Explicit Lyapunov functions for SIR and SEIR compartmental epidemic models with nonlinear incidence of the form $\beta I_p S_q$ for the case p<1 are constructed.  Global stability of the models is thereby established

Korobeinikov, A.

Maini, P. K.

Mathematical Biosciences & Engineering

English

Philip K. Maini

Andrei Korobeinikov

Crossref

Mathematical Biosciences and Engineering

A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence

Mathematical Institute Eprints Archive

MATHEMATICAL BIOSCIENCES http://math.asu.edu/˜mbe/
AND ENGINEERING
Volume 1, Number 1, June 2004 pp. 57–60
A LYAPUNOV FUNCTION AND GLOBAL PROPERTIES FOR
SIR AND SEIR EPIDEMIOLOGICAL MODELS WITH
NONLINEAR INCIDENCE
Andrei Korobeinikov and Philip K. Maini
Centre for Mathematical Biology, Mathematical Institute
University of Oxford
24-29 St Giles’, Oxford, OX1 3LB, UK
(Communicated by Yang Kuang)
Abstract. Explicit Lyapunov functions for SIR and SEIR compartmental
epidemic models with nonlinear incidence of the form βIpSq for the case p ≤ 1
are constructed. Global stability of the models is thereby established.
1. Introduction. It is traditionally postulated that the spread of an infection
occurs according to the principle of mass action and associated with it the bilinear
incidence rate. However, there are a variety of reasons why this standard bilinear
incidence rate may require modification. The incidence rate of the form βIpSq,
were S and I are respectively the number of susceptible and infective individuals
in the population (or the fractions of susceptible and infective), and β, p and q are
positive constants, is the most common nonlinear incidence rate. In recent years
models with this incidence rate were considered by several authors, for example Liu
et al. [11, 12], Hethcote et al. [6], Hethcote and van den Driessche [7], Derrick and
van den Driessche [2, 3], Glendinning and Perry [4], Lizana and Rivero [13] and
others. With very few exceptions, these authors focused on the local properties
and bifurcations of equilibrium states.
In this note applying the direct Lyapunov method, we consider global properties
of SIR and SEIR models with the incidence rate of the form βIpSq for the partic-
ular case p ≤ 1. We construct a Lyapunov function which is a generalization of the
Lyapunov functions constructed earlier for models with bilinear incidence [10, 8].
We show that the condition p ≤ 1 is a sufficient condition for global stability, and
that the global properties of the systems do not depend on the value of q.
2. Main result. Following the classical assumption [1, 5], we divide a population
of a constant size into a number of classes of epidemiological significance denoting
respectively by S,E, I and R the fractions of susceptible, exposed, infective and
recovered (or removed) individuals in the population, so that S + (E) + I + R = 1
holds. For the incidence rate of the form βIpSq, the basic SIR model is
S˙ = b− βIpSq − µS,
I˙ = βIpSq − δI, (1)
2000 Mathematics Subject Classification. 92D30, 34D20.
Key words and phrases. Direct Lyapunov method, endemic equilibrium state, global stability,
nonlinear incidence.
57
58 A. KOROBEINIKOV AND P.K. MAINI
and the basic SEIR model is
S˙ = b− βIpSq − µS,
E˙ = βIpSq − σE, (2)
I˙ = θE − δI.
Here b is the birth rate, µ is the susceptible death rate, δ is the infective removal
rate (including the mortality rate) and θ is the rate with which the exposed popu-
lation moves into the infective class; σ ≥ θ includes also mortality of the exposed
individuals. We omit the equations for the recovered population R; the constant
population size assumption enables us to do so.
If 0 < p < 1 holds, then each of these systems has two equilibrium states: an
infection-free equilibrium Q0 with the coordinates S0 = b/µ,E0 = I0 = 0, and an
endemic equilibrium state Q∗ = (S∗, E∗, I∗), such that
BδI∗ = βI∗
p
S∗
q
, µS∗ + BδI∗ = b, δI∗ = θE∗. (3)
Here B = σ/θ for the SEIR system, or B = 1 for the SIR system. (In the case
p > 1, these systems have either one, two, or no equilibrium states [11].)
The following Theorem provides global properties of the systems (1) and (2).
Theorem 2.1. If p ≤ 1, then the endemic equilibrium states Q∗ of the models (1)
and (2) are globally asymptotically stable. The stability does not depend on the
value of the parameter q.
Proof. Assume that p, q 6= 1. Then for these SIR and SEIR models we
construct a Lyapunov function of the form
V (S,E, I) = AS
(
1 +
1
q − 1
(
S∗
S
)q)
+BI
(
1 +
1
p− 1
(
I∗
I
)p)
+ C (E − E∗ ln E) . (4)
Here A = 1, B = σ/θ and C = 1 for the SEIR system, or A = B = 1 and C = 0
for the SIR system. This function is defined and continuous for all S,E, I > 0.
The function V (S,E, I) satisfies
∂V
∂S
= A
(
1−
(
S∗
S
)q)
,
∂V
∂E
= C
(
1−
E∗
E
)
,
∂V
∂I
= B
(
1−
(
I∗
I
)p)
,
and hence it is easy to see that Q∗ is the only extremum and the global minimum of
the function in the positive octant R3+. Consequently, the function (4) is indeed a
Lyapunov function [14]. This function is a generalization of the Lyapunov functions
constructed in [10, 8] for the case of the nonlinear incidence rate βIpSq.
In the case of the SEIR system (2), using the equalities
b = µS∗ + BδI∗, βI∗
p
S∗
q
= BδI∗, δI∗ = θE∗, Bθ = σ
LYAPUNOV FUNCTION FOR EPIDEMIC MODELS 59
for the equilibrium state Q∗, the function V (S,E, I) satisfies
V˙ = b− βIpSq − µS − b
S∗
q
Sq
+ βIpS∗
q
+ µ
S∗
q
Sq−1
+βIpSq − σE − βIpSq
E∗
E
+ σE∗ + B
(
θE − δI − θ
I∗
p
Ip
E + δ
I∗
p
Ip−1
)
= BδI∗
(
2−
S∗
q
Sq
+
Ip
I∗p
−
E∗
E
SqIp
S∗qI∗p
−
I
I∗
−
E
E∗
I∗
p
Ip
+
(
I
I∗
)1−p)
+µS∗
(
1−
S
S∗
−
S∗
q
Sq
+
(
S
S∗
)1−q)
= −BδI∗
(
v1−p − 1
)
(vp − 1) + BδI∗
(
3−
1
uq
− w −
uq
w
)
+µS∗ (1− u)
(
1−
1
uq
)
.
Here u = S/S∗, v = I/I∗ and w = EI∗p/E∗Ip. If p < 1, then
h(v) = (vp − 1)(v1−p − 1) ≥ 0 for all v > 0.
Since the arithmetical mean is greater than, or equal to the geometrical mean,
g(u,w) = 3−
1
uq
− w −
uq
w
≤ 0 for all u,w, q > 0
(where the equality holds only if u = w = 1). Furthermore,
(1− u)
(
1−
1
uq
)
≤ 0 for all u, q > 0.
Therefore, the condition p < 1 ensures that
dV
dt
≤ 0 for all S,E, I > 0, where the
equality holds only at the equilibrium point Q∗ = (S∗, E∗, I∗). By the Lyapunov
asymptotic stability theorem [14], the equilibrium point Q∗ is globally asymptoti-
cally stable. This result is valid for the whole positive octant R3+.
If either p = 1, or q = 1, then we have to replace in the equation (4) the
corresponding term, I
∗
p
p−1I
1−p, or S
∗
q
q−1S
1−q respectively, by the term −I∗ ln I, or
−S∗ ln S. If p = q = 1, that is in the case of the bilinear incidence rate βSI, the
function (4) takes the form
V (S, I) = A (S − S∗ ln S) + B (I − I∗ ln I) + C (E − E∗ ln E) .
This Lyapunov function has been constructed for three- and four-compartmental
models [10, 8] with the standard bilinear incidence.
The case of the SIR system (1) is analogous.
It is obvious that this result does not depend on the value of the parameter q. 
3. Discussion and conclusion. In this paper we proved global stability of the
SIR and SEIR models with the incidence rate βIpSq. This conclusion is valid
under the constant population size assumption, and under the condition p ≤ 1.
In general, this global stability is associated with concavity of the incidence rate
with respect to the variable I (see [9] for details). We have to note that in the
case p > 1 the systems can have either one, two, or no equilibrium states, and lose
stability [12, 11].
60 A. KOROBEINIKOV AND P.K. MAINI
The condition p ≤ 1 may arise as a consequence of saturation effects: when
the number of the infectives is very high, so that exposure to the disease agent is
virtually certain, the disease transmission or the incidence rate will respond more
slowly than linearly to the increase in I. The case of the incidence rate being
convex with respect to I (that is the case p > 1) is hardly common: it may arise in
some particular cases as a consequence of community effects (we discuss this matter
in [9]).
Finally, we point out one particular feature of the incidence rate βIpSq : for all
p < 1 the basic reproduction number R0 of a model is equal to zero, whereas for
p > 1 it tends to infinity.
Acknowledgments. A. K. is supported by the Foundation for Research, Science
and Technology of New Zealand, through Project contract UOXX0101.
REFERENCES
[1] R.M. Anderson and R.M. May, Infectious Diseases in Humans: Dynamics and Control,
Oxford University Press, Oxford, 1991.
[2] W.R. Derrick and P. van den Driessche, emphA disease transmission model in a nonconstant
population, J. Math. Biol., 31 (1993), 495–512.
[3] W.R. Derrick and P. van den Driessche, Homoclinic orbits in a disease transmission model
with nonlinear incidence and nonconstant population, Discreate and Contin. Dyn. Syst. Ser.
B, 3 (2003) 299–309.
[4] P. Glendinning and L.P. Perry, Melnikov analysis of chaos in a simple epidemiological model,
J. Math. Biol., 35 (1997), 359–373.
[5] H.W. Hethcote, The Mathematics of infectious diseases, SIAM Rev., 42 (4), (2000), 599–653.
[6] H.W. Hethcote, M.A. Lewis and P. van den Driessche, An epidemiological model with delay
and a nonlinear incidence rate, J. Math. Biol., 27 (1989), 49–64.
[7] H.W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear inci-
dence, J. Math. Biol., 29 (1991), 271–287.
[8] A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic
models, Math. Med. Biol. J. IMA, to appear.
[9] A. Korobeinikov and P.K. Maini, Non-linear incidence and stability of infectious disease
models, preprint (2004).
[10] A. Korobeinikov and G.C. Wake, Lyapunov functions and global stability for SIR, SIRS and
SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955–961.
[11] W.M. Liu, H.W. Hethcote and S.A. Levin, Dynamical behavior of epidemiological models
with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359–380.
[12] W.M. Liu, S.A. Levin and Y. Isawa, Influence of nonlinear incidence rates upon the behaviour
of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187–204.
[13] M. Lizana and J. Rivero, Multiparametric bifurcations for a model in epidemiology, J. Math.
Biol., 35 (1996), 21–36.
[14] A.M. Lyapunov, The General Problem of the Stability of Motion, Taylor & Francis, London,
1992.
Received on Feb. 11, 2004. Revised on Feb. 15, 2004.
E-mail address: korobein@maths.ox.ac.uk
E-mail address: maini@maths.ox.ac.uk


Oxford University Research Archive (ORA)

Explicit Lyapunov functions for SIR and SEIR compartmental epidemic models with nonlinear incidence of the form βIpSq for the case p≤1 are constructed. Global stability of the models is thereby established

Korobeinikov, Andrei

Maini, Philip K.

Hokkaido University Collection of Scholarly and Academic Papers

Explicit Lyapunov functions for SIR and SEIR compartmental epidemic models with nonlinear incidence of the form betaI(p)S(q) for the case 

Korobeinikov, A

Maini, Philip

Oxford University Research Archive

A lyapunov function and global properties for sir and seir epidemiological models with nonlinear incidence.

Explicit Lyapunov functions for SIR and SEIR compartmental epidemic models with nonlinear incidence of the form betaI(p)S(q) for the case p </= 1 are constructed. Global stability of the models is thereby established

Maini, PK

MATHEMATICAL BIOSCIENCES http://math.asu.edu/˜mbe/AND ENGINEERINGVolume 1, Number 1, June 2004 pp. 57–60A LYAPUNOV FUNCTION AND GLOBAL PROPERTIES FORSIR AND SEIR EPIDEMIOLOGICAL MODELS WITHNONLINEAR INCIDENCEAndrei Korobeinikov and Philip K. MainiCentre for Mathematical Biology, Mathematical InstituteUniversity of Oxford24-29 St Giles’, Oxford, OX1 3LB, UK(Communicated by Yang Kuang)Abstract. Explicit Lyapunov functions for SIR and SEIR compartmentalepidemic models with nonlinear incidence of the form βIpSq for the case p ≤ 1are constructed. Global stability of the models is thereby established.1. Introduction. It is traditionally postulated that the spread of an infectionoccurs according to the principle of mass action and associated with it the bilinearincidence rate. However, there are a variety of reasons why this standard bilinearincidence rate may require modification. The incidence rate of the form βIpSq,were S and I are respectively the number of susceptible and infective individualsin the population (or the fractions of susceptible and infective), and β, p and q arepositive constants, is the most common nonlinear incidence rate. In recent yearsmodels with this incidence rate were considered by several authors, for example Liuet al. [11, 12], Hethcote et al. [6], Hethcote and van den Driessche [7], Derrick andvan den Driessche [2, 3], Glendinning and Perry [4], Lizana and Rivero [13] andothers. With very few exceptions, these authors focused on the local propertiesand bifurcations of equilibrium states.In this note applying the direct Lyapunov method, we consider global propertiesof SIR and SEIR models with the incidence rate of the form βIpSq for the partic-ular case p ≤ 1. We construct a Lyapunov function which is a generalization of theLyapunov functions constructed earlier for models with bilinear incidence [10, 8].We show that the condition p ≤ 1 is a sufficient condition for global stability, andthat the global properties of the systems do not depend on the value of q.2. Main result. Following the classical assumption [1, 5], we divide a populationof a constant size into a number of classes of epidemiological significance denotingrespectively by S,E, I and R the fractions of susceptible, exposed, infective andrecovered (or removed) individuals in the population, so that S + (E) + I + R = 1holds. For the incidence rate of the form βIpSq, the basic SIR model isṠ = b − βIpSq − µS,İ = βIpSq − δI, (1)2000 Mathematics Subject Classification. 92D30, 34D20.Key words and phrases. Direct Lyapunov method, endemic equilibrium state, global stability,nonlinear incidence.5758 A. KOROBEINIKOV AND P.K. MAINIand the basic SEIR model isṠ = b − βIpSq − µS,Ė = βIpSq − σE, (2)İ = θE − δI.Here b is the birth rate, µ is the susceptible death rate, δ is the infective removalrate (including the mortality rate) and θ is the rate with which the exposed popu-lation moves into the infective class; σ ≥ θ includes also mortality of the exposedindividuals. We omit the equations for the recovered population R; the constantpopulation size assumption enables us to do so.If 0 < p < 1 holds, then each of these systems has two equilibrium states: aninfection-free equilibrium Q0 with the coordinates S0 = b/µ,E0 = I0 = 0, and anendemic equilibrium state Q∗ = (S∗, E∗, I∗), such thatBδI∗ = βI∗pS∗q, µS∗ + BδI∗ = b, δI∗ = θE∗. (3)Here B = σ/θ for the SEIR system, or B = 1 for the SIR system. (In the casep > 1, these systems have either one, two, or no equilibrium states [11].)The following Theorem provides global properties of the systems (1) and (2).Theorem 2.1. If p ≤ 1, then the endemic equilibrium states Q∗ of the models (1)and (2) are globally asymptotically stable. The stability does not depend on thevalue of the parameter q.Proof. Assume that p, q 6= 1. Then for these SIR and SEIR models weconstruct a Lyapunov function of the formV (S,E, I) = AS(1 +1q − 1(S∗S)q)+BI(1 +1p − 1(I∗I)p)+ C (E − E∗ ln E) . (4)Here A = 1, B = σ/θ and C = 1 for the SEIR system, or A = B = 1 and C = 0for the SIR system. This function is defined and continuous for all S,E, I > 0.The function V (S,E, I) satisfies∂V∂S= A(1 −(S∗S)q),∂V∂E= C(1 −E∗E),∂V∂I= B(1 −(I∗I)p),and hence it is easy to see that Q∗ is the only extremum and the global minimum ofthe function in the positive octant R3+. Consequently, the function (4) is indeed aLyapunov function [14]. This function is a generalization of the Lyapunov functionsconstructed in [10, 8] for the case of the nonlinear incidence rate βIpSq.In the case of the SEIR system (2), using the equalitiesb = µS∗ + BδI∗, βI∗pS∗q= BδI∗, δI∗ = θE∗, Bθ = σLYAPUNOV FUNCTION FOR EPIDEMIC MODELS 59for the equilibrium state Q∗, the function V (S,E, I) satisfiesV̇ = b − βIpSq − µS − bS∗qSq+ βIpS∗q+ µS∗qSq−1+βIpSq − σE − βIpSqE∗E+ σE∗ + B(θE − δI − θI∗pIpE + δI∗pIp−1)= BδI∗(2 −S∗qSq+IpI∗p−E∗ESqIpS∗qI∗p−II∗−EE∗I∗pIp+(II∗)1−p)+µS∗(1 −SS∗−S∗qSq+(SS∗)1−q)= −BδI∗(v1−p − 1)(vp − 1) + BδI∗(3 −1uq− w −uqw)+µS∗ (1 − u)(1 −1uq).Here u = S/S∗, v = I/I∗ and w = EI∗p/E∗Ip. If p < 1, thenh(v) = (vp − 1)(v1−p − 1) ≥ 0 for all v > 0.Since the arithmetical mean is greater than, or equal to the geometrical mean,g(u,w) = 3 −1uq− w −uqw≤ 0 for all u,w, q > 0(where the equality holds only if u = w = 1). Furthermore,(1 − u)(1 −1uq)≤ 0 for all u, q > 0.Therefore, the condition p < 1 ensures thatdVdt≤ 0 for all S,E, I > 0, where theequality holds only at the equilibrium point Q∗ = (S∗, E∗, I∗). By the Lyapunovasymptotic stability theorem [14], the equilibrium point Q∗ is globally asymptoti-cally stable. This result is valid for the whole positive octant R3+.If either p = 1, or q = 1, then we have to replace in the equation (4) thecorresponding term, I∗pp−1I1−p, or S∗qq−1S1−q respectively, by the term −I∗ ln I, or−S∗ ln S. If p = q = 1, that is in the case of the bilinear incidence rate βSI, thefunction (4) takes the formV (S, I) = A (S − S∗ ln S) + B (I − I∗ ln I) + C (E − E∗ ln E) .This Lyapunov function has been constructed for three- and four-compartmentalmodels [10, 8] with the standard bilinear incidence.The case of the SIR system (1) is analogous.It is obvious that this result does not depend on the value of the parameter q. 3. Discussion and conclusion. In this paper we proved global stability of theSIR and SEIR models with the incidence rate βIpSq. This conclusion is validunder the constant population size assumption, and under the condition p ≤ 1.In general, this global stability is associated with concavity of the incidence ratewith respect to the variable I (see [9] for details). We have to note that in thecase p > 1 the systems can have either one, two, or no equilibrium states, and losestability [12, 11].60 A. KOROBEINIKOV AND P.K. MAINIThe condition p ≤ 1 may arise as a consequence of saturation effects: whenthe number of the infectives is very high, so that exposure to the disease agent isvirtually certain, the disease transmission or the incidence rate will respond moreslowly than linearly to the increase in I. The case of the incidence rate beingconvex with respect to I (that is the case p > 1) is hardly common: it may arise insome particular cases as a consequence of community effects (we discuss this matterin [9]).Finally, we point out one particular feature of the incidence rate βIpSq : for allp < 1 the basic reproduction number R0 of a model is equal to zero, whereas forp > 1 it tends to infinity.Acknowledgments. A. K. is supported by the Foundation for Research, Scienceand Technology of New Zealand, through Project contract UOXX0101.REFERENCES[1] R.M. Anderson and R.M. May, Infectious Diseases in Humans: Dynamics and Control,Oxford University Press, Oxford, 1991.[2] W.R. Derrick and P. van den Driessche, emphA disease transmission model in a nonconstantpopulation, J. Math. Biol., 31 (1993), 495–512.[3] W.R. Derrick and P. van den Driessche, Homoclinic orbits in a disease transmission modelwith nonlinear incidence and nonconstant population, Discreate and Contin. Dyn. Syst. Ser.B, 3 (2003) 299–309.[4] P. Glendinning and L.P. Perry, Melnikov analysis of chaos in a simple epidemiological model,J. Math. Biol., 35 (1997), 359–373.[5] H.W. Hethcote, The Mathematics of infectious diseases, SIAM Rev., 42 (4), (2000), 599–653.[6] H.W. Hethcote, M.A. Lewis and P. van den Driessche, An epidemiological model with delayand a nonlinear incidence rate, J. Math. Biol., 27 (1989), 49–64.[7] H.W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear inci-dence, J. Math. Biol., 29 (1991), 271–287.[8] A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemicmodels, Math. Med. Biol. J. IMA, to appear.[9] A. Korobeinikov and P.K. Maini, Non-linear incidence and stability of infectious diseasemodels, preprint (2004).[10] A. Korobeinikov and G.C. Wake, Lyapunov functions and global stability for SIR, SIRS andSIS epidemiological models, Appl. Math. Lett., 15 (2002), 955–961.[11] W.M. Liu, H.W. Hethcote and S.A. Levin, Dynamical behavior of epidemiological modelswith nonlinear incidence rates, J. Math. Biol., 25 (1987), 359–380.[12] W.M. Liu, S.A. Levin and Y. Isawa, Influence of nonlinear incidence rates upon the behaviourof SIRS epidemiological models, J. Math. Biol., 23 (1986), 187–204.[13] M. Lizana and J. Rivero, Multiparametric bifurcations for a model in epidemiology, J. Math.Biol., 35 (1996), 21–36.[14] A.M. Lyapunov, The General Problem of the Stability of Motion, Taylor & Francis, London,1992.Received on Feb. 11, 2004. Revised on Feb. 15, 2004.E-mail address: korobein@maths.ox.ac.ukE-mail address: maini@maths.ox.ac.uk

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A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence

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