Mathematics for Life Sciences and Medicine - Takeuchi Iwasa and Sato

52 Yasuhiro Takeuchi and Wanbiao Ma

vital dynamics (see, for example, Kermack and McKendrick (1927); Thieme (2003); Ruan (2005)):

dS = −βS(t)I(t) , dt

dI = βS(t)I(t) − γI(t) , (1) dt

dR = γI(t) . dt

In the model, it is assumed that there is no birth and death in the population and that the total numbers of population is constant, i. e., N (t) = S(t)+I(t)+ R(t) = const.. The positive constants β and γ are called the daily contact rate and the daily recovery rate, respectively. A complete theoretical analysis on dynamics properties of (1) can be done easily. A simple but very primary SIR epidemic model with vital dynamics is proposed and studied by Hethcote (1976):

where the total number of population is assumed to be N (t) = S(t) + I(t) + R(t) = 1, the positive constant µ is the birth and death rate. For the SIR model (2), the following result is well known:

Theorem 1 (Hethcote 1976). Let

β

σ = µ + γ .

If σ ≤ 1, then the disease free equilibrium

E0 = (1, 0, 0)

is globally asymptotically stable.

If σ > 1, then the disease free equilibrium E0 becomes unstable and the

is globally asymptotically stable.

Theorem 1 implies that, for a disease with vital dynamics where recovery gives permanent immunity to the disease, if the infectious contact number σ exceeds one, then the susceptible, infective and recovered fractions eventually approach constant positive endemic values. If the infectious contact number is

less than or equal to one, then the infective and recovered fractions eventually approach zero, and hence, the all population member becomes eventually susceptible.

Based on the Kermack-McKendrick model, various epidemic models have been developed in recent decades, such as SI models, SIS models, SIRS models, SIR models, SEIRS models, SEIR models with or without time delays (see, for example, Smith (1983); Liu et al. (1986); Mena-Lorca and Hethcote (1996); Cooke and van den Driessche (1996); Cooke et al. (1999); Hethcote and van den Driessche (2000); Chen et al. (2002); Ma et al.(2004) and the references there in).

The purpose of the chapter is to give a survey on the recent researches on SIR models with time delays for epidemic which spreads in a human population via a vector. In the next section, based on Hethcote model (2) and Cooke’s SIS model with a time delay, we introduce SIR models with time delays and a constant population size. Further, the SIR models are modified in such a way that the death rates for three population classes are di erent. In Sect. 3.3, the models are revised to assume that the birth rate is not independent of the total population size. We assume the the number of newborns is proportional to the total population size. In the final section, the number is assumed to saturate when the total population size increases. For all models, we summarize the known mathematical results on stability of the equilibria and permanence. We also give some open problems and our conjectures on the threshold for an epidemic to occur.

3.2 SIR epidemic models with constant birth rate and time delays

3.2.1 Stability analysis of SIR epidemic models with constant population

It is well known that Cooke (1979) proposed an SIS model for epidemic which spreads in a human population via a vector (such as mosquito etc). The model considers the case where susceptible individuals (denoted by S(t)) receive the infection from an infectious vector, and susceptible vectors receive the infection from infectious individuals. If it is assumed that when a susceptible vector is infected by an infectious person, there is a time τ > 0 during which the infectious agents develop in the vector and it is only after that time that the infected vector becomes itself infectious. Cooke’s epidemic models with time delays are stated as follows:

where y(t) denotes the infective individuals of population who are infectious at time t. b > 0 and c > 0 are the contact rate and the recovery rate,

∞

sf (s) ds < ∞

0

for (5). It is assumed that the total number of populations is constant, i. e.,

N (t) = S(t) + I(t) + R(t) = 1 .

Note that models (4)–(6) completely have the same equilibria as system (2), i. e., there always exists the disease free equilibrium E0 = (1, 0, 0); if the infectious contact number is larger than one,

β

σ = µ + γ > 1 ,

then, there also exists the endemic equilibrium

1 µ(σ − 1) γ(σ − 1)

E+ = σ , β , β .

From the mathematical and biological points of view, it is usually an important problem to give a complete analysis for the local stability and global stability of the disease free equilibrium E0 and the endemic equilibrium E+. The models (4) and (5) belong to a class of functional di erential equations with infinite time delays, their stability analysis is closely related to the choice of phase spaces. However, with help of the well known linear chain technique, the model (4) can be easily induced into a high-dimensional nonlinear ordinary di erential equations, and hence, the global stability of its nonnegative equilibira can be discussed based on the method of Liapunov functions (Beretta, Capasso and Rinaldi (1988)). For the models (5) and (6), the following results are obtained by Beretta and Takeuchi (1995) by constructing proper Liapunov functionals:

Theorem 2 (Beretta and Takeuchi 1995). For the model (5), if σ < 1, then the disease free equilibrium E0 = (1, 0, 0) is locally asymptotically stable; if σ > 1, then the disease free equilibrium E0 becomes unstable, and the endemic equilibrium E+ is locally asymptotically stable.

Theorem 3 (Beretta and Takeuchi 1995). For the model (6), if σ < 1, then the disease free equilibrium E0 = (1, 0, 0) is globally asymptotically stable; if σ > 1, then the disease free equilibrium E0 becomes unstable, and the endemic equilibrium E+ is locally asymptotically stable.

3.2.2 Stability analysis of SIR epidemic models with varying population

Beretta and Takeuchi (1997) remove some unrealistic assumptions on the parameters in the model (6). It is assumed that the death rates of three classes

56 Yasuhiro Takeuchi and Wanbiao Ma

of the population are di erent, that is susceptibles, infectives and recovered have µi(i = 1, 2, 3) as their death rates respectively. It is further assumed that the birth rate b > 0 is di erent from the death rate. All newborns are assumed to be susceptible, again. Then we have the following SIR epidemic models with distributed time delays:

where the positive constants β and γ have same biological meanings as in the model (1). The constant h ≥ 0 is a time delay. The kernel f (s) is continuous, nonnegative and assumed to satisfy the condition

h

f (s) ds = 1 .

0

For the models (7), there always exists the disease free equilibrium E0 =

For the models (7), the following results are obtained by Beretta and Takeuchi (1997):

Theorem 4 (Beretta and Takeuchi 1997). If S0 < S , then the disease free equilibrium E0 = (S0, 0, 0) is globally asymptotically stable; if S0 > S , then the disease free equilibrium E0 becomes unstable, and the endemic equilibrium E+ = (S , I , R ) is locally asymptotically stable. Furthermore, if S0 > S , it is possible to construct an explicit region Ω such that, for any initial function belonging to Ω, the solutions (S(t), I(t), R(t)) of (7) tends to E+ = (S , I , R ) as t tends infinity.

Takeuchi, Ma and Beretta (2000) further considered attractivity of the disease free equilibrium E0 of (7) while S0 = S , by using Liapunov-LaSalle invariant principle (Hale, 1977; Kuang, 1993), and had the following result:

Theorem 5 (Takeuchi, Ma and Beretta 2000). If S0 = S , then the disease free equilibrium E0 = (S0, 0, 0) is globally attractive.

Theorem 5 shows that the disease will eventually disappear under any length of time delay h whenever the endemic equilibrium does not exist. For global stability of the endemic equilibrium E+ = (S , I , R ) of (7), based on some inequality techniques, the following results are obtained:

Theorem 6 (Takeuchi, Ma and Beretta 2000). If

then the endemic equilibrium E+ = (S , I , R ) is globally asymptotically stable.

Theorem 7 (Takeuchi, Ma and Beretta 2000). If

then the endemic equilibrium E+ = (S , I , R ) is globally asymptotically stable.

In mathematics, Theorems 6–7 give the su cient conditions to ensure the global asymptotic stability of the endemic equilibrium E+ whenever it exists. In biology, Theorems 6–7 show that, while E+ exists, the disease always remains endemic if the time delay h is short enough and the product βb of

58 Yasuhiro Takeuchi and Wanbiao Ma

the contact constant β and the birth rate b is relatively large (or the death rates µ1 and µ2, and the recovery rate γ are small enough).

Based on Hethcote’s result, i. e., Theorem 1 for the SIR epidemic model (2) without delay, and general properties for delay di erential equations, the following natural conjecture was proposed by Takeuchi, Ma and Beretta:

Problem 1 (Takeuchi, Ma and Beretta, 2000). For su ciently small time delay h ≤ h0, condition (8) should imply the global asymptotic stability of the endemic equilibrium E+, i. e. condition (8) should be the threshold of (7) for an epidemic to occur.

In fact, note that Theorems 4 and 5 for the SIR epidemic model (7) with time delays and numerical simulations given by Ma, Takeuchi, Hara and Beretta (2002), it is strongly suggested that the follwoing more general conclusion may be true:

Problem 2 (Ma, Takeuchi, Hara and Beretta 2002). For any time delay h, condition (8) should imply the global asymptotic stability of the endemic equilibrium E+.

Unfortunately, Theorems 6 and 7 need more restrictive conditions in order to ensure the global asymptotic stability of the endemic equilibrium E+.

Under the assumption of permanence of model, Beretta, Hara, Ma and Takeuchi (2001) gave a positive answer for the above Problem 1 for a class of SIR epidemic model with more general time delays by constructing a complicated Liapunov functional.

Consider the following SIR epidemic model with distributed time delays:

where the positive constants b, β, µi(i = 1, 2, 3) and γ have same biological meanings as in the model (7). The constant h ≥ 0 is a time delay. The function η(s) is nondecreasing and has bounded variation such that

h

dη(s) = η(h) − η(0) = 1 .

0

The model (9) has the same the equilibria as the model (7). The following result was proved by Beretta et al.:

Theorem 8 (Beretta et al. 2001). Suppose that S0 > S , and the model (9) is permanent, that is, there are positive constants νi and Mi(i =

hold. Then the endemic equilibrium E+ = (S , I , R ) of (9) is globally asymptotically stable, if the time delay h is small enough such that

h

H ≡ s dη(s) < H0 ,

0

where the positive constant H0 can be explicitly expressed by the parameters in (9) and the constants νi and Mi(i = 1, 2, 3).

3.2.3 Permanence of SIR epidemic models with varying population

It is well known that permanence of dynamical systems plays a very important role in the studying of population dynamical systems. Based on some known techniques on limit sets of di erential dynamical systems developed by Butler et al. (1986); Hale and Wlatman (1989) and Freedman and Ruan (1995), Ma et al. (2002) give a partial answer to Problem 1.

Theorem 9 (Ma et al. 2002). For any time delay h ≥ 0, (8) is necessary and su cient for permanence of (9).

In biology, Theorem 9 implies that, for any time delay h ≥ 0, (8) gives the threshold of the model (9) for an endemic to occur.

It should be pointed out here that, in the proof of Theorem 9, the positive constant ν2 appeared in Theorem 8 is only shown to exist in a theoretical form.

Based on some known results for epidemic models with time delays by Cooke and van den Driessche (1996) and Hethcote and van den Driessche (2000,1995) and Wang (2002) proposed two classes of SEIRS epidemic model with time delays and also discussed global behavior of the equilibria of the models. With help of some analysis techniques developed by Wang (2002), Song et al. (2005, 2006) recently further considered permanence of a class of more general SIR epidemic model with non-constant birth rate and time delays, and gave an explicit expression for the positive constant ν2, which plays an important role in the application of Theorem 8. We shall give a detailed description in the following section.

60 Yasuhiro Takeuchi and Wanbiao Ma

3.3 SIR epidemic models with time delays and non-constant birth rate bN (t)

Takeuchi and Ma (1999) considered the following delayed SIR epidemic model with density dependent birth rate:

where S(t) + I(t) + R(t) ≡ N (t) denotes the total number of population at time t. The positive constants b, µ1, µ2, µ3, β and γ have the same biological meanings as in model (7). The nonnegative constant h is a time delay. Note that, in (10), the birth rate of population is dependent on the total number of population N (t).

For (10), we have the following classification on the existence of its equilibria.

(i)Equation (10) always has a trivial equilibrium E0 = (0, 0, 0).

(ii)If b = µ1, then for any S > 0, ES = (S, 0, 0) is the boundary equilibrium (the disease free equilibrium) of (10).

(iii)If

b = µ1 = µ3(µ2 + γ) , µ3 + γ

then for any I > 0 and R > 0 such that γI = µ3R, ESR = (S , I, R) is the positive equilibrium (the endemic equilibrium) of (10), where

S ≡ µ2 + γ .

β

(iv) If

then (10) has a unique positive equilibrium E+ = (S , I , R ), where

3.3.1 Global asymptotic properties

For the model (10), it has the following results:

Theorem 10 (Takeuchi and Ma 1999).

(a)If µ1 > b, then the boundary equilibrium E0 is globally asymptotically stable.

(b) If b > µ1, then E0 is unstable. Further, if

b > µ3(µ2 + γ) , µ3 + γ

then, for any solution (S(t), I(t), R(t))T of (10),

lim N (t) = lim (S(t) + I(t) + R(t)) = +∞ .

t→+∞ t→+∞

Theorem 11 (Takeuchi and Ma 1999). If

µ1 = b < µ3(µ2 + γ) , µ3 + γ

then, for any solution (S(t), I(t), R(t))T of (10), there is some constant c ≥ 0 such that

c ≤ S = (µ2 + γ)/β

and

Moreover, we also have the following

Theorem 12 (Ma and Takeuchi 2004). If

b > µ1

and

b ≥ µ3(µ2 + γ) , µ3 + γ

then, for any solution (S(t), I(t), R(t))T of (10),

lim (S(t) + I(t)) = +∞ .

t→+∞

In [29], the convergence of the positive equilibrium E+ had also been considered by using Liapunov functionals.

3.3.2 Hopf bifurcation and local stability on E+

Ma and Takeuchi (2004) further considered local asymptotic stability of the positive equilibrium E+ and Hopf bifurcation of (10) based on well-known Hopf bifurcation theorem (see, for example, Hale (1977) and Kuang (1993)). We have the following: