Existence of periodic solutions for the periodically forced SIR model
We prove that the seasonally-forced SIR model with a T-periodic forcing has a periodic solution with period T whenever the basic reproductive number R₀ > 1. The proof uses Leray – Schauder degree theory.
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irk-123456789-1771262021-02-11T01:28:48Z Existence of periodic solutions for the periodically forced SIR model Katriel, G. We prove that the seasonally-forced SIR model with a T-periodic forcing has a periodic solution with period T whenever the basic reproductive number R₀ > 1. The proof uses Leray – Schauder degree theory. Доведено, що для моделi SIR з T-перiодичною iнфекцiєю iснує перiодичний розв’язок з перiодом T, якщо основне репродуктивне число R₀ > 1. При цьому використано теорiю Лере – Шаудера. 2013 Article Existence of periodic solutions for the periodically forced SIR model / G. Katriel // Нелінійні коливання. — 2013. — Т. 16, № 3. — С. 359-366. — Бібліогр.: 19 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177126 517.9 en Нелінійні коливання Інститут математики НАН України |
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We prove that the seasonally-forced SIR model with a T-periodic forcing has a periodic solution with period T whenever the basic reproductive number R₀ > 1. The proof uses Leray – Schauder degree theory. |
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Katriel, G. |
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Katriel, G. Existence of periodic solutions for the periodically forced SIR model Нелінійні коливання |
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Katriel, G. |
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Katriel, G. |
| title |
Existence of periodic solutions for the periodically forced SIR model |
| title_short |
Existence of periodic solutions for the periodically forced SIR model |
| title_full |
Existence of periodic solutions for the periodically forced SIR model |
| title_fullStr |
Existence of periodic solutions for the periodically forced SIR model |
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Existence of periodic solutions for the periodically forced SIR model |
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existence of periodic solutions for the periodically forced sir model |
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Інститут математики НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/177126 |
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Existence of periodic solutions for the periodically forced SIR model / G. Katriel // Нелінійні коливання. — 2013. — Т. 16, № 3. — С. 359-366. — Бібліогр.: 19 назв. — англ. |
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Нелінійні коливання |
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AT katrielg existenceofperiodicsolutionsfortheperiodicallyforcedsirmodel |
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2025-07-15T15:08:54Z |
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2025-07-15T15:08:54Z |
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1837726052571414528 |
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UDC 517.9
EXISTENCE OF PERIODIC SOLUTIONS
FOR THE PERIODICALLY FORCED SIR MODEL*
IСНУВАННЯ ПЕРIОДИЧНИХ РОЗВ’ЯЗКIВ
ДЛЯ МОДЕЛI SIR З ПЕРIОДИЧНОЮ IНФЕКЦIЄЮ
G. Katriel
Ort Braude College
Karmiel, Israel
e-mail: katriel@braude.ac.il
We prove that the seasonally-forced SIR model with a T -periodic forcing has a periodic solution with
period T whenever the basic reproductive numberR0 > 1.The proof uses Leray – Schauder degree theory.
Доведено, що для моделi SIR з T -перiодичною iнфекцiєю iснує перiодичний розв’язок з перiодом
T, якщо основне репродуктивне числоR0 > 1. При цьому використано теорiю Лере – Шаудера.
1. Introduction. The periodically forced SIR model
S′ = µ(1− S)− β(t)SI, (1)
I ′ = β(t)SI − (γ + µ)I, (2)
R′ = γI − µR, (3)
and variants of it, are extensively used to model seasonally recurrent diseases [1, 2, 4, 5, 7 –
14, 16]. Here S, I, R are the fractions of the population which are susceptible, infective, and
recovered, µ denotes the birth and death rate, γ the recovery rate, and β(t), which we assume
is a positive continuous T -periodic function, is the seasonally-dependent transmission rate (so
that T is the yearly period). Since S(t), I(t), R(t) are fractions of the population we require
S(t) + I(t) +R(t) = 1. (4)
Note that by adding the equations (1) – (3) we have (S(t) + I(t) + R(t))′ = 0, so that if the
initial conditions satisfy S(0) + I(0) +R(0) = 1, (4) will hold for all t.
When simulating this model numerically, it is observed that:
(i) IfR0 ≤ 1, where
R0 =
β̄
γ + µ
,
β̄ =
1
T
T∫
0
β(t) dt,
∗ This research was supported by the EU-FP7 (grant Epiwork).
c© G. Katriel, 2013
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 3 359
360 G. KATRIEL
then all solutions tend to the disease-free equilibrium S0 = 1, I0 = 0, R0 = 0. This fact can be
rigorously proved, see [15].
(ii) If R0 > 1, then depending on the values of the parameters, one observes convergence
to T -periodic orbits, or to nT -periodic orbits with n > 1 (subharmonics), or chaotic behavior.
A fundamental question, that is addressed here, is the existence of a T -periodic solution
of the system. We demand, of course that the components S(t), I(t), R(t) of this solution will
be positive. Obviously when R0 ≤ 1, a positive periodic solution cannot exist, because such a
solution would not converge to the disease-free equilibrium. We will prove, however, that the
following theorem.
Theorem 1. Whenever R0 > 1, there exists at least one T -periodic solution (S(t), I(t), R(t))
of (1) – (3), all of whose components are positive and satisfy (4).
Thus, when T -periodic behavior is not observed in simulations, this is not due to the fact that
such a solution does not exist, but rather to the fact that all T -period solutions are unstable. Such
unstable periodic solutions cannot be found by direct simulation, but they can be computed by
numerical continuation techniques.
Despite the fact that the existence of a T -periodic solution of the T -periodically forced SIR
model is a fundamental issue, the only paper in the literature of which we are aware to have
dealt with this question is the recent paper of Jódar, Villanueva and Arenas [9]. They treated
a more general system than we do here (including loss of immunity and allowing other coeffi-
cients besides β(t) to be T -periodic). Restricting their existence result to the case of the SIR
model (1) – (3), they proved, using Mawhin’s continuation theorem, that a T -periodic solution
exists whenever the condition
min
t∈R
β(t) > γ + µ (5)
holds. Note that the condition (5) implies that β̄ > γ + µ, that is R0 > 1, but it is a stronger
condition. Theorem 1 uses only the condition R0 > 1, so that together with the fact noted
above, that when R0 ≤ 1 a T -periodic solution does not exist, we have that R0 > 1 is
a necessary and sufficient condition for the existence of a T -periodic solution with positive
components.
Our technique for proving Theorem 1 relies on nonlinear functional analysis, for which we
refer to the textbooks [3, 19]. Reformulating the problem as one of solving an equation in an
infinite dimensional space of periodic functions, we define a homotopy between the periodi-
cally forced problem and the autonomous problem in which β(t) is replaced by the mean β̄.
The autonomous problem has an endemic equilibrium, which is a trivial periodic solution. We
then employ Leray – Schauder degree theory to continue this solution along the homotopy. The
challenge here lies in the fact that there we always have a trivial periodic solution, given by the
disease-free equilibrium, which lies on the boundary of the relevant domainD in the functional
space, which requires us to construct a smaller domain U ⊂ D excluding the trivial solution,
and to show that the conditions for applying the Leray – Schauder theory hold for the domainU.
We note that our proof of Theorem 1 is easily extended to give the same result for the SIR
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 3
EXISTENCE OF PERIODIC SOLUTIONS FOR THE PERIODICALLY FORCED SIR MODEL 361
model, which includes loss of immunity
S′ = αS + µ(1− S)− β(t)SI, (6)
I ′ = β(t)SI − (γ + µ)I, (7)
R′ = γI − (µ+ α)R. (8)
We present the proof for the SIR model (α = 0) in order to avoid notational clutter.
In Section 2 we prove Theorem 1. In the discussion Section 3 we mention some other works
providing rigorous mathematical results on forced SIR models, beyond numerical simulation.
2. Proof of Theorem 1. Since R does not appear in (1), (2) and can be determined from (4)
once S and I are known, the equation (3) can be ignored, and it suffices to proved the existence
of a periodic solution of (1), (2) satisfying
S(t) > 0, I(t) > 0, S(t) + I(t) < 1 ∀t, (9)
where the third condition is equivalent to R(t) = 1− I(t)− S(t) > 0.
We decompose β(t) as
β(t) = β̄ + β0(t),
T∫
0
β0(t) dt = 0.
Setting, for λ ∈ [0, 1],
βλ(t) = β̄ + λβ0(t), (10)
we consider the system
S′ = µ(1− S)− βλ(t)SI, (11)
I ′ = βλ(t)SI − (γ + µ)I, (12)
which is homotopy between an unforced system with β0(t) = β̄ and our system (1), (2), which
corresponds to λ = 1. For λ = 0, (11), (12) has exactly two periodic solutions, which are
constant, given by
S0 = 1, I0 = 0, (13)
S∗ =
γ + µ
β̄
, I∗ = µ
(
1
γ + µ
− 1
β̄
)
. (14)
We note that (S0, I0) (the disease-free equilibrium) is in fact a (trivial) periodic solution of (11),
(12) for all λ. Our aim is to continue the solution (S∗, I∗) with respect to λ in order to prove
the existence of a periodic solution for λ = 1. To this end, we now reformulate the problem in
a functional-analytic setting, which will enable us to employ degree theory.
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 3
362 G. KATRIEL
We rewrite (11), (12) as
S′ + µS = µ− βλ(t)SI, (15)
I ′ + (γ + µ)I = βλ(t)SI. (16)
Let X, Y be the Banach spaces
X = {(S, I)|S, I ∈ C1(R), S(t+ T ) = S(t), I(t+ T ) = I(t)},
Y = {(S, I)|S, I ∈ C0(R), S(t+ T ) = S(t), I(t+ T ) = I(t)}.
Define the linear operator L : X → Y by
L(S, I) = (S′ + µS, I ′ + (γ + µ)I)
and the nonlinear operator N : Y → Y
Nλ(S, I) = (µ− βλ(t)SI, βλ(t)SI).
Then the periodic problem for (15), (16) can be rewritten as
L(S, I) = Nλ(S, I). (17)
It is easy to check that L is invertible, that is the equations S′ + µS = f and I ′ + (γ + µ)I = g
have unique C1 T -periodic solutions S, I for any f, g ∈ Y, and the mapping L−1 : Y → X
given by L−1(f, g) = (S, I) is bounded. We can thus rewrite (17) as
Fλ(S, I) = 0, (18)
where Fλ : Y → X is given by
Fλ(S, I) = (S, I)− L−1 ◦Nλ(S, I). (19)
Since L−1 : Y → X is bounded, and since, by the Arzela – Ascoli theorem, X is compactly
embedded in Y,we can consider L−1 as a compact operator from Y to itself, and sinceN : Y →
→ Y is continuous, L−1 ◦Nλ is compact as an operator from Y to itself. We therefore consider
(18) in the space Y , and we note that any solution in Y will in fact be in X , hence a classical
solution of (15), (16). Since Fλ is a compact perturbation of the identity on Y, Leray – Schauder
theory is applicable. Since we want our solution to satisfy (9), we want to solve (18) in the subset
D ⊂ Y given by
D = {(S, I) ∈ Y | S(t) > 0, I(t) > 0, S(t) + I(t) < 1}.
Note that for λ = 0 the solution (S∗, I∗) given by (14) lies in D. Our aim is to continue this
solution in λ up to λ = 1.
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EXISTENCE OF PERIODIC SOLUTIONS FOR THE PERIODICALLY FORCED SIR MODEL 363
We recall that the Leray – Schauder degree theory (see, e.g., [3, 19]) implies that, given a
bounded open set U ⊂ Y, the existence of a solution (S, I) of (18) for all λ ∈ [0, 1] will be
assured if the following conditions hold:
(I) (S∗, I∗) ∈ U,
(II) deg(F0, (S
∗, I∗)) 6= 0,
(III) Fλ(S, I) 6= 0 for all (S, I) ∈ ∂U, λ ∈ [0, 1].
The most obvious choice for U would be U = D. However, this will not do, since (S0, I0)
(given by (13)) satisfies (S0, I0) ∈ ∂D and Fλ(S0, I0) = 0, so that (III) does not hold. To satisfy
(III) we will need to choose U so as to exclude (S0, I0) from its boundary. We take U to be the
open subset of D given by
U = {(S, I) ∈ D | min
t∈R
S(t) < δ}, (20)
where δ ∈ (0, 1) is fixed. Note that (S0, I0) 6∈ Ū . We will show below that U satisfies (I) – (III)
if δ is chosen so that δ ∈ (R−10 , 1).
We first show that (S0, I0) is the only solution of (18) on ∂D.
Lemma 1. If (S, I) ∈ ∂D is a solution of (18) for some λ ∈ [0, 1], then (S, I) = (S0, I0), as
given by (13).
Proof. Assume that (S, I) ∈ ∂D is a solution of (15), (16). Note that (S, I) ∈ ∂D, if an
only if
S(t) ≥ 0, I(t) ≥ 0, S(t) + I(t) ≤ 1, (21)
and at least one of the following conditions holds:
(i) there exists t0 ∈ R so that I(t0) = 0,
(ii) there exists t0 ∈ R so that S(t0) = 0,
(iii) there exists t0 ∈ R so that S(t0) + I(t0) = 1.
We now consider each of these three cases:
(1) Assume (i) holds. Let S̃ be a solution of
S̃′ = µ(1− S̃), S̃(t0) = S(t0)
and let Ĩ(t) = 0. Then S̃, Ĩ is a solution of the initial-value problem (15), (16) with initial
condition
S̃(t0) = S(t0), Ĩ(t0) = 0.
By uniqueness of the solution for the initial-value problem, we conclude that S = S̃, I = 0.
Thus S satisfies S′ = µ(1 − S), and since the only periodic solution of this equation is S = 1,
we conclude that (S, I) = (1, 0), as we wanted to prove.
(2) Assume now that (ii) holds. Then from (15) we get S′(t0) = µ. But this implies that
S(t) < 0 for t < t0 sufficiently close to t0, which contradicts (21). Thus this case is impossible.
(3) Assume now that (iii) holds. Moreover since we have already proven the result in the
case that (i) holds, we may assume that I(t) > 0 for all t. Adding (11) and (12) we get
(S + I)′(t0) = µ(1− S(t0)− I(t0))− γI(t0) = −γI(t0) < 0.
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 3
364 G. KATRIEL
Therefore we conclude that S(t)+I(t) > 1 for t < t0 sufficiently close to t0, contradicting (21).
Therefore this case is impossible.
Lemma 1 is proved.
We can now show that U, defined by (20), satisfies (III).
Lemma 2. Assume R0 > 1. If
1
R0
< δ < 1 then, for any λ ∈ [0, 1] there are no solutions
(S, I) of (18) with (S, I) ∈ ∂U.
Proof. Suppose (S, I) ∈ ∂U. Then either (S, I) ∈ ∂D or (S, I) ∈ D and
min
t∈R
S(t) = δ. (22)
In the first case, Lemma 1 and the fact that (S0, I0) 6∈ ∂U imply that (S, I) is not a solution of
(18). We therefore assume that (S, I) ∈ D and (22) holds, which implies that
S(t) ≥ δ ∀t. (23)
Assume by way of contradiction that (S, I) solves (18), or equivalently (S, I) solves (15), (16).
Using the assumption (S, I) ∈ D, we have that I is everywhere positive, so we can divide (16)
by I, and integrate over [0, T ], to obtain
1
T
T∫
0
βλ(t)S(t) dt = γ + µ. (24)
But from (23) we get
1
T
T∫
0
βλ(t)S(t) dt ≥ δ
1
T
T∫
0
βλ(t) dt = δβ̄. (25)
By the assumption δ >
1
R0
we have δβ̄ > γ + µ, so that (25) implies
1
T
T∫
0
βλ(t)S(t) dt > γ + µ,
contradicting (24).
Lemma 2 is proved.
To apply the Leray – Schauder degree it remains to verify that (I) and (II) hold. Since S∗ =
=
1
R0
, the condition δ >
1
R0
implies (S∗, I∗) ∈ U , so (I) holds. To prove (II), it suffices to
show that the Fréchet derivative DF0(S
∗, I∗) is invertible. Since F is a compact perturbation of
the identity so that DF0(S
∗, I∗) is Fredholm, it suffices to prove that the kernel of DF0(S
∗, I∗)
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 3
EXISTENCE OF PERIODIC SOLUTIONS FOR THE PERIODICALLY FORCED SIR MODEL 365
is trivial. Indeed, let us assume that (V,W ) ∈ ker (DF0(S
∗, I∗)), and prove that (V,W ) = 0.
We have DF0(S
∗, I∗)(V,W ) = 0, or, equivalently,
L(V,W ) = DN0(S
∗, I∗)(V,W ). (26)
Note that
DN0(S
∗, I∗)(V,W ) = (−β̄(S∗W + I∗V ), β̄(S∗W + I∗V )),
so that (26) is equivalent to(
V
W
)′
=
(
−µR0 −(γ + µ)
µ(R0 − 1) 0
) (
V
W
)
. (27)
The characteristic polynomial of the above matrix is
p(x) = x2 + µR0x+ (γ + µ)µ(R0 − 1).
Noting that p(0) > 0 and that, for ω ∈ R, Im (p(ωi)) = µR0ω, we see that the matrix has no
imaginary or 0 eigenvalues, so that (27) has no periodic solutions except (V,W ) = (0, 0), and
the claim is proved.
We have thus proven that (I) – (III) hold, which completes the proof of Theorem 1.
3. Discussion. The forced SIR model is a beautiful example of a simple nonlinear dynami-
cal system which displays complicated behaviors which are difficult to understand in intuitive
terms. Moreover, these complicated behaviors are relevant to explaining the epidemiology of
infectious diseases in humans, as studies comparing the behavior of the SIR and variants of it
to surveillance data have shown [2, 5, 12]. We have proven the fundamental result that a T -
periodic solution exists for the T -periodically forced SIR model wheneverR0 > 1. As we have
stressed, this does not mean that the dynamics of the model is periodic, since the periodic soluti-
on whose existence is proved need not be stable, although one can use standard perturbation
theory to prove that the T -periodic solution is stable provided the seasonality parameter λ in
(10) is sufficiently small. Numerical simulations show that complex dynamics — subharmonic
and chaotic behavior — is very common in the forced SIR model. It is interesting to ask to what
extent the complex dynamics of the forced SIR model can be rigorously understood, beyond
numerical simulations. While we do not expect to be able to precisely characterize the dynamics
of the model for different parameter values, it is of great interest even to be able to rigourously
prove that complicated dynamics occurs for at least some parameter values. In this context we
mention the work of H. L. Smith [17, 18], who proved that the forced SIR model can have multi-
ple stable subharmonic oscillations in certain parameter ranges. Chaotic behavior has been ri-
gorously established by Glendinning and Perry [6] for a variant of the forced SIR model, in
which the dependence of the incidence term on I is nonlinear. For the standard SIR model (1) –
(3), we are not aware of a proof of chaotic behavior. Classifying and explaining the dynamical
patterns observed in simulations of the forced SIR model is still very challenging, so that, li-
ke other well-known ‘simple’ models such as the forced pendulum equation, the forced SIR
model can serve as a stimulus and as a benchmark problem for new developments in nonlinear
analysis.
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 3
366 G. KATRIEL
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Received 19.04.12,
after revision — 25.07.12
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