kokotović et al. (1986)), we obtain the following result: If a
solution (x(τ ), Y (τ )) of system (14) starts from a point (x0 , Y0 ) such that 0 < x0 < x±F
and Y0 > 0, then Y (τ ) decreases quickly towards stable point (x0 , 0) then slow motion
takes place in the vicinity of the slow curve L.
More precisely, by applying Tikhonov’s theory, we can state the following result.
Proposition 1 Suppose hypothesis 1 holds, 0 < x0 < x±F and Y0 > 0. Let ϕ(τ ) be
the solution of equation (17) such that ϕ(0) = Y0 , let ψ(t) be the solution of equation
(18) such that ψ(0) = x0 and defined for t ∈ [0, t F ] such that ψ(t F ) = x±F . Then the
solution (x(t), Y (t)) of system (15) starting at point (x0 , Y0 ) verifies, when μ goes to
0, the following asymptotic properties
x(t) = ψ(t) + O(μ) f or 0 ≤ t ≤ t F
Y (t) = ϕ( μt ) + O(μ) f or 0 ≤ t ≤ t F
One can see in Fig. 4 that the solution of system (15) follows the attractive part
for a while until its reaches point (x±F , 0); it then follows the repulsive part of this
123
Fig. 4 Canard solutions of system (15) with β± = β+ and measles parameters in plane (x,Y) (a) and in plane (x,z = μ ln(Y )) (b)
Seasonal dynamics in an SIR epidemic system
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E. Augeraud-Véron, N. Sari
slow curve. This is a situation of canard solutions (French duck solutions). These are
solutions of slow-fast systems which, for a long time, remain close to the stable part of
the slow manifold then to the unstable part. Canard solutions were first discovered and
studied, using nonstandard singular perturbation tools, by Benoît et al. (1981); these
results are gathered in Diener and Diener (1995). A few years later classical studies
of canard solutions were performed by Dumortier and Roussarie (1996) and Eckhaus
(1983).
4.1 The slow dynamics
In order to study canard solutions in the slow dynamics of system (15), we make the
following change of variable
z
Y = eμ
This change of variable maps the Y -interval (0, 1) into the z-interval (−∞, 0). System
(15) becomes
dx
dt
dz
dt
= 1 − x − β± x exp
z
μ
(20)
= β± x − γ − μ
This system appears as a regular perturbation of the system obtained for μ = 0,
dx
dt
dz
dt
= 1−x
(21)
= β± x − γ
Hence we compute solution component z with respect to solution component x starting
from point (x0 , 0) with x0 < x±F . So, the slow motion of system (15) near slow curve
L, is approximated, in (x, z)-coordinates, by
x
z ± (x) =
x0
β± σ − γ
dσ
1−σ
z ± (x) = β± (x0 − x) + (β± − γ ) ln
1 − x0
1−x
4.2 The averaging: the slow dynamics of the seasonally switched system
Remember that in time τ , the original seasonal SIR system has period Tτ and, in time
t, the corresponding period is Tt = μTτ . Hence period Tt is small, so that in one period
the solution has a small displacement when the solution is close to the slow curve L.
To determine the long run behavior of the solution of the seasonally switched system
(8) we apply an averaging method:
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Seasonal dynamics in an SIR epidemic system
Let (tn , xn , z n )n be the sequence of the positions at each period of the trajectory of
the system
dx
dt
dz
dt
= 1 − x − β(t)x exp
= β(t)x − γ − μ
z
μ
(22)
starting from point (x0 , z 0 ) = (x0 , 0) at time t0 = 0.
That is, if we denote the solution starting from point (x0 , z 0 ) at t0 = 0
of system (20) with β± = β+ by (x+ (t, x0 , z 0 ), z + (t, x0 , z 0 )) (respectively
(x− (t, x0 , z 0 ), z − (t, x0 , z 0 )) with β± = β− ), we have
⎧
⎨ tn = nTt
xn = x− ((1 − θ )Tt , 0, (x+ (θ Tt , xn−1 , z n−1 ), z + (θ Tt , xn−1 , z n−1 ))
(23)
⎩
z n = z − ((1 − θ )Tt , 0, (x+ (θ Tt , xn−1 , z n−1 ), z + (θ Tt , xn−1 , z n−1 ))
with (x0 , z 0 ) = (x0 , 0). With the mean value theorem, the rate of variations with
respect to (tn ), of (xn ) and (z n ) are
x
n+1 −x n
tn+1 −tn
z n+1 −z n
tn+1 −tn
= 1 − xn + O(μ)
= β0 (θ δ+ + (1 − θ )δ− )xn − γ + O(μ)
(24)
Hence the solution of the seasonal system is approximated in the vicinity of slow
manifold L by the solution starting from (t0 , x0 , z 0 ) = (0, x0 , 0) of the averaged
system:
dx
dt
dz
dt
= 1−x
(25)
= β0 δm x − γ
This system is of the same type as system (21), its solution is given by
z(x) = β0 δm (x0 − x) + (β0 δm − γ ) ln
1 − x0
1−x
(26)
Let xm = δmγβ0 . It is clear to see that if x0 is in the interval (0, xm ) then there is a
unique solution x 0 > xm such that z(x 0 ) = 0. Let t 0 > 0 such that x(t 0 ) = x 0 .
We can therefore state the following result.
Proposition 2 The solution (x(t), z(t)) of seasonally switched system (22) starting
from point (x0 , z 0 ) = (x0 , 0) at time t0 = 0 is approximated by (26) in the sense
z(t) = β0 δm (x0 − x(t)) + (β0 δm − γ ) ln
1 − x0
+ O(μ) f or 0 ≤ t ≤ t 0
1 − x(t)
Figure 5 shows some solutions of the seasonally switched system (20) and their approximations given by the corresponding solutions of system (25). We can see that these
approximations are only valid when z < 0 (i.e. near the slow manifold L).
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E. Augeraud-Véron, N. Sari
Fig. 5 Averaging: Solutions
(polylines) of system (20) and
their approximations (smooth
lines) given by solution (26)
with measles parameters
4.3 The fast dynamics
To study the fast dynamics of system (14) we return to system (2), which is a regular
perturbation of the following system obtained by allowing μ = 0
dx
dτ
dy
dτ
= −β± x y
(27)
= (β± x − γ ) y
the non-trivial solution of this system starting at time τ0 = 0 from point (x 0 , 0) is
given by
y± (x) = x±F ln
x
− x + x0
x0
(28)
It is easy to show that if x 0 > x±F there is a unique point x00 ∈ 0, x±F such that
y± (x00 ) = 0. Let τ 0 > 0 such that x(τ 0 ) = x00 .
We can therefore state the following result
Proposition 3 The solution (x(τ ), y(τ )) of system (2) starting from point (x 0 , 0) at
time τ0 = 0 is approximated by (28) in the sense
y± (τ ) = x±F ln
x(τ )
− x(τ ) + x 0 + O(μ) f or 0 ≤ τ ≤ τ 0
x0
4.4 Proof of theorem 4
Using dynamical properties of averaged system (25) and system (5) we construct a
macroscopically and positively invariant compact set K delimited by a trajectory of
each system. Equation
0 = β0 δm (x0 − x1 ) + (β0 δm − γ ) ln
1 − x0
1 − x1
defines an implicit function x1 = φ (x0 ), which is a bijection from 0, x+F onto
φ 0, x+F ⊂ x+F , 1 .
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Seasonal dynamics in an SIR epidemic system
Indeed
β0 δm − γ
β0 δm − γ
0 = β0 δm −
d x0 +
− β0 δm d x1
1 − x0
1 − x1
γ − x0 β0 δm
φ (x0 ) =
<0
γ − x1 β0 δm
In the same way, equation
x+F ln
x2
− x2 + x1 = 0
x1
defines an implicit function x2 = ψ (x1 ) on the interval φ
ψ (x1 ) =
x+F x11 − 1
x+F x12 − 1
0, x+F and
<0
So ψ is a bijection on φ 0, x+F to ψ φ 0, x+F
⊂ 0, x+F .
We now consider the composed function Φ = ψ ◦ φ defined on 0, x+F into itself,
it verifies Φ(x+F ) < x+F and
Φ (x)) > 0, ∀x ∈ 0, x+F
Hence Φ has a unique fixed point x D ∈ 0, x+F .
Let x K1 = φ(x K ), then the arc of trajectory, denoted by Γ+ , of system (27) with
β± = β+ , starting from (x K1 , 0) to (x K , 0) and, the corresponding arc of trajectory, in
the (x, y) coordinates, denoted by Γ− , of system (25) starting from (x K , 0) to (x K1 , 0),
delimitate a compact domain K . This domain is also convex since the trajectories
delimiting the domain below are convex and concave respectively: indeed the second
derivatives
z (x) = (β0 δm − γ )
1
(1 − x)2
and
y (x) = −x+F
1
x2
are positive and negative respectively since β0 δm − γ > 0.
Now we show that K is a macroscopically and positively invariant set of the seasonally switched system (2).
Indeed, let us consider (x0 , y0 ) ∈ ∂ K . The trajectory (x(τ ), y(τ )), during the fast
dynamics, remains below arc Γ+ until it jumps into the vicinity of slow manifold L
with a component x(τ ) ≥ x K , since on the frontier ∂ K of K the vector field defined by
123
E. Augeraud-Véron, N. Sari
system (2) is directed, during the fast dynamics, towards the interior of K or is tangent
to ∂ K . Then slow motion takes place and it is easy to see, in the (x, z) -coordinates,
that the trajectory remains above Γ− until it once again reaches an escape point in the
vicinity of L, with a component x(τ ) ≤ x K1 . In the case where x(τ ) = x K trajectory
(x(τ ), y(τ )) remains in the vicinity of ∂ K at a distance of O(μ).
4.5 The dynamics in the neighborhood of point (xm , ym )
In this section we shall describe the behavior of the trajectories of system (2) with the
aim of proving Theorem 5. First of all, by the change of variable y = exp(z) system
(5) and system (5) become respectively
dx
dτ = μ (1 − x) − β+ x exp(z)
(29)
dz
dτ = β+ x − γ − μ
and
Let Ym± =
1−xm
β ± xm
dx
dτ
dz
dτ
= μ (1 − x) − β− x exp(z)
= β− x − γ − μ
(30)
± = ln(Y ± ), under the following change of variables and time
and z m
m
⎧
⎨ X = x − xm
√
±)
Z = μ(z − z m
√
⎩
s = μτ
(31)
system (30) and system (30) appear, respectively, as regular perturbations of the following integrable systems
1−xm
dX
ds = − xm Z (x m + X )
(32)
dZ
ds = β+ (x m + X ) − γ
and
dX
ds
dZ
ds
m
= − 1−x
xm Z (x m + X )
= β− (xm + X ) − γ
(33)
These systems have unique steady states which are centers, defined, respectively, by
+ − γ
γ
Xm , 0 =
− xm , 0
and
Xm , 0 =
− xm , 0
β+
β−
Let us denote the first integrals of systems (33) and (33) as F± (X, Z ), shown as
Z2
+ F± (X )
2
123
(34)
Seasonal dynamics in an SIR epidemic system
where
X
γ
X
F± (X ) = ± −
ln 1 +
xm
Ym
β± Ym±
(35)
Hence a trajectory of system (33) or system (33), starting from point (X 0 , 0), is defined
by
Z2
+ F± (X ) = F± (X 0 )
2
(36)
As
±
d F±
1 X − Xm
= 0
(X ) = ±
dX
Ym xm + X
± , we can define solutions of (36) as implicit functions X = ϕ ± (X , Z )
for X = X m
0
for values of X such that ddFX± (X ) = 0. Since the trajectories are symmetric with
respect to the X -axis and in order to have a one-to-one relationship between X and Z ,
we restrict ϕ ± to be defined only for Z ∈ R+ .
Denoted by T± (X 0 , Z ), times spent along the trajectories of system (33) and (33)
between the points (X 0 , 0) and ϕ ± (X 0 , Z ) , Z are given by
Z
T± (X 0 , Z ) =
0
β± xm
dζ
+ ϕ ± (X
0, ζ )
−γ
− Now consider trajectory Σ+ of system
+ (33)
starting from X m , 0 and trajectory Σ−
of system (33) starting from center X m , 0 . It is easy to see that these two trajectories
+ < X ∗ < X −.
intersect at the unique point denoted by (X ∗ , Z ∗ ) such that X m
m
∗
−
∗
∗
+
∗
Let T+ = T+ (X m , Z ) and T− = T− (X m , Z ).
Proof of theorem 5 By considering switching between systems (33) and (33), which
approximate systems (30) and (30), we can state
Lemma 1 The switched dynamics between systems (33) and (33) have a unique peri√
odic orbit of period μTμ .
Proof Remember that at time τ the period switching is equal to Tτ , hence at time s
√
this period becomes μTμ . Since T+∗ and T−∗ are of order O(1) then
T+∗ >
√ θ
μ Tμ
2
and
T−∗ >
√ 1−θ
Tμ
μ
2
+ −
Let X 0+ ∈ X m
, X m and consider the arc of trajectory γ+ starting from (X 0+ , 0) and
+
ending at X , Z + defined by
= ϕ + X 0+ , Z +
X+
√
T+ (X 0+ , Z + ) =
μ θ2 Tμ
123
E. Augeraud-Véron, N. Sari
+ −
In the same way, for X 0− ∈ X m
, X m , consider the arc of trajectory γ− starting from
(X 0− , 0) and ending at X − , Z − such that
X−
= ϕ − X 0− , Z −
√
T− (X 0− , Z − ) =
μ 1−θ
2 Tμ
√
√
Time spent on γ+ is thus μ θ2 Tμ and time spent on γ− is μ 1−θ
2 Tμ . The Implicit
function theorem implies that
d Z+
d X+
>0
+ > 0 and
d X0
d X 0+
Indeed, by definition of X + and Z + ,
+
−
1 + Xxm
+
∂ϕ + + X ,Z = =
∂ X 0+ 0
X 0+
γ
X+
xm 1 + xm − β+ 1 + xm
+
+ x + X+
X0 − Xm
m
>0
= +
+ X − Xm
xm + X 0+
d X+
d X 0+
xm 1 +
X 0+
xm
γ
β+
and
∂ T+ +
X0 , Z +
d Z+
∂ X 0+
= − ∂T +
+
d X 0+
X0 , Z +
∂ Z+
where
∂ T+
X +, Z
∂ X 0+ 0
+
Z +
=−
0
∂ϕ + X 0+ ,Z +
dz
+
∂ X0
2 < 0
β+ xm + ϕ + X 0+ , Z + − γ
∂ T+ + + 1
+
X0 , Z = +
+
∂Z
β+ xm + ϕ X 0 , Z + − γ
1
1
=
+
+ >0
β+ X 0 − X m
+ +
+ , X − ], is
We thus deduce that theset of points
X , Z , obtained as X 0+ describes [X m
m
+
+
+
+
where η is an increasing continuous function.
the graph of a function X , η X
the same
function
way, one can show that there is a decreasing continuous
In
X − , η− X − whose graph corresponds to the set of points X − , Z − obtained as
+ , X − ]. So the graphs of these two functions intersect at a unique
X 0− describes [X m
m
point. Figure 6 provides an illustration of Σ + , Σ − , γ + , γ − η+ , η− graphs and the
different points used in the proof.
123
Seasonal dynamics in an SIR epidemic system
Fig. 6 Σ + , Σ − , γ + , γ − , η+
and η− graphs and
corresponding intersection
points
Thus there is a unique solution X 0 such that η− X 0 = η+ X 0 . Let Z 0 = η+ X 0
and X 00 verifying X 0 = ϕ + X 00 , Z 0 . Hence, by symmetry of the trajectories with
respect to the X -axis, point X 0 , Z 0 is an initial condition of a periodic solution of
√
dynamics
between
systems (33) and (33).
period μTμ of the switched
√
As times T+ X 00 , Z 0 and T− X 00 , Z 0 are of order O( μ), X 0 and Z 0 are also of
√
the same order. So this periodic solution is in a ball of radius of order O( μ) centered
at the origin.
Now we complete the proof of Theorem 5. Since the solutions of systems (30) and
(30) are O(μ)-close to solutions of systems (33) and (33), by continuity of the solutions with respect to μ, a unique point (x 0 , y 0 ) exists such that x 0 − xm = O(μ)
and y 0 − ym = O(μ) which is an initial condition of a Tτ -periodic solution of
system (2).
5 Extension to SIR model with disease-induced mortality
The aim of this section is to prove that all the results we obtained can be easily
extended when a disease-induced mortality rate α > 0 is considered. Indeed, system
(1) is rewritten as
⎧ dS
SI
⎪
⎨ dτ = μ (N − S) − β(τ ) N
SI
dI
dτ = β(τ ) N − (γ + μ) I − α I
⎪
⎩ dR
dτ = γ I − μR
(37)
Population size is now variable as ddτN = −α I . Here again, the R variable is irrelevant
in this study. Letting x = NS , and y = NI , the per-capita dynamics are described by
dx
dτ
dy
dτ
= μ (1 − x) − (β(τ ) − α)x y
= β(τ )x y − (γ + μ + α) y + αy 2
(38)
It is well known that in this case, when β(τ ) = β for all τ , the reproduction number
β
R0 is R0 = γ +μ+α
. Busenberg and Van Den Driessche (1990) have provedthat if
123
E. Augeraud-Véron, N. Sari
R0 > 1 a unique endemic steady state exists which is globally asymptotically stable.
We now consider a switched model where, as before, the switching is defined on the
contact rate by (3).
dx
dτ
dy
dτ
= μ (1 − x) − (β± − α)x y
= β± x y − (γ + μ + α) y + αy 2
(39)
Hypothesis 1 is now replaced by the following assumption
Hypothesis 6
γ + μ + α < β−
and μ is small enough so that the two endemic states of systems (39) are stable focuses.
Theorem 2 relies on the existence of fixed points for application P, which is defined
on two parts of a curve where the vector fields of systems (39) are collinear. The
equation of this curve is given by
αy 2 + y(γ + α + μ − αx) + μ(1 − x) = 0
The existence of periodic solutions is obtained in the same way as in Theorem 3.
Now considering μ as a small parameter, the fast dynamics of system (39) are
studied, in the same way as for (27), by using
dx
dτ
dy
dτ
= −(β± − α)x y
(40)
= β± x y − (γ + α) y + αy 2
The non-trivial solution of this system starting at time τ0 = 0 from the point (x 0 , 0)
is given by
y(x) =
α
γ +α
x 0 β± − α
γ +α
0
−x+
x −
x
α
α
To study the slow dynamics, we need to make the change of variable and time as in
(9) in order to write the analogue of (20)
⎧
⎨ d x = (1 − x) − (β± − α) x exp z
dt
μ
⎩ dz = β± x − (γ + μ + α) + μα exp
dt
This system is a regular perturbation of
123
z
μ
(41)
Seasonal dynamics in an SIR epidemic system
dx
dt
dz
dt
= 1−x
(42)
= β± x − (γ + α)
which is similar to (21). Hence using the same method of averaging as the one carried
out in Sect. 4, we can conclude that solution (x (t) , y (t)) of the seasonally switched
system, starting from point (x0 , 0) at time t0 = 0, is approximated by
z(x) = β0 δm (x0 − x) + (β0 δm − γ − α) ln
1 − x0
1−x
The remainder of the study is similar to the previous case for establishing the existence
of the macroscopically invariant domain K and the existence of the harmonic solution
(annual epidemic).
6 Epidemiological interpretations and simulations
Under hypothesis 1, the determination of K enables us to give a good approximation
of order O (μ) of the prevalence bounds, for each set of parameters (β0 , δ, γ ) characterizing a disease, whatever the initial condition of the epidemics. Indeed since any
periodic solution of the system remains, at most, at a distance of order O (μ) from
K , this maximum can be computed, with a precision of O (μ). Using the following
formula, obtained in Theorem 4 by computing the maximum value of Γ+ where Γ+
is the upper edge of K .
ymax = x F+ ln
x F+
x K1
− x F+ + x K1
In the same way, the highest and lowest values of susceptible parts of a population
correspond respectively to
xmax = x K1
xmin = x K
Proof of Theorem 4 is based on the construction of an approximate solution obtained
by an averaging method. This averaged solution gives a macroscopical description of
epidemic spikes in such a system (Olinky et al. 2008). Time spent on this averaged
solution gives a good prediction of the time period between two major outbreaks of
the epidemics. Hence, time spent along Γ− is the upper bound of the period between
two major outbreaks for any periodic solution. Thus, for any disease, this maximum
time Tmax , at initial timescale τ , depends only on the parameters of the epidemics, as
previously, for xmax , xmin and ymax and can be computed as
123
E. Augeraud-Véron, N. Sari
Tmax =
1 1 − xK
ln
μ 1 − x K1
Therefore our theoretical results, obtained in the previous part, give qualitative characteristics of the epidemics, such as the existence of periodic solutions and the existence
of invariant domains, as well as quantitative information. In the following Table 1, we
summarize the invariant values of each epidemic, where we consider the parameters
taken from Keeling et al. (2001). All the simulations are performed with μ = 3.65 10−5
and θ = 0.65 and the parameters given in this table.
Figure 7 shows the periodic motion of the part of susceptibles (x(t)) and infectives
(y(t)). It can be seen that the solutions essentially remain in the bounds [xmin , xmax ]
and [0, ymax ] respectively. One can see that for measles, the solution is slightly over
bound ymax , but, as expected from our theoretical result (Theorem 4), it does not
exceed ymax more than O(μ).
In Fig. 8 we have represented the macroscopically invariant domain K by a dotted
line. In each figure we have also represented the small Tμ -periodic solution (harmonic
√
solution of order O( μ)). This solution surrounds point (xm , ym ). It seems likely
that a family of subharmonic solutions exists, since we can see, for example, that for
whooping cough (see Fig. 8c’) some of these subharmonic solutions remain uniformly
close to the border of domain K (see Fig. 8a’ for measles, and 8b’ for rubella).
In Fig. 9 we have also represented a simulation for system (39) to illustrate the
same phenomena as in the previous figures (macroscopically invariant domain K ,
subharmonic solution close to ∂ K , annual periodic solution, etc).
Table 1 Extremal values for susceptible, infective and the longest period between major outbreaks of
Measles, Rubella and Whooping cough
(β0 , δ, γ ) (days −1 )
xmax
xmin
ymax
Measles
(1.175, 0.25, 0.0769)
0.0967
0.0243
0.0122
5.78
Rubella
(0.311, 0.5981, 0.055)
0.241
0.046
0.039
17.15
Whooping cough
(0.664, 0.25, 0.04347)
0.0966
0.243
0.0122
5.77
Tmax (year s)
Fig. 7 Periodic solutions x(t) and y(t) of system (2) with the parameters given in days −1 for measles (a),
rubella (b) and whooping cough (c)
123
Seasonal dynamics in an SIR epidemic system
Fig. 8 Periodic solution of system (2) with the parameters given in days −1 for measles (a) in coordinates
(x, y) and (a’) in coordinates (x, z), for rubella (b) in coordinates (x, y) and (b’) in coordinates (x, z), for
whooping cough (c) in coordinates (x, y) and (c’) in coordinates (x, z)
Fig. 9 Periodic solutions of system (39) with the parameters given in days −1 for measles , and α = 0.001
123
E. Augeraud-Véron, N. Sari
Acknowledgments We would like to thank the anonymous referees most sincerely for their fruitful
comments and suggestions. We also acknowledge the support of the PRES Limousin Poitou-Charentes.
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