An Explicit Periodic Solution of a Delay Differential Equation

Abstract

In this paper we prove that the following delay differential equation

$$\begin{aligned} \frac{d}{dt}x(t)=rx(t)\left( 1-\int _{0}^{1}x(t-s)ds\right) , \end{aligned}$$

has a periodic solution of period two for $r>\frac{\pi ^{2}}{2}$ (when the steady state, $x=1$, is unstable). In order to find the periodic solution, we study an integrable system of ordinary differential equations, following the idea by Kaplan and Yorke (J Math Anal Appl 48:317–324, 1974). The periodic solution is expressed in terms of the Jacobi elliptic functions.

Appendices

Appendix A. Elliptic Functions

We briefly introduce the Jacobi elliptic functions. See [3] for detail. See also [20], where the Jacobi elliptic functions are defined as the solutions of a system of ordinary differential equations. We then show that the solution of the Duffing equation is expressed in terms of the Jacobi elliptic function.

The incomplete elliptic integrals of the first kind and second kind are respectively given as

$$\begin{aligned} F\left( \varphi ,k\right)&=\int _{0}^{\varphi }\frac{1}{\sqrt{1-k^{2}\sin ^{2}\theta }}d\theta ,\\ E\left( \varphi ,k\right)&=\int _{0}^{\varphi }\sqrt{1-k^{2}\sin ^{2}\theta }d\theta \end{aligned}$$

for $\varphi \in \mathbb {R}$ and $0\le k<1$. Here k is a parameter called the modulus. Then the complete elliptic integrals of the first kind and second kind introduced in Sect. 4 are

$$\begin{aligned} K(k):=F\left( \frac{\pi }{2},k\right) ,\ E(k):=E\left( \frac{\pi }{2},k\right) . \end{aligned}$$

The amplitude function $\text {am}$ is defined as the inverse function of the elliptic integral of the first kind, fixing the modulus k, i.e.,

$$\begin{aligned} \text {am}\left( F\left( \varphi ,k\right) ,k\right) =\varphi . \end{aligned}$$

Then the Jacobi elliptic functions $\text {sn},\text {cn}:\mathbb {R}\rightarrow \left[ -1,1\right]$ are respectively defined as

$$\begin{aligned} \text {sn}\left( t,k\right)&=\sin \left( \text {am}\left( t,k\right) \right) ,\\ \text {cn}\left( t,k\right)&=\cos \left( \text {am}\left( t,k\right) \right) . \end{aligned}$$

One then sees that the period of $\text {sn}$ and $\text {cn}$ is given as 4K(k). Then the Jacobi elliptic function $\text {dn}$ is defined by

$$\begin{aligned} \text {dn}\left( t,k\right) =\sqrt{1-k^{2}\text {sn}\left( t,k\right) }. \end{aligned}$$

The period of $\text {dn}$ is 2K(k).

The Duffing equation can be solved by the Jacobi elliptic function (see [3, 20]). Let us consider the following ordinary differential equation

$$\begin{aligned} \frac{d^{2}}{dt^{2}}y(t)&=-py(t)+qy(t)^{3} \end{aligned}$$

(A.1)

with initial condition

$$\begin{aligned} \frac{d}{dt}y(0)&=d, \end{aligned}$$

(A.2a)

$$\begin{aligned} y(0)&=0. \end{aligned}$$

(A.2b)

Here $p,\ q$ and d are assumed to be real. One obtains the equation (3.4) with the initial condition (3.5) from (A.1) with (A.2) by

$$\begin{aligned} p=r\left( a+b\right) ,\ q=\frac{r^{2}}{2},\ d=a-b. \end{aligned}$$

(A.3)

For (A.1) with the initial condition (A.2), we consider the following ansatz

$$\begin{aligned} y(t)=\alpha \text {sn}\left( \beta t,k\right) \end{aligned}$$

with $\alpha >0$ and $\beta >0$, noting that $\text {sn}$ is an odd function. Differentiating the Jacobi elliptic functions, we have

$$\begin{aligned} y^{\prime }(t)&=\alpha \beta \text {cn}\left( \beta t,k\right) \text {dn}\left( \beta t,k\right) ,\\ y^{\prime \prime }(t)&=-\alpha \beta ^{2}\text {sn}\left( \beta t,k\right) \left( \text {dn}^{2}\left( \beta t,k\right) +k^{2}\text {cn}^{2}\left( \beta t,k\right) \right) \\&=-\alpha \beta ^{2}\text {sn}\left( \beta t,k\right) \left( \left( 1+k^{2}\right) -2k^{2}\text {sn}^{2}\left( \beta t,k\right) \right) , \end{aligned}$$

thus

$$\begin{aligned} y^{\prime \prime }(t)=-\beta ^{2}y(t)\left( 1+k^{2}-2\frac{k^{2}}{\alpha ^{2}}y^{2}(t)\right) \end{aligned}$$

follows. We then obtain the following three equations

$$\begin{aligned} d&=\alpha \beta , \end{aligned}$$

(A.4a)

$$\begin{aligned} p&=\beta ^{2}\left( 1+k^{2}\right) , \end{aligned}$$

(A.4b)

$$\begin{aligned} q&=\frac{2\beta ^{2}k^{2}}{\alpha ^{2}}. \end{aligned}$$

(A.4c)

Let us now solve the Eq. (A.4) in terms of $\alpha ,\ \beta$ and k. One obtains

$$\begin{aligned} k^{2}-ck+1=0, \end{aligned}$$

(A.5)

where

$$\begin{aligned} c:=\frac{p}{d}\sqrt{\frac{2}{q}}. \end{aligned}$$

(A.6)

Since the equation (A.5) has a root in $\left[ 0,1\right)$ if and only if $2<c$, assume that $2<c$ holds. We then get

$$\begin{aligned} k=\frac{1}{2}\left( c-\sqrt{c^{2}-4}\right) \in \left[ 0,1\right) . \end{aligned}$$

Now it follows

$$\begin{aligned} \alpha =d\sqrt{\frac{1+k^{2}}{p}},\ \beta =\sqrt{\frac{p}{1+k^{2}}}. \end{aligned}$$

For the equation (3.4) with the initial condition (3.5), substituting (A.3) to (A.6), we obtain

$$\begin{aligned} c=2\left( \frac{a+b}{a-b}\right) >2. \end{aligned}$$

Then one can easily obtain $\alpha ,\ \beta$ and k as in (3.7) and (3.8).

Appendix B. An Epidemic Model with Temporary Immunity

The delay differential equation (1.3) can be related to an epidemic model that accounts for temporary immunity ([2, 8, 11, 30, 36]). Let us derive the delay differential equation (1.3) as a limiting case of the following SIRS type epidemic model with temporary immunity

$$\begin{aligned} \frac{d}{dt}S(t)&=-\beta S(t)I(t)+\gamma I(t-\tau ), \end{aligned}$$

(B.1a)

$$\begin{aligned} \frac{d}{dt}I(t)&=\beta S(t)I(t)-\gamma I(t), \end{aligned}$$

(B.1b)

$$\begin{aligned} \frac{d}{dt}R(t)&=\gamma I(t)-\gamma I(t-\tau ). \end{aligned}$$

(B.1c)

The model (B.1) is equivalent to the model studied in Section 3 of [11] (see (B.4) below) and is a special case of the model considered in [8]. As in [8, 11], ignoring birth and death of individuals, transitions of susceptible, infective and recovered populations are described. Here $S(t),\ I(t)$ and R(t) respectively denote the fraction of susceptible, infective and recovered populations at time t. The model (B.1) has three parameters: transmission coefficient $\beta >0$, the recovery rate $\gamma >0$ and the immune period $\tau >0$. See also [2, 30] for SIRS models with demographic turn-over.

The initial condition is given as follows

$$\begin{aligned} S(0)&=S_{0}>0,\\ I(s)&=\psi (s),\ s\in \left[ -\tau ,0\right] ,\\ R(0)&=\gamma \int _{0}^{\tau }\psi (-s)ds, \end{aligned}$$

where $\psi$ is a positive continuous function. We now require that

$$\begin{aligned} S_{0}+\psi (0)+\gamma \int _{0}^{\tau }\psi (-s)ds=1, \end{aligned}$$

so that

$$\begin{aligned} S(t)+I(t)+R(t)=1,\ t\ge 0 \end{aligned}$$

(B.2)

implying the constant total population. It also follows

$$\begin{aligned} R(t)=\gamma \int _{0}^{\tau }I(t-s)ds,\ t\ge 0. \end{aligned}$$

(B.3)

From (B.2) and (B.3) we get

$$\begin{aligned} S(t)=1-I(t)-\gamma \int _{0}^{\tau }I(t-s)ds. \end{aligned}$$

Then from (B.1b) we obtain the following scalar delay differential equation

$$\begin{aligned} \frac{d}{dt}I(t)=I(t)\left\{ \beta \left( 1-I(t)-\gamma \int _{0}^{\tau }I(t-s)ds\right) -\gamma \right\} . \end{aligned}$$

(B.4)

We let $x(t)=\frac{I(t)}{I_{e}},$ where $I_{e}$ is a nontrivial equilibrium of (B.4) given as

$$\begin{aligned} I_{e}=\frac{1-\frac{\gamma }{\beta }}{1+\gamma \tau }. \end{aligned}$$

It is assumed that $\beta >\gamma$ to ensure $I_{e}>0$. Considering a nondimensional time so that the immune period is 1, we obtain

$$\begin{aligned} \frac{d}{dt}x(t)=\left( \beta -\gamma \right) x(t)\left( 1-\frac{x(t)+\gamma \tau \int _{0}^{1}x(t-s)ds}{1+\gamma \tau }\right) . \end{aligned}$$

We now fix $r=\beta -\gamma$ and let $\gamma \tau \rightarrow \infty$ to formally obtain the equation (1.3). Local stability analysis for (B.4) can be found in [8, 11]. See also [25] for the application of the mathematical model to explain the periodic outbreak of a childhood disease.