2.1 Equivalence to a functional differential equation with infinite delay

In what follows, we consider the following delay differential-algebraic system:

(2.1) $$ \begin{align} \left\{\begin{array}{@{}l} I'(t)=\beta(t)g(I(t))-rI(t), \\ \beta(t)=f(\beta(t-\tau),g(I(t-\tau))), \end{array}\right. \end{align} $$

where we assume:

1. (H1) $g: \mathbb {R}_{+}\to \mathbb {R}_{+}$ is $\mathbb {C}^2$ -smooth;

2. (H2) $f:\mathbb {R}_+^2\to \mathbb {R}_+$ is given by $f(x, y)=F(xy)$ , where $F:\mathbb {R}_{+}\to \mathbb {R}_{+}$ is $\mathbb {C}^2$ -smooth, $F(\xi )$ is strictly decreasing on behavioral adaption of incidence $\xi \geq 0$ with $F(\infty ):=\lim _{\xi \to \infty }F(\xi )\geq 0$ and $F(0):=\beta _c>0$ .

Assumption (H2), more explicitly, the multiplicative assumption on f, is motivated by the epidemic models in [Reference Cheng and Wu12, Reference Zhang, Scarabel, Murty and Wu39], where the incidence rate depends on the product of the transmission rate and the infectivity term. Although some of the principal techniques we developed, such as the reduction of the delay differential-algebraic system to a delay differential equation with infinite delay (for solutions defined for all t), can be applied to general system (2.1) without this multiplicative structure imposed on f, much of the analysis and calculations we present below to exclude periodic solutions with a certain period is detailed with this particular form of f in mind for the sake of simplicity.

Examples arising from the aforementioned studies on disease outbreaks include the linear function

$$ \begin{align*}g(I)=g_1(I):=I \end{align*} $$

during the early phase of a pandemic [Reference Zhang, Scarabel, Murty and Wu39], with susceptible populations remaining constant; and the quadratic function

$$ \begin{align*}g(I)=g_2(I):=I(1-I/N) \end{align*} $$

post a pandemic [Reference Cheng and Wu12], where $N>0$ is a constant for the total population and susceptibility in populations will decrease due to infection-generated immunity. The second component of the coupled system describes the adaptation behavior to the perceived risk of infection of the incidence term $\beta (t-\tau )g(I(t-\tau ))$ with $\tau>0$ being the sum of the delay from exposure to infection to perception of risk and behavioral changes. The monotonicity assumption in (H2) reflects the reduction of transmission effective contact when the perceived risk increases. Assumption (H1) is clearly satisfied by $g_1$ and $g_2$ .

An equilibrium $E^*=(I^*, \beta ^*)$ of the coupled system is given by

(2.2) $$ \begin{align} \left\{\begin{array}{@{}l} \beta^*g(I^*)=rI^*,\\ \beta^*=f(\beta^*,g(I^*)). \end{array}\right. \end{align} $$

It is obvious that if $g(I^*)$ is given, then $\beta ^*$ is a fixed point of $f(\beta ^*, g(I^*))$ . Assumption (H2) implies the following lemma.

In what follows, we use $f_1(x, y ):=\frac {\partial f(x, y)}{\partial x }$ and $ f_2(x, y ):=\frac {\partial f(x,y )}{\partial y}$ .

The linearization at a given equilibrium $(\beta ^*,I^*)$ of (2.1) is

(2.3) $$ \begin{align} \left\{\begin{array}{@{}l} I'(t)=g(I^*)\beta(t) +(\beta^*g'(I^*)-r)I(t), \\ \beta(t)=f_1(\beta^*,g(I^*))\beta(t-\tau)+f_2(\beta^*,g(I^*))g'(I^*)I(t-\tau). \end{array}\right. \end{align} $$

Substituting $I=e^{\lambda t}I_{\mathbf {o}}$ , $\beta =e^{\lambda t}\beta _{\mathbf {o}}$ leads to

(2.4) $$ \begin{align} \left\{\begin{array}{@{}l} \lambda I_{\mathbf{o}}=g(I^*)\beta_{\mathbf{o}}+(\beta^*g'(I^*)-r)I_{\mathbf{o}}, \\ \beta_{\mathbf{o}}(1-f_{1}(\beta^*,g(I^*))e^{-\lambda\tau})=f_{2}(\beta^*,g(I^*))g'(I^*)I_{\mathbf{o}}e^{-\lambda\tau}. \end{array}\right. \end{align} $$

Since $f_{1}(x,y) )=F'(xy))y \leq 0$ for $y\ge 0$ , we have that $1-f_{1}(\beta ^*,g(I^*))e^{-\lambda \tau }\neq 0$ for any $\lambda \in \mathbb {C}$ . So a characteristic equation of (2.3) is written as

(2.5) $$ \begin{align} \Delta_{\mathrm{S}}(E^*,\tau)(\lambda):=(\lambda+r-\beta^*g'(I^*))-\frac{g'(I^*)g(I^*)f_{2}(\beta^*,g(I^*))e^{-\lambda\tau}}{(1-f_{1}(\beta^*,g(I^*))e^{-\lambda\tau})}=0. \end{align} $$

We will explore the onset and global continuation of Hopf bifurcation from a positive equilibrium by using the $S^1$ -degree-based global Hopf bifurcation theory developed in [Reference Wu38]. Such a global Hopf bifurcation theorem has been developed for delay differential equations [Reference Erbe, Krawcewicz, Geba and Wu21] and for neutral equations with condensing fields [Reference Krawcewicz, Wu and Xia27]. Although system (2.3) can be considered as a neutral equation with initial conditions constrained in a sub-manifold of the phase space, the corresponding $\mathcal {D}$ -operator defined by

$$ \begin{align*}\mathcal{D}(\beta_t,I_t)=\beta(t)-f(\beta(t-\tau),g(I(t-\tau))) \end{align*} $$

is not necessarily stable; thus, the $ S^1$ -degrees for condensing fields cannot be applied directly.

On the other hand, periodic solutions of (2.1) are defined for all $t\in \mathbb {R}$ . For these kinds of solutions, we can use the algebraic equation to solve $\beta (t)$ iteratively in terms of $I(t-k\tau )$ with $k=1,2,\ldots $ After substituting the iteration to the first equation of (2.1), we obtain a delay differential equation so that the right-hand side depends on $I(t-k\tau )$ with $\tau \geq 0$ , namely, a delay differential equation with infinite delay. To be more precise, we note

(2.6) $$ \begin{align} \begin{aligned} \beta(t)&=f(\beta(t-\tau),g(I(t-\tau)))\\ &=f(f(\beta(t-2\tau),g(I(t-2\tau)),g(I(t-\tau))))\\ &=f(f(f(\beta(t-3\tau),g(I(t-3\tau))),g(I(t-2\tau)),g(I(t-\tau))))\\ &\vdots\\ &=\underbrace{f\big(f \big(\dots f \big(}_{k} \beta(t-k\tau),\, g(I(t-k\tau)) \big),\, g(I(t-(k-1)\tau))\big)\dots,\, g(I(t-\tau))\big).\\ \end{aligned} \end{align} $$

Define

(2.7) $$ \begin{align} \left\{ \begin{array}{@{}l} f_{[1]}(x_1,y_1) := f(x_1, g(y_1)), \\[0.5em] f_{[2]}(x_2,y_2,y_1) := f\big(f(x_2,y_2), g(y_1)\big), \\[0.5em] \vdots \\[0.5em] f_{[k]}(x_k,y_k,\ldots,y_1):= \underbrace{f\big(f\big(\dots f\big(}_{k} x_k,\, g(y_k) \big),\, g(y_{k-1})\big)\ldots,\, g(y_1)\big),\ldots, g(y_{1})\big), \end{array} \right. \end{align} $$

with

(2.8) $$ \begin{align} f_{[k+1]}(x_{k+1},y_k,\ldots,y_2,y_1)=f(f_{[k]}(x_{k+1},y_{k+1},\ldots,y_2),g(y_1)). \end{align} $$

According to the definition of $f_{[k]}(x_{k}, y_k,\ldots , y_2, y_1)$ , $ f_{[\infty ]}(\ldots , y_k,\ldots , y_2, y_1) $ is well defined (in the sense of existence, not necessarily convergence) and satisfies the following property:

1. (C0) $f_{[\infty ]}(\ldots ,y_k,\ldots ,y_2,y_1)=f(f_{[\infty ]}(\ldots ,y_{k+1},y_k,\ldots ,y_2),g(y_1))$ .

We get

(2.9) $$ \begin{align} \begin{aligned} \beta(t)=f_{[1]}(\beta(t-\tau),I(t-\tau))&=f_{[2]}(\beta(t-2\tau),I(t-2\tau),I(t-\tau))\\ &=f_{[k]}(\beta(t-k\tau),I(t-k\tau),\ldots,I(t-\tau))\\ &\vdots\\ &=f_{[\infty]}(\ldots,I(t-3\tau),I(t-2\tau),I(t-\tau)):=\tilde{f}(I^{t}), \end{aligned} \end{align} $$

where $I^t(s)=I(t+s)$ for $s\in \mathbb {R}$ . Substituting (2.9) into the first equation of (2.1), we get

(2.10) $$ \begin{align} I'(t)=\tilde{f}(I^t)g(I(t))-rI(t). \end{align} $$

This motivates us to consider the global Hopf bifurcation theorem for functional differential equations with infinite delay formulated in [Reference Wu38], which we describe below.

2.2 Preliminary: Global Hopf bifurcation theory for FDEs with infinite delay

Let $\mathbb {X}$ denote the Banach space of bounded continuous mappings $x:\mathbb {R}\to \mathbb {R}$ equipped with the supremum norm. For $x\in \mathbb {X}$ and $t\in \mathbb {R}$ , we will use $x^t$ to denote an element in $\mathbb {X}$ defined by $x^t(s)=x(t+s)$ for $s\in \mathbb {R}$ .

Consider the following functional differential equation:

(2.11) $$ \begin{align} \dot{x}=\mathcal{F}(x^t,a,p) \end{align} $$

parameterized by two real numbers $(a,p)\in \mathbb {R}\times \mathbb {R}_+$ , where $\mathcal {F}:=\mathbb {X}\times \mathbb {R}\times \mathbb {R}_+\to \mathbb {R}^n$ is completely continuous. Identifying the subspace of $\mathbb {X}$ consisting of all constant mapping with $\mathbb {R}^n$ , we get a mapping $\hat {\mathcal {F}}=\mathcal {F}|_{\mathbb {R}^n\times \mathbb {R}\times \mathbb {R}_{+}}:\mathbb {R}^n\times \mathbb {R}\times \mathbb {R}_{+}\to \mathbb {R}^n$ . We assume

1. (A1). $\hat {\mathcal {F}}$ is twice continuously differentiable.

Denote by $\hat {x}\in \mathbb {X}$ the constant mapping with the value $x_0\in \mathbb {R}$ . $(\hat {x}_0,a_0,p_0)$ is a stationary solution of (2.11) if $\hat {\mathcal {F}}(x_0,a_0,p_0)=0$ .

1. (A2). At each stationary solution $(\hat {x}_0,a_0,p_0)$ , the derivative of $\hat {\mathcal {F}}$ with respect to the first variable $x$ , evaluated at $(\hat {x}_0,a_0,p_0)$ is an isomorphism of $\mathbb {R}^n$ .

Under (A1) and (A2), for each stationary solution $(\hat {x}_0,a_0,p_0)$ there exists $\epsilon _0>0$ and a continuously differential mapping $y:B_{\epsilon _0}(a_0,p_0)\to \mathbb {R}^n$ such that $\hat {\mathcal {F}}(\hat {x}(a,p),a,p)=0$ for $(a,p)\in B_{\epsilon _0}(a_0,p_0)=(a_0-\epsilon _0,a_0+\epsilon _0)\times (p_0-\epsilon _0,p_0+\epsilon _0)$ . We assume the following smoothness condition:

1. (A3). $\mathcal {F}(\varphi ,a,p)$ is differentiable with respect to $\varphi $ , and the $n\times n$ complex matrix function $\Delta (\varphi ,a,p)(\lambda )$ is continuous in $(a,p,\lambda )\in B_{\varepsilon _0}(a_0,p_0)\times \mathbb {C}$ . Here, for each stationary solution $(\hat {x}_0,a_0,p_0)$ , we have

   $$\begin{align*}\Delta(\hat{x}_0,a_0,p_0)(\lambda) = \lambda I_d -D\mathcal{F}(\hat{x}_0,a_0,p_0)(e^{\lambda \cdot}\mathrm{id}), \end{align*}$$

   where $D\mathcal {F}(\hat {x}_0,a_0,p_0)$ is the complexification of the derivative of $F(\varphi ,a,p)$ with respect to $\varphi $ , evaluated at $(\hat {x}_0,a_0,p_0)$ .

We call $\Delta _{\hat {x}_0,a_0,p_0}(\lambda )$ the characteristic matrix and the zeros of $\det \Delta _{(\hat {x}_0,a_0,p_0)} (\lambda )=0$ the characteristic values of $(\hat {x}_0,a_0,p_0)$ . A stationary solution $(\hat {x}_0,a_0,p_0)$ is called a center if it has purely imaginary characteristic values of the form $i m \tfrac {2\pi }{p_0}$ for some positive integer $m$ . A center $(\hat {x}_0,a_0,p_0)$ is said to be isolated if (i) it is the only center in some neighborhood of $(\hat {x}_0,a_0,p_0)$ ; (ii) it has only finitely many purely imaginary characteristic values of the form $i m \tfrac {2\pi }{p_0}$ , $m$ an integer.

Assume $(\hat {x}_0,a_0,p_0)$ is an isolated center. Let $\mathcal {J}(\hat {x}_0,a_0,p_0)$ denote the set of all positive integers $m$ such that $i m \tfrac {2\pi }{p_0}$ is a characteristic value of $(\hat {x}_0,a_0,p_0)$ . We assume there exists $m\in \mathcal {J}(\hat {x}_0,a_0,p_0)$ such that

1. (A4) There exist $\varepsilon \in (0,\varepsilon _0)$ and $\delta \in (0,\varepsilon _0)$ such that on $[a_0-\delta ,a_0+\delta ]\times \Omega _{\varepsilon ,p_0}$ , we have $\det \Delta (\hat {x}(a,p),a,p)(\mu +i m \tfrac {2\pi }{p})=0$ , if and only if $a=a_0$ , $\mu =0$ , and $ p=p_0$ , where $\Omega _{\varepsilon ,p_0} = \{ (\mu ,p);\ 0<\mu <\varepsilon ,\ p_0-\varepsilon < p < p_0+\varepsilon \}$ .

Define

$$\begin{align*}\mathcal{H}^{\pm}(\hat{x}_0,a_0,p_0)(\mu,p) = \det \Delta(\hat{x}_0(a_0\pm \delta),a_0\pm \delta,p_0) \left(\mu + i m \tfrac{2\pi}{p}\right). \end{align*}$$

Then (A4) implies $\mathcal {H}_m^{\pm }(\hat {x}_0,a_0,p_0)\neq 0$ on $\Omega _{\varepsilon ,p_0}$ . Consequently, the following integer:

$$\begin{align*}\gamma_m(\hat{x}_0,a_0,p_0) := \deg_B\big(\mathcal{H}_m^{-}(\hat{x}_0,a_0,p_0),\Omega_{\varepsilon,p_0}\big) - \deg_B\big(\mathcal{H}_m^{+}(\hat{x}_0,a_0,p_0),\Omega_{\varepsilon,p_0}\big) \end{align*}$$

is well defined, where $\deg _B$ is the Boruwer degree. $\gamma _m(\hat {x}_0,a_0,p_0)$ is called the mth crossing number of $(\hat {x}_0,a_0,p_0)$ . Further, $\gamma \left (\hat {x}_0, a_0,p_0\right )\neq 0$ implies the existence of a local Hopf bifurcation of periodic solutions with period $p_0/m$ . More precisely, if (A1)–(A3) are satisfied, and there exists an isolated center $(\hat {x}_0,a_0,p_0)$ and an integer $m\in \mathcal {J}(\hat {x}_0,a_0,p_0)$ such that (A4) holds and $\gamma _m(\hat {x}_0,a_0,p_0)\neq 0$ . Then there exists a sequence $(a_k,p_k)\in \mathbb {R}\times \mathbb {R}_+$ such that

1. (i) $ \lim _{k\to \infty } (a_k,p_k) = (a_0,p_0)$ ;

2. (ii) at each $(a,p)=(a_k,p_k)$ , (2.11) has a non-constant periodic solution $x_k(t)$ with a period $p_k/m$ ; $\lim _{k\to \infty }x_k(t)=\hat {x}_0$ , uniformly for $t\in \mathbb {R}$ .

We can now state the global Hopf bifurcation theorem (Theorem 3.3 in [Reference Wu38]).

Theorem 2.3 Assume that

1. (GH1) All centers of (2.11) are isolated and (A4) holds for each center $(\hat {x}_0,a_0,p_0)$ and each $m\in \mathcal {J}(\hat {x}_0,a_0,p_0)$ .

2. (GH2) For each bounded set $W\subset \mathbb {X}\times \mathbb {R}\times \mathbb {R}_{+}$ , there exist a constant $L>0$ such that $|\mathcal {F}(\varphi ,a,p)-\mathcal {F}(\psi ,a,p))|\leq L\cdot \sup _{s\in \mathbb {R}}|\varphi (s)-\psi (s)|$ for $(\varphi ,a,p), (\psi ,a,p)\in W$ .

Let

$$ \begin{align*} & \Sigma (\mathcal{F}):= Cl\{ x, a, p) : x \text{ is a } p\text{-periodic solution of system}\, (2.11) \} \subseteq \mathbb{X} \times \mathbb{R} \times \mathbb{R},\\ & N(\mathcal{F})=\{(x, a, p), \mathcal{F}(\hat{x},a,p)=0\}. \end{align*} $$

Assume that $(\hat {x},a,p)$ is an isolated center satisfying (A1)–(A4). Denote by $ C(\hat {x}_0,a_0,p_0)$ the connected component of $(\hat {x}_0,a_0,p_0)$ in $\Sigma $ , we have either

1. (i) $C(\hat {x}_0,a_0,p_0) $ is unbounded; or

2. (ii) $C(\hat {x}_0,a_0,p_0)$ is bounded, $C(\hat {x}_0,a_0,p_0)\cap N(\mathcal {F})$ is finite, and

   $$ \begin{align*}\sum_{(\hat{x},a,p) \in C(\hat{x}_0,a_0,p_0)\cap N(\mathcal{F})} \gamma\left(\hat{x},a,p \right) =0, \end{align*} $$

   for all $m=1,2,\ldots $ where $\gamma _m(\hat {x},a,p)$ is the $m$ -th crossing number of $(\hat {x},a,p)$ if $m\in \mathcal {J}(\hat {x},a,p)$ or it is zero if otherwise.

2.3 Reformulation of the global Hopf bifurcation for delay-algebraic systems

We now reformulate this theorem in the setting of a delay-algebraic equation that is equivalent to (as far as solutions are defined for all $t\in \mathbb {R}$ ) the functional differential equation (2.10) with infinite delay.

The normalization of period by $I(t)=x(\frac {2\pi }{T} t)$ , where $T>0$ the period of periodic solutions $I(t)$ , gives $I(t-k\tau )=x(\theta -k\frac {2\pi }{T} \tau )$ with $\theta =\frac {2\pi }{T} t$ , and $I^t=x(\theta +\frac {2\pi }{T}s)$ . Denote

$$ \begin{align*}\tilde{f}_{\omega}(x^{\theta}):=f_{[\infty]}(\dots,x(\theta-\frac{2\pi}{T}k\tau),\ldots,x(\theta-\frac{2\pi}{T}2\tau),x(\theta-\frac{2\pi}{T}\tau)),\end{align*} $$

where $x^{\theta }=x(\theta +\frac {2\pi }{T} s)$ for $s\in \mathbb {R}$ .

Re-denoting $\theta $ as $t$ , system (2.10) is transformed into the following form:

(2.12) $$ \begin{align} x'(t)=\frac{T}{2\pi}\left(\tilde{f}_{\omega}(x^{t})g(x(t))-rx(t)\right), \end{align} $$

where $x^{t}=x(t+\frac {2\pi }{T} s)$ for $s\in \mathbb {R}$ .

The normalized system (2.12) has two bifurcation parameters, $\tau $ and $T$ . $x(t)$ is the periodic solution $2\pi $ -period for (2.12) if and only if $I(t)$ is a periodic solution $T$ -periodic for (2.10) (equivalently, $(I(t),\beta (t))$ is a periodic solution $T$ -periodic for (2.1)). Let $ C_{2\pi } \times \mathbb {R}_+ \times \mathbb {R}_+$ be the Fuller space, where $C_{2\pi }$ denotes the Banach space of $2\pi $ -periodic functions equipped with the sup-norm.

Then the corresponding $\mathcal {F}(\cdot )$ is as follows:

(2.13) $$ \begin{align} \mathcal{F}(x^t,\tau,T)=\frac{T}{2\pi}\left(\tilde{f}_{\omega}(x^t)g(x^t)-rx^t\right), \end{align} $$

where $x^t \in \mathbb {X}$ . Identifying the subspace of $\mathbb {X}$ consisting of all constant functions with $\mathbb {R}_{+}$ , we obtain a restricted function $\hat {\mathcal {F}}:=\mathcal {F}|_{\mathbb {R}_{+}\times \mathbb {R}\times \mathbb {R}_{+}}$ .

Denote $H_{[\infty ]}(x):= f_{[\infty ]}(x,x,x,\ldots )$ . The property (C0) and Lemma 2.1 yield that $H_{[\infty ]}(x)$ is well defined, and it is the unique fixed point of the map $u \mapsto f(u,g(x))$ for $u> 0$ , that is, $f \big (H_{[\infty ]}(x), g(x)\big ) = H_{[\infty ]}(x)$ .

Additionally, we assume the following:

1. (S0) $H_{[\infty ]} (x)$ is $C^2$ smooth as a function of $x\ge 0$ .

Further, the following stronger condition ensures that the condition (S0) is satisfied (by the implicit function theorem).

1. (S0)^′ $f_1(H_{[\infty ]}(x),g(x))\neq 1$ for all $x\ge 0$ .

Since $p\tau (H_{[\infty ]}(x)g(x)-rx)$ is twice continuously differentiable, we have the following result.

The stationary points $\hat {x}$ of (2.10) (or (2.12)) can be obtained by solving

$$ \begin{align*}f(u,g(x))=u,\,\,ug(x)-rx=0\quad \text{for}\,\,x.\end{align*} $$

Then the set of stationary solutions of system (2.12) is

$$ \begin{align*}\mathcal{N}(\mathcal{F}):=\{(\hat{x}, \tau, T):f(H_{[\infty]}(\hat{x}),g(\hat{x}))=H_{[\infty]}(\hat{x}),\,\,H_{[\infty]}(\hat{x})g(\hat{x})-r\hat{x}=0\}. \end{align*} $$

Obviously, as long as $(\hat {x},\tau ,T)\in \mathcal {N}(\mathcal {F})$ , it necessarily follows that $H_{[\infty ]}(\hat {x})=F(r\hat {x})$ .

We define

$$ \begin{align*}\Sigma(\mathcal{F}) :=Cl\{ (\hat{x},\tau,T) : \hat{x} \text{ is a } 2\pi\text{-periodic solution of system (2.12) }\}\subseteq C_{2\pi} \times \mathbb{R}_+ \times \mathbb{R}_+. \end{align*} $$

For any stationary solution $(\hat {x},\tau ,T)\in \mathcal {N}(\mathcal {F})$ , the characteristic function is

$$ \begin{align*}\Delta_{(\hat{x},\tau,T)}(\lambda):=\lambda-D\mathcal{F}(\hat{x},\tau,T)(e^{\lambda }). \end{align*} $$

Let

(2.14) $$ \begin{align} \det \Delta_{(\hat{x},\tau,T)}(\lambda):=\lambda-\frac{T}{2\pi}(g(\hat{x})\sum_{j=1}^\infty \partial_{j}f_{[\infty]}(\ldots,\hat{x},\hat{x})e^{-j \frac{2\pi}{T}\lambda\tau}+H_{\infty}(\hat{x})g'(\hat{x})-r), \lambda\in \mathbb{C}. \end{align} $$

Define $\partial _{k} f_{[\infty ]}(\ldots ,y_3,y_2,y_1):=\frac {\partial f_{[\infty ]}(\ldots ,y_k,\ldots ,y_2,y_1)}{\partial y_k}$ , where $y_k$ is the $k$ -th argument from the right (with $y_1$ the outermost). Then from (C0) it follows that

(2.15) $$ \begin{align} \partial_{k} f_{[\infty]}(\ldots,y_3,y_2,y_1)=\left\{ \begin{array}{@{}l} f_1(f_{[\infty]}(\ldots,y_3,y_2),g(y_1))\partial_{k-1}f_{[\infty]}(\ldots,y_3,y_2), k\geq2,\\ f_2(f_{[\infty]}(\ldots,y_3,y_2),g(y_1))g'(y_1), k=1. \end{array} \right. \end{align} $$

We get

(2.16) $$ \begin{align} \begin{aligned} &\sum_{j=1}^\infty \partial_{j}f_{[\infty]}(\ldots,\hat{x},\hat{x})e^{-j \frac{2\pi}{T}\lambda\tau}\\ &\quad=e^{- \frac{2\pi}{T}\lambda\tau}f_1(H_{[\infty]}(\hat{x}),g(\hat{x}))\sum_{j=1}^{\infty} \partial_{j}f_{[\infty]}(\ldots,\hat{x},\hat{x})e^{-j \frac{2\pi}{T}\lambda\tau}\\&\qquad+f_2(H_{[\infty]}(\hat{x}),g(\hat{x}))g'(\hat{x})e^{- \frac{2\pi}{T}\lambda\tau}. \end{aligned} \end{align} $$

We impose the following assumption to ensure that $\sum _{j=1}^\infty \partial _{j}f_{[\infty ]}(\ldots ,\hat {x},\hat {x})e^{-j \frac {2\pi }{T}\lambda \tau }$ is well defined.

1. (S1) For any $(\hat {x},\tau ,T)\in \mathcal {N}(\mathcal {F})$ , $1-e^{- \frac {2\pi }{T}\lambda \tau }f_1(H_{[\infty ]}(\hat {x}),g(\hat {x}))\neq 0$ holds for all $\lambda \in \mathbb {C}$ .

We have the following result about the characteristic function $ \det \Delta _{(\hat {x},\tau ,T)}(\lambda )$ .

Lemma 2.5 If condition (S1) holds, then

$$ \begin{align*}\sum_{j=1}^\infty \partial_{j}f_{[\infty]}(\ldots,\hat{x},\hat{x})e^{-j \frac{2\pi}{T}\lambda\tau}=\frac{e^{-\frac{2\pi}{T}\lambda\tau}f_2(H_{[\infty]}(\hat{x}),g(\hat{x}))g'(\hat{x})}{1-e^{- \frac{2\pi}{T}\lambda\tau}f_1(H_{[\infty]}(\hat{x}),g(\hat{x}))}. \end{align*} $$

Moreover, we have

(2.17) $$ \begin{align} \begin{aligned} \det \Delta_{(\hat{x},\tau,T)}(\lambda)=\lambda-\frac{T}{2\pi}\left(\frac{e^{-\frac{2\pi}{T}\lambda\tau}f_2(H_{[\infty]}(\hat{x}),g(\hat{x}))g'(\hat{x})g(\hat{x})}{1-e^{-\frac{2\pi}{T}\lambda\tau}f_1(H_{[\infty]}(\hat{x}),g(\hat{x}))}+H_{[\infty]}(\hat{x})g'(\hat{x})-r\right). \end{aligned} \end{align} $$

To ensure that there is no root of $\det \Delta _{(\hat {x},\tau ,T)}(\lambda )=0$ entering the right (left) half of the complex plane from the left (right) half through infinities, we give the following assumption.

1. (S1a) For any $(\hat {x},\tau ,T)\in \mathcal {N}(\mathcal {F})$ , $\left |f_1(H_{[\infty ]}(\hat {x}),g(\hat {x}))\right |<1$ holds.

Clearly, the assumption (S1a) guarantees that (S1) holds for all $\operatorname {Re}\lambda \ge 0$ .

From formula (2.2), we find that the equilibrium component $I^*$ of (2.1) is identical to the stationary point of (2.10), with $H_{[\infty ]}(I^*)=\beta ^*$ . And by comparing the characteristic equation $\Delta _{\mathrm {S}}(E^*,\tau )(\lambda )$ defined in (2.5) and $ \det \Delta _{(\hat {x},\tau ,T)}(\lambda )$ in (2.17), we obtain the following result.

Lemma 2.6 If $(\hat {x},\tau ,T)\in \mathcal {N}(\mathcal {F})$ , then $\hat {E}:=(\hat {x},H_{[\infty ]}(\hat {x}))$ is an equilibrium of the delay-algebraic system (2.1). Further,

$$ \begin{align*}\frac{T}{2\pi}\Delta_{\mathrm{S}}(\hat{E},\tau)\left(\frac{2\pi}{T}\lambda\right)\equiv \det \Delta_{(\hat{x},\tau,T)}(\lambda)\; \text{for any}\; \lambda\in \mathbb{C}. \end{align*} $$

Consequently, we have the following lemma.

In addition, it can be easily checked from (2.13) that the smoothness condition (A3) is also satisfied.

Denote the set of centers of (2.12) by

$$ \begin{align*}\mathcal{N}_c(\mathcal{F}):=\{(\hat{x},\tau,T)|(\hat{x},\tau,T)\in \mathcal{N}(\mathcal{F}), \mathcal{J}(\hat{x},\tau,T)\neq \varnothing \}, \end{align*} $$

where $\mathcal {J}(\hat {x},\tau ,T)$ is the set of all positive integers $m$ such that $\Delta _{\mathrm {S}}(\hat {E},\tau )(im\frac {2\pi }{T})=0$ . (Equivalently, $im$ is a characteristic value of $(\hat {x},\tau ,T)$ .)

For a given isolated center $(\hat {x}_0, \tau _0, T_0)\in \mathcal {N}_c(\mathcal {F})$ , the corresponding $\hat {E}_0:=(\hat {x}_0,H_{[\infty ]}(\hat {x}_0))$ is an equilibrium of (2.1) from Lemma 2.6, we assume that there exists $m\in \mathcal {J}(\hat {x}_0, \tau _0, T_0)$ such that

1. (S3) There exist $\epsilon ,\delta \in (0,\epsilon _0)$ such that $[\tau _0-\delta ,\tau _0+\delta ]\times \partial \Omega _{\epsilon , T_0}$ with $\partial \Omega _{\epsilon , T_0}=\{(\mu ,T):0<\mu <\epsilon , T_0-\epsilon <T< T_0+\epsilon \}$ , $\Delta _{\mathrm {S}}(\hat {E}(\tau ,T),\tau )\left (\frac {2\pi }{T}\mu +im\frac {2\pi }{T}\right )=0$ (i.e., $\det \Delta _{(\hat {x}(\tau ,T),\tau ,T)}(\mu +im)=0$ ) if and only if $\mu =0$ , $T=T_0$ , and $\tau =\tau _0$ .

In what follows, we denote $\hat {E}(\tau _0\pm \delta ,T):=\left (\hat {x}(\tau _0\pm \delta ,T),H_{[\infty ]}\left (\hat {x}(\tau _0\pm \delta ,T)\right )\right )$ , and let

$$ \begin{align*} \begin{array}{@{}cl} \mathcal{H}^{\pm}(\hat{x}_0,\tau_0,T_0)(\mu,T)&:=\det \Delta_{(\hat{x}(\tau_0\pm\delta,T),\tau_0\pm \delta,T)}\left(\mu+im\right)\\ &=\frac{T}{2\pi}\Delta_{\mathrm{S}}\left(\hat{E}(\tau_0\pm\delta,T),\tau_0\pm\delta\right)\left(\frac{2\pi}{T}\mu+im\frac{2\pi}{T}\right). \end{array} \end{align*} $$

Hence, we obtain the following result.

Lemma 2.8 Supposed that (S3) holds (i.e., (A4) is satisfied), implying no zero of $\Delta _{\mathrm {S}}(\hat {E}_0,\tau )\left (\frac {2\pi }{T}\mu +im\frac {2\pi }{T}\right )=0$ (i.e., $\det \Delta _{(\hat {x}_0,\tau _0,T_0)}(\mu +im)= 0$ ) in $\partial \Omega _{\epsilon ,T_0}$ . Then the mth crossing number of $(\hat {x}_0,\tau _0,T_0)$ :

$$ \begin{align*}\gamma_m \left(\hat{x}_0,\tau_0,T_0\right) := \deg_B\left(\mathcal{H}_m^{-}(\hat{x}_0,\tau_0,T_0),\Omega_{\epsilon,T_0}\right)-\deg_B\left(\mathcal{H}_m^{+}(\hat{x}_0,\tau_0,T_0),\Omega_{\epsilon,T_0}\right), \end{align*} $$

is well defined.

Lemma 2.9 For every bounded set $W\subset \mathbb {X}\times \mathbb {R}\times \mathbb {R}_{+}$ , there exists $L_{\mathcal {F}}>0$ such that

$$ \begin{align*}\|\mathcal{F}(\varphi,\tau,T)-\mathcal{F}(\psi,\tau,T)\|_\infty \le \frac{T}{2\pi}L_{\mathcal{F}}\,\|\varphi-\psi\|_\infty\; \text{for all }(\varphi,\tau,T),(\psi,\tau,T)\in W, \end{align*} $$

where

$$ \begin{align*}L _{\mathcal{F}}:= \sup_{\substack{(\phi,\tau,T)\in W \\ s\in\mathbb{R}}} \Bigl(|\tilde f_\omega(\phi(s))|\cdot|g'(\phi(s))|+|g(\phi(s))|\cdot|\tilde {f}_\omega'(\phi(s))|+r\Bigr)<\infty. \end{align*} $$

That is, (GH2) is satisfied.

Proof By the triangle inequality, we have

(2.18) $$ \begin{align} \begin{aligned} &\|\hspace{1pt}\tilde f_\omega(\varphi)g(\varphi)-r\varphi-(\tilde f_\omega(\psi)g(\psi)-r\psi)\|_\infty\\ &\le \|\hspace{1pt}\tilde f_\omega(\varphi)\|_\infty\|g(\varphi)-g(\psi)\|_\infty +\|g(\psi)\|_\infty\|\tilde f_\omega(\varphi)-\tilde f_\omega(\psi)\|_\infty+r\|\varphi-\psi\|. \end{aligned} \end{align} $$

Since $f,g$ is $\mathbb {C}^2$ -smooth, by the mean value theorem, we can get the constant $L_{\mathcal {F}}<\infty $ directly.

We can now state the following local and global Hopf bifurcation theorems for delay-algebraic system (2.1).

To examine the boundedness of global Hopf branches of delay-algebraic system (2.1), we need to explore the boundedness of the projection of $C(\hat {x}_0,\tau _0,T_0) $ onto the parameter ( $\tau $ ) space, the $\mathbb {X}$ -space, and the $T$ -space, respectively. If the projections of $C(\hat {x}_0,\tau _0,T_0) $ onto the parameter ( $\tau $ ) space, the $\mathbb {X}$ -space, and $T$ -space are all bounded, then (ii) of Theorem 2.11 must hold, implying that the sum of crossing numbers along the bounded branch must be zero. Conversely, if this sum along any such branch cannot be zero, then one of the projections of a given connected component must be maximal.

3.1 Preliminary

We describe some preliminary results as preparation and introduce a few notions:

(3.1) $$ \begin{align} G(u;x_{n},x_{n-1},\ldots x_1):=\underbrace{f(f(\dots f}_{n}(u, x_n), x_{n-1}), \ldots, x_1) ,\,\, j\in N^+ \end{align} $$

with $(x_1,\ldots ,x_{n})\in \mathbb {R}^n_{+}$ . Since $G(u;x_n,x_{n-1} \ldots ,x_1)$ is monotonic with respect to $u\geq 0$ , increasing if $n$ is even and decreasing if $n$ is odd (due to $f_1(x,y) < 0$ ), and

$$ \begin{align*}G(0;x_n,x_{n-1} \ldots,x_1)>0,\,\,\, G(\beta_c;x_n,x_{n-1} \ldots,x_1)<\beta_c,\end{align*} $$

then

(3.2) $$ \begin{align} G(u;x_n,x_{n-1} \ldots,x_1)=u \end{align} $$

must have a positive fixed point in $(0,\beta _c)$ . Assume that

1. (C1). The positive fixed point of (3.2) in $(0,\beta _c)$ is unique, denoted by

   (3.3) $$ \begin{align} u^*:=H(x_{1},x_{2},\ldots,x_{n}). \end{align} $$

$H(\cdot )$ can be regarded as a $n$ variable functions with $(x_{1},x_{2},\ldots ,x_{n})\in \mathbb {R}^n_+$ . When $n$ is odd, (C1) holds naturally.

For brevity, denote

$$ \begin{align*}u_j :=\underbrace{f \big(f\big(\dots f\big(}_{\,n-j} u,\, x_n \big),\, x_{n-1}\big)\dots,\, x_{j+1}\big), \; j=0,\ldots,n-1 \end{align*} $$

and $u_n:=u^*$ . Then $u_0=G(u;x_1,\ldots ,x_n)$ and $u_{j-1}=f(u_{j},x_{j})$ for $j=1,\ldots ,n$ .

To ensure existence of $\partial u^*/\partial x_j$ , we assume that

1. (C2). $ 1-\prod _{k=1}^{n} f_1(u_k,x_k)\neq 0. $

When $n$ is odd, (C2) holds naturally.

Lemma 3.1 (Monotonicity of the function $H$ )

If (C2) holds, then

$$\begin{align*}\operatorname{sign} \Big(H_{j+1}(x_1,x_2,\ldots,x_n)\Big) = -\,\operatorname{sign} \Big( H_j(x_1,x_2,\ldots,x_n)\Big)\ ,\; j=1,\ldots,n-1, \end{align*}$$

where $H_j(x_1,x_2,\ldots ,x_n):=\frac {\partial H(x_1,x_2,\ldots ,x_n)}{\partial x_{j}}$ .

Proof Taking the derivative of (3.2) with respective to $x_j$ on both sides, we obtain

$$ \begin{align*}\left(1-\frac{\partial G(u^*;x_n,x_{n-1}\ldots x_1)}{\partial u}\right)\frac{\partial u^*}{\partial x_j}=\frac{\partial G(u^*;x_n,\ldots,x_1)}{\partial x_j}, \qquad j=1,\ldots,n, \end{align*} $$

with

$$ \begin{align*}\frac{\partial G(u^*;x_n,x_{n-1}\ldots x_1)}{\partial u}=\prod_{k=1}^{n} f_1(u_k,x_k), \end{align*} $$

$$ \begin{align*} \frac{\partial G(u^*;x_n,\ldots,x_1)}{\partial x_j}= \begin{cases} f_2 \big(u_1,x_1\big),\,\,\, j=1,\\ \Bigg(\prod_{k=1}^{j-1} f_1 \big(u_k,x_k\big)\Bigg)\, f_2 \big(u_j,x_j\big),\,\,\, j=2,\ldots,n. \end{cases} \end{align*} $$

Hence, we obtain

(3.4) $$ \begin{align} H_j(x_1,x_2,\ldots,x_n)=\dfrac{\partial u^*}{\partial x_j}=\begin{cases} \dfrac{f_2\big(u_j,x_j\big)}{1-\prod_{k=1}^{n} f_1(u_k,x_k)},\,\,\, j=1,\\ \dfrac{\Bigg(\prod_{k=1}^{j-1} f_1\big(u_k,x_k\big)\Bigg)\, f_2\big(u_j,x_j\big)}{1-\prod_{k=1}^{n} f_1(u_k,x_k)},\,\,\, j=2,\ldots,n. \end{cases} \end{align} $$

Since $f_1(x,y)=F'(xy)y<0$ , $f_2(x,y)=F'(xy)x<0$ , we obtain

$$ \begin{align*} \operatorname{Sign} \Big(\frac{\partial u^*}{\partial x_{j+1}}\Big) = -\,\operatorname{Sign} \Big(\frac{\partial u^*}{\partial x_j}\Big)\ ,\quad j=1,\ldots,n-1.\\[-34pt] \end{align*} $$

For the cases $n=2,3$ , we have

Lemma 3.3 Let $\{y_k\}_{k\in \mathbb {Z}}$ be an $n$ -cyclic sequence, that is, $y_{k+n} = y_k$ for all $k \in \mathbb {Z}$ , $y_k \in \mathbb {R}_+$ . Then

$$ \begin{align*}f_{[\infty]}(\ldots, y_n, y_{n-1}, \ldots, y_1) = H\big(g(y_n), \dots, g(y_1)\big).\end{align*} $$

In the particular case, if $n=1$ , we have $H_{[\infty ]}(y_1)=H(g(y_1),g(y_1),\ldots ,g(y_1))$ .

Proof From property (C0) of $f_{[\infty ]}(\ldots ,y_n,y_{n-1},\ldots ,y_1)$ , it follows that

$$ \begin{align*}&f_{[\infty]}(\ldots,y_n,y_{n-1},\ldots,y_1)\\&\quad=\underbrace{f \big(f \big(\dots f}_{n}\big(\,\, f_{[\infty]}(\ldots,y_{n+2},y_{n+1}),\,g(y_{n}) \big),\,g(y_{n-1})\big)\ldots,\,g(y_{1})\big). \end{align*} $$

Since $(y_k)_{k\in \mathbb {Z}}$ is cyclical, we have

$$ \begin{align*}f_{[\infty]}(\ldots,y_n,y_{n-1},\ldots,y_1)=f_{[\infty]}(\dots,y_{n+2},y_{n+1}),\end{align*} $$

which holds if and only if $f_{[\infty ]}(\ldots ,y_{n+1},y_n,y_{n-1},\ldots ,y_1)$ is the unique positive fixed point of

$$ \begin{align*}&\underbrace{f \big(f \big(\dots f}_{n}\big( u,\,g(y_{n}) \big),\,g(y_{n-1})\big)\ldots,\,g(y_{1})\big)=u,\\&\quad (\text{i.e.},\,G(u;g(y_{n}),g(y_{n-1}),\dots,g(y_1))=u). \end{align*} $$

Hence, $ f_{[\infty ]}(\ldots ,y_n,y_{n-1},\ldots ,y_1)=H(g(y_1),\dots ,g(y_n)), $ where the function $H(\cdot )$ of $ n $ -variables is given as the positive solution of (3.2).

Theorem 3.4 For a nontrivial periodic solution of period $ n\tau $ for system (2.10), denoted by $\tilde {I}(t)$ , we have

$$ \begin{align*}f_{[\infty]}(\ldots,\tilde{I}(t-n\tau),\tilde{I}(t-(n-1)\tau),\ldots,\tilde{I}(t-\tau))=H(g(\tilde{I}(t-\tau)),\dots,g(\tilde{I}(t-n\tau))),\end{align*} $$

where $H(\cdot )$ is given as a positive solution of (3.2). Furthermore, the existence of such an $ n\tau $ -periodic solution for system (2.10) implies the following n-dimensional autonomous ODE system:

(3.5) $$ \begin{align} \begin{aligned} I_j' &= H(g(I_{j+1 \bmod n}),\ g(I_{j+2 \bmod n}),\ \dots,\ g(I_{j+n-1 \bmod n}),g(I_j))g(I_j) - r I_j,\\ &\quad \text{for } j = 0, 1, \dots, n-1 \end{aligned} \end{align} $$

has a nontrivial $n\tau $ -periodic solution $(\tilde {I}_0,\ldots ,\tilde {I}_{n-1})$ , where $\tilde {I}_k=\tilde {I}(t-k\tau )$ , $k=0, 1,\ldots n-1$ . In particular, if $\tilde {I}(t)$ is a $\tau $ -periodic solution of (2.10), then is also a $\tau $ - periodic solution of the following ODE:

(3.6) $$ \begin{align} I'=H_{[\infty]}(I)g(I)-rI. \end{align} $$

Proof Suppose $\tilde {I}(t)$ is an $n\tau $ -periodic solution of (2.10) and define $ \tilde {I}_j(t):=\tilde {I}(t-j\tau )$ , $j=0,\ldots ,n-1$ , and $ \tilde {I}_j(t-\tau )=\tilde {I}_{j+1}(t),\,\,\tilde {I}_0(t)=\tilde {I}(t),\,\, \tilde {I}_k(t-n\tau )=\tilde {I}_{k+n}(t)=\tilde {I}_k(t). $ Then

(3.7) $$ \begin{align} \begin{aligned} \tilde{I}_j'= \tilde{I}'(t-j\tau)&=f_{[\infty]}(\dots,\tilde{I}(t-(j+1)\tau))g(\tilde{I}(t-j\tau))-r \tilde{I}(t-j\tau)\\ &=f_{[\infty]}(\dots, \tilde{I}_{j+1})g(\tilde{I}_j)-r \tilde{I}_j,\quad j\in N^{+}. \end{aligned} \end{align} $$

Because $\tilde {I}(t)$ is $n\tau $ -periodic, the sequence $\{\tilde {I}_k(t)\}_{k\ge 1}$ is $n$ -cyclic in $k$ .

By Lemma 3.3, we have

$$ \begin{align*}f_{[\infty]}(\ldots, \tilde{I}_n, \tilde{I}_{n-1}, \ldots, \tilde{I}_1) = H\big(g(\tilde{I}_1), \dots, g(\tilde{I}_n)\big). \end{align*} $$

Then we get the following $n$ -dimensional autonomous ODE system:

(3.8) $$ \begin{align} \begin{aligned} \tilde{I}_j'&= H(g(\tilde{I}_{j+1 \bmod n}),\ g(\tilde{I}_{j+2 \bmod n}),\ \dots,\ g(\tilde{I}_{j+n-1 \bmod n}),g(\tilde{I}_j))g(\tilde{I}_j) - r \tilde{I}_j,\\ &\quad \text{for } j = 0, 1, \dots, n-1, \end{aligned} \end{align} $$

which means that $(\tilde {I}_0,\tilde {I}_1,\ldots ,\tilde {I}_{n-1})$ is a $n\tau $ -periodic solution of system (3.5).

3.2 Some properties of the equivalent ODE system

We start with listing a few propositions.

Proposition 3.7 (Cyclic permutation operator on $\mathbb {R}^n$ )

Let

$$ \begin{align*}\gamma(t) = (I_0(t), I_1(t), I_2(t),\ldots,I_{n-1}(t) )\subset \mathcal{L}_n:=\mathbb{R}_+^n /\mathcal{D}_n \end{align*} $$

be an arbitrary solution of system (3.5) with the initial condition $ \gamma (0) $ . Let $ \boldsymbol {x} = (x_0, x_1, \dots , x_{n-1})^\top \in \mathbb {R}^n $ and define the cyclic shift operator $ \sigma \colon \mathcal {L}_n\to \mathcal {L}_n $ by $ \sigma (\boldsymbol {x}) := (x_1, x_2, \dots , x_{n-1}, x_0)^\top , $ which is bijection with $ \sigma ^n = \mathrm {id} $ , the identity map. Then, for each $ k = 1, 2\ldots ,n$ , the permuted $ \sigma ^k(\gamma (t)) $ , is also a solution of system (3.5) with initial condition $ \sigma (\gamma (0))$ . Moreover, $\gamma (t)$ , $\sigma (\gamma (t))$ , $\sigma ^2(\gamma (t))$ ,…, $\sigma ^n(\gamma (t))$ are isomorphic in $ \mathcal {L}_n $ .

Proposition 3.8 (Isomorphism)

$\mathfrak {S}_n$ is the symmetric group on the set $\{0, 1, \dots , n-1\}$ . Let $ \pi \in \mathfrak {S}_n $ and define the corresponding relabeled system by:

(3.9) $$ \begin{align} \tilde{I}_j := I_{\pi(j)} \, \Rightarrow \, \tilde{I}_j' = H(g(\tilde{I}_{j+1}), \dots, g(\tilde{I}_{j+n-1}), g(\tilde{I}_j))g(\tilde{I}_j) - r \tilde{I}_j, \, j = 0, 1, \dots, n-1, \end{align} $$

which is topologically conjugate to system (3.5) via the coordinate relabeling:

$$ \begin{align*}\Phi_\pi : (I_0, I_1, \dots, I_{n-1}) \mapsto (I_{\pi(0)}, I_{\pi(1)}, \dots, I_{\pi(n-1)}). \end{align*} $$

Then $ (\tilde {I}_0, \dots , \tilde {I}_{n-1}) = \Phi _\pi (I_0, \dots , I_{n-1}) $ . That is, if $ \gamma (t) = (I_0(t), \dots , I_{n-1}(t)) $ is a solution of the original system, then $ \Phi _\pi (\gamma (t)) $ is a solution of the permuted system (3.9). In addition, trajectories $ \gamma (t)$ and $ \Phi _\pi (\gamma (t)) $ are isomorphic in $ \mathcal {L}_n $ .

These results are natural consequence of the equivalence of the ODE vector field with respect to the cyclic group (see [Reference Balanov, Krawcewicz, Rachinskii, Wu and Yu3, Reference Golubitsky, Stewart and Schaeffer23]). For the sake of completeness, we provide some direct proof for Propositions 3.6–3.8, which are given in the Appendix.

Specifically, system (3.5) for $n = 2$ reduces to the following planar system:

(3.10) $$ \begin{align} \left\{\begin{array}{@{}l} I_0'=H(g(I_1),g(I_0))g(I_0)-rI_0, \\ I_1' =H(g(I_0),g(I_1))g(I_1)-rI_1. \\ \end{array}\right. \end{align} $$

By Proposition 3.6 and using the uniqueness of solutions, we obtain the following result.

To exclude periodic solutions of system (3.10), we need an additional assumption:

1. (H1a) $g'(x)>0$ for $x\ge 0$ .

To apply Lemma 3.2-(i), we need the following assumption on $g$ .

1. (H1b) $\operatorname {Sign}(H_2(y_1,y_2))(g'(x)x-g(x)) \ge 0$ for $y_1,y_2,x\ge 0$ .

Proof By Proposition 3.7 and Lemma 3.9, we only need to exclude the existence of non-constant periodic solution within one of $\Omega _+$ or $\Omega _-$ . We state the proof for $\Omega _+$ .

Define the scalar function:

$$ \begin{align*}V(x, y) := \log\left( \frac{x}{y} \right), \quad \text{for}\, x, y> 0. \end{align*} $$

We compute the derivative of $V(t):=V(I_0(t), I_1(t))$ along trajectories of the system as follows:

(3.11) $$ \begin{align} \begin{aligned} \frac{dV(t)}{dt} = \frac{\dot{I_0}}{I_0} - \frac{\dot{I_1}}{I_1} &= \left( H(g(I_1), g(I_0))\frac{g(I_0)}{I_0} - r \right) - \left( H(g(I_0), g(I_1))\frac{g(I_1)}{I_1} - r \right)\\ &= H(g(I_1), g(I_0))\frac{g(I_0)}{I_0} - H(g(I_0), g(I_1))\frac{g(I_1)}{I_1}.\\ \end{aligned} \end{align} $$

Therefore, if $H_2<0, \text { then } H_1>0$ by Corollary 3.2 (i), and if $g'(x)x-g(x)\le 0,$ we have

$$ \begin{align*} H(g(I_1),g(I_0))\frac{g(I_0)}{I_0}<H(g(I_0),g(I_0))\frac{g(I_0)}{I_0}<H(g(I_0),g(I_1))\frac{g(I_1)}{I_1}, \end{align*} $$

for all $ (I_0,I_1)\in \Omega _+.$ Consequently, $ \frac {dV(t)}{dt}<0.$ Similarly, if $ H_2>0$ and $ g'(x)x-g(x) \ge 0,$ using the same argument we obtain $ \frac {dV(t)}{dt}>0$ . Thus, $ V(I_0(t),\, I_1(t)) $ strictly monotonic on $\Omega _+$ , given that (H1a) and (H1b) hold. Now suppose, for contradiction, that there exists a nontrivial periodic solution $ (I_0(t), I_1(t)) $ with period $ T> 0 $ . Then $ V(0) = V(T) $ must hold. But this contradicts with the strict monotonicity of $ V(t) $ . Hence, no limit cycle exists in any region of the positive quadrant $ \mathbb {R}_+^2$ .

We also note the following Lemma.

The global dynamics of ODE system (3.5) with high dimension ( $n \geq 3$ ) is more delicate than the two-dimensional case and poses substantial analytical challenges. Li–Muldowney’s technique [Reference Li and Muldowney28] has been widely used to rule out nontrivial solutions, which is critical for us to use the global Hopf bifurcation theory. To use this approach, we first note that the Jacobian matrix associated with a general solution of (3.5) is $J_n := \big ( (J_n)_{k\ell } \big )_{0\le k,\ell \le n-1}$ with entries

(3.12) $$ \begin{align} \begin{aligned} & (J_n)_{k\ell}=\\ & \begin{cases} H_n(g(I_{k+1\bmod n}), g(I_{k+2 \bmod n}), \ldots, g(I_{k+n-1 \bmod n}),\\\quad g(I_k))\, g(I_k) + \beta_k g'(I_k) - r, & \ell = k, \\[0.6ex] H_m(g(I_{k+1 \bmod n}), g(I_{k+2 \bmod n}), \ldots, g(I_{k+n-1 \bmod n}), g(I_k))\, g(I_k), & \ell \neq k, \end{cases} \end{aligned} \end{align} $$

where $m$ is the position of $I_\ell $ in the ordered list $(I_{k+1\bmod n}, I_{k+2\bmod n}, \ldots , I_{k+n-1\bmod n}, I_k)$ and

(3.13) $$ \begin{align} \beta_k :=H\big(g(I_{k+1 \bmod n}), g(I_{k+2 \bmod n}), \ldots, g(I_{k+n-1 \bmod n}), g(I_k)\big), \,\, k=0,1,\ldots,n-1. \end{align} $$

From the definition of the function $H(x_1,x_2,\ldots ,x_0)$ , we obtain

$$ \begin{align*} \beta_k = \underbrace{f \big(f \big(\dots f \big(}_{n} \beta_k,\, g(I_k) \big),\, g(I_{(k+n-1)\bmod n})\big)\ldots,\, g(I_{(k+1)\bmod n})\big). \end{align*} $$

Therefore,

$$ \begin{align*}f(\beta_k, g(I_k)) = \underbrace{f \big(f \big(\dots f }_{n}\,\big( f(\beta_k, g(I_k)),\, g(I_{(k+n-1)\bmod n}) \big)\ldots,\, g(I_{(k+1)\bmod n})\big),\, g(I_k)\big).\end{align*} $$

It follows the following lemma.

Denote by $J_n^{[2]}$ the second addictive compound matrix of $J_n$ and by $\mu (\cdot )$ the Lozinskii measure [Reference Coppel13] corresponding to an absolute norm $|\cdot |$ . An application of Corollary 3.5 in [Reference Li and Muldowney28] yields the following lemma.

4.1 The local bifurcation of the Zhang–Scarabel–Murty–Wu outbreak model

The endemic equilibrium of system (4.1) is

(4.3) $$ \begin{align} I^* = \frac{R_0 - 1}{\kappa r}> 0,\,\, \beta^* = \frac{\beta_c}{1 + k \beta^* I^*}=r. \end{align} $$

It is easy to verify that $ H_{[\infty ]}(I^*)=\beta ^*=r$ .

It was shown in [Reference Zhang, Scarabel, Murty and Wu39] that $E_0:=(I_0,\beta _0)=(0,\beta _c)$ is stable if $R_0<1$ ; and if $R_0>1$ , then $E_0$ is unstable and system (4.1) has a unique inner equilibrium $E^*=(I^*,\beta ^*)$ given by (4.3). Hence,

$$ \begin{align*}\mathcal{N}(\mathcal{F})=\{(0,\tau,T),(I^*,\tau,T)\}. \end{align*} $$

Obviously, $\left ( 1 + \frac {R_0 - 1}{R_0} e^{-\lambda \tau } \right )\neq 1$ for any $\lambda \in \mathbb {C}$ . The characteristic equation for the linearization of system (4.1) at the endemic equilibrium $E^*$ is

(4.4) $$ \begin{align} \Delta_{\mathrm{S}}(E^*,\tau)(\lambda):=\lambda +\frac{\frac{R_0 - 1}{R_0^2} \, \beta_c e^{-\lambda \tau}}{ \left( 1 + \frac{R_0 - 1}{R_0} e^{-\lambda \tau} \right) }=0. \end{align} $$

We note that $\Delta _{\mathrm {S}}(E^*,\tau )(0)\neq 0$ if $R_0>1$ , hence, (S2) is satisfied.

In [Reference Zhang, Scarabel, Murty and Wu39], the issue of local Hopf bifurcation was considered when $R_0>1$ . It was shown that there exists a critical $\tau ^*$ such that: when $\tau \in [0,\tau ^*)$ , all real parts of the eigenvalue are negative, and $E^*$ is locally asymptotically stable. However, if $\tau $ passes through the critical value: $\tau _0$ and other $\tau _k, k\ge 1$ , there will be local Hopf bifurcations of periodic solutions. More precisely, at each $\tau =\tau _k$ , there appears a pair of purely imaginary eigenvalues $\pm i \omega ^*$ . When $\tau \in (\tau _k,\tau _{k+1}]$ , there are $k+1$ pairs of eigenvalues with positive real part, so $E^*$ is unstable for $\tau>\tau ^*$ ; system (4.1) undergoes a Hopf bifurcation of periodic solutions at $E^*$ when $\tau $ passes through $\tau _k$ for each $k=0,1,2,\ldots $ , where

(4.5) $$ \begin{align} \begin{aligned} \omega^* = \frac{\beta_c / R_0}{\sqrt{\left( \frac{R_0}{R_0 - 1} \right)^2 - 1}}>0, \end{aligned} \end{align} $$

(4.6) $$ \begin{align} \tan(\omega^* \tau) = \frac{-\beta_c / R_0}{\omega^*} = - \sqrt{\left( \frac{R_0}{R_0 - 1} \right)^2 - 1}<0, \end{align} $$

and then

(4.7) $$ \begin{align} \begin{aligned} \tau^*= \frac{\pi+ \arctan \left( - \sqrt{ \left( \frac{R_0}{R_0 - 1} \right)^2 - 1 } \right) }{\omega^*},\,\,\,\tau_k=\tau^*+\frac{2k\pi}{\omega^*},\,\,\ k=0,1,2,\dots \end{aligned} \end{align} $$

satisfying $\Delta _{\mathrm {S}}(E^*,\tau _k)(\omega ^* i)=0$ ,

$$ \begin{align*}\frac{d\lambda}{d\tau}=\frac{-\lambda^2}{\frac{-\frac{R_0-1}{R_0}\lambda}{\frac{R_0-1}{R_0}\lambda+\frac{(R_0-1)\beta_c}{R_0^2}}+\lambda\tau+1}, \end{align*} $$

and

(4.8) $$ \begin{align} \operatorname{Re}\left(\left.\frac{\mathrm{d} \lambda}{\mathrm{d} \tau}\right|_{\tau=\tau^{k}} ^{-1}\right)>0,\,\,\ k=0,1,2,\ldots \end{align} $$

Since

$$ \begin{align*}1-e^{-\frac{2\pi}{T}\lambda\tau}f_1(H_{[\infty]}(I^*),g(I^*))=1-e^{-\frac{2\pi}{T}\lambda\tau}f_1(\beta^*,I^*)=1+e^{-\frac{2\pi}{T}\lambda\tau}\frac{R_0-1}{R_0}\neq1, \end{align*} $$

condition (S1) is satisfied. Moreover, because $(R_0-1)/R_0\in (0,1)$ , condition (S1a) also holds. Then, by Lemma 2.5, we have

(4.9) $$ \begin{align} \begin{aligned} \det \Delta_{(I^*,\tau,T)}(\lambda)&=\lambda-\frac{T}{2\pi}\left(\frac{e^{-\frac{2\pi}{T}\lambda\tau}f_2(H_{[\infty]}(I^*),I^*)I^*}{1-e^{-\frac{2\pi}{T}\lambda\tau}f_1(H_{[\infty]}(I^*),g(I^*))}\right)\\ &=\lambda+\frac{T}{2\pi}\frac{\frac{R_0 - 1}{R_0^2} \, \beta_ce^{-\frac{2\pi}{T}\lambda\tau}}{ \left( 1 + \frac{R_0 - 1}{R_0}e^{-\frac{2\pi}{T}\lambda\tau} \right) }=\frac{T}{2\pi} \Delta_{\mathrm{S}}(E^*,\tau)\left(\frac{2\pi}{T}\lambda\right), \end{aligned} \end{align} $$

for any $\lambda \in \mathbb {C}$ , which also verifies conditions in Lemma 2.6. Based on the local Hopf bifurcation result of system (4.1), we obtain that $T^*=\frac {2\pi }{\omega ^*}$ , $m=1\in \mathcal {J}(I^*,\tau _k,T^*)$ and $\det \Delta _{(I^*,\tau _k,T^*)}\left (i\right )=0$ , which means that $\mathcal {N}_c(\mathcal {F})=\{(I^*,\tau _k,T^*),k=0,1,2\ldots \}$ and any $(I^*,\tau _k,T^*)$ is isolated.

For any fixed $k$ , there exist $\delta>0$ and a smooth curve $\lambda :(\tau _k-\delta ,\tau _k+\delta )\to \mathbb {C}$ such that $\Delta _{\mathrm {S}}(E^*,\tau )(\lambda (\tau ))=0$ , $ |\lambda (\tau ) - i\omega ^*| < \epsilon $ for all $ \tau \in (\tau _k - \delta , \tau _k+ \delta )$ , and

$$ \begin{align*}\lambda(\tau_k) = i\omega^*, \quad \left. \frac{d}{d\tau} \operatorname{Re} \lambda(\tau) \right|_{\tau = \tau_k}> 0. \end{align*} $$

Let $\Omega _{\epsilon ,T^*} =(0,\epsilon )\times (T^*-\epsilon ,T^*+\epsilon )$ . If $|\tau - \tau _k| \leq \delta $ and $(\mu , T) \in \partial \Omega _\epsilon $ such that $\Delta _{\mathrm {S}}(E^*,\tau )(\lambda (\tau )) (\frac {2\pi }{T}\mu + \frac {2\pi }{T}i) = 0$ , then $\mu = 0$ , $T = T^*$ (i.e., $\frac {2\pi }{T}=\omega ^*$ ), and $\tau = \tau _k$ . Hence, condition (S3) is satisfied for any $k=0,1,2,\ldots $ Furthermore, it follows from Lemma 2.8, that the first crossing number of $(I^*,\tau _k,T^*)$ is

$$ \begin{align*}\gamma_1(I^*,\tau_k,T^*)=-\operatorname{Sign} \left(\operatorname{Re}\left(\left.\frac{\mathrm{d} \lambda}{\mathrm{d} \tau}\right|_{\tau=\tau^{k}} ^{-1}\right)\right)=-1. \end{align*} $$

In summary, for system (4.1), given that $R_0>1$ , then any $(I^*,\tau _k,T^*)\in \mathcal {N}_c(\mathcal {F})=\{(I^*,\tau _k,T^*),k=0,1,2,\ldots \}$ is isolated and condition (S3) holds for any $(I^*,\tau _k,T^*)$ with $m=1$ . In addition, the isolated center $(I^*,\tau ^*,T^*)$ satisfies (S0), (S1), (S1a), and (S2). Denote by $C(I^*, \tau _k, T^*)$ the connected component through $(I^*, \tau _k, T^*)$ in $\Sigma (\mathcal {F})$ , for $k = 0, 1, 2, \dots $ Then, we have

$$ \begin{align*}\sum_{(\hat{x},\tau,T) \in C(I^*,\tau_k, T^*)\cap \mathcal{N}(\mathcal{F})} \gamma_1\left(\hat{x}, \tau, T \right) <0. \end{align*} $$

This allows us to apply Theorem 2.11 to obtain the following result.

From (4.6), we know $\frac {\pi }{2}<\omega ^*\tau ^*<\pi ,\,\ 2\pi <\omega ^*\tau _k<(2k+1)\pi ,\,k\geq 1.$ Therefore, $2<\frac {T^*}{\tau ^*}<4,\,\,\,\frac {1}{k+1}<\frac {T^*}{\tau _k}<1,\,\, k\geq 1, $ and $\tau _k>T^*,\,\,k=1,2,\ldots $ Then for $\tau>\tau _k$ , there exists an integer $n$ such that $\tau /n<T^*<\tau $ . Thus, we obtain the following remark.

Remark 4.2 Periodic solutions of (4.1) on the first Hopf branch based at $\tau ^*$ have period $T^*$ bounded between $2\tau ^*$ and $4\tau ^*$ . Moreover, if $R_0>2$ , period $T^*$ is bounded between $2\tau ^*$ and $3\tau ^*$ (see Figure 1-(a,b)), and thus periodic solutions on the first Hopf branch are slowly varying. Meanwhile, periodic solutions on other Hopf branches $\tau _k$ with $k>1$ have periods smaller than $\tau _k$ and thus are fast oscillating. In particular, if $ R_0> 2 $ (and hence, $ 2 < T/\tau ^* < 3$ ), and if system (4.1) has no $2\tau $ and $3\tau $ nontrivial periodic solution, then the period $T$ of a periodic solution on the connected component $C(I^*,\tau ^*,T^*)$ satisfies $T\in (2\tau ,3\tau )$ . These periodic solutions bifurcating from the first branch, slowly oscillating periodic solutions, resemble those slowly oscillating periodic solutions, intensively studied [Reference Diekmann, van Gils, Verduyn Lunel and Walther14, Reference Guo and Wu24], for delay differential equations.

Figure 1:

An illustration of the difficulty in obtaining lower and upper bounds of periodic solutions of a delay-algebraic system when varying the delay. (a) The $R_0$ – $T/\tau ^*$ plane with $r=0.5$ . (b) Ratio of period to delay for periodic solutions of (4.1) while varying $\tau $ , with $\kappa =0.05$ and $\beta _c=0.7143$ , for $R_0=2$ ( $\!\! \tau ^*=10.1571$ , yellow), $R_0=3$ ( $\!\! \tau ^*=10.8025$ , purple), $R_0=1.0753$ ( $\tau ^*=35.2005$ , blue), and $R_0=1.3889$ ( $\!\! \tau ^*=12.3638$ , orange); These numerical simulations demonstrate that a relatively large $R_0$ , specifically $R_0>2$ , ensures that the ratio $T/\tau $ is below $3$ . The solution curve of $ I(t) $ when $ \tau ^*=12.3638$ : upper panel shows the long-term periodic pattern, bottom panel shows the periodic wave comes as a spike with large amplitude in short time (within 90 days). Here, the delay $ \tau $ and the period T are measured in days, consistent with the parameterization adopted in [Reference Zhang, Scarabel, Murty and Wu39].

Figure 1 Long description

Panel a is a line graph titled R sub 0 minus T all over tau super asterisk for r = 0.5. The x-axis is R sub 0 ranging from 0 to 60, and the y-axis is T all over tau super asterisk ranging from 2 to 4. A red curve shows a steep exponential decay from R sub 0 = 1 to R sub 0 = 10, then levels off asymptotically toward 2.1. An inset graph shows a close-up for R sub 0 between 1 and 2.

Panel b is a scatter plot with tau on the x-axis (0 to 130) and T sub min all over tau on the y-axis (0 to 14). It contains four colored data series. A purple line remains flat near y = 2.5. A yellow line remains flat near y = 3. An orange line increases slightly then plateaus near y = 4.5. A blue line shows a steep linear increase starting from tau = 35, reaching y = 12.

Panel c consists of two vertically stacked time-series plots for I(t). The top plot shows a long-term periodic pattern with sharp blue spikes of equal height (1.6 times 10 super 6) occurring at regular intervals across a time t from 0 to 10,000. The bottom plot is a zoom-in on the interval 0 to 90, showing a single blue spike that begins to rise sharply after t = 70, reaching 3.5 times 10 super 5 by t = 90.

So our next goal is to exclude the existence of $2\tau -$ and $3\tau -$ nontrivial periodic solutions of system (4.1). Based on Theorem 3.4, it is sufficient to rule out the existence of $2\tau -$ and $3\tau -$ nontrivial periodic solutions of the corresponding ODE system (3.5) with $n=2,3$ .

4.2 Exclusion of nontrivial periodic solution of the equivalent ODE system

Since $g(x)=x$ satisfies (H1a) and (H1b), from Theorem 3.10 and Lemma 3.11, it follows the following lemma.

Next, we discuss the case $n=3$ . The corresponding function $H(x_1,x_2,x_3)$ can then be explicitly computed as follows:

(4.10) $$ \begin{align} \begin{aligned} H(x_1,x_2,x_3) &=\frac{1}{2\kappa x_3(x_1\kappa\beta_c+ 1)}\Big(-1 + \kappa\beta_c(x_3 - x_1 - x_2)\\ &+\sqrt{(\kappa\beta_c(x_3- x_1 - x_2) - 1)^2+4\kappa\beta_cx_3(x_2\kappa\beta_c + 1)(x_1\kappa\beta_c + 1)}\Big) , \end{aligned} \end{align} $$

which is positive in its domain. Moreover, $H(x_1,x_2,\infty )=\frac {\kappa \beta _c}{\kappa (\kappa \beta _c x_1+1)}<\beta _c$ , $H(0,0,0)=\beta _c$ , and

(4.11) $$ \begin{align} H(x, x, x) = \frac{\sqrt{4 \kappa\beta_c x + 1} - 1}{2 \kappa x}=H_{[\infty]}(x), \end{align} $$

which is strictly decreasing in $x$ with $\lim _{x\to \infty }H_{[\infty ]}(x)=0$ and $H(I^*,I^*,I^*)=H_{[\infty ]}(I^*)=r$ .

Then, system (A.1) is specified as follows:

(4.12) $$ \begin{align} \frac{d I_i}{dt} = \left(H(I_{i+1}, I_{i+2}, I_i) - r\right) I_i, \quad i = 0,1,2 \ (\mathrm{mod}\ 3). \end{align} $$

To rule out existence of periodic solutions of system (4.12), we first obtain the Jacobian matrix associated with a general solution of (4.12) as follows:

(4.13) $$ \begin{align} J_3:=\begin{pmatrix} H_3(I_1,I_2,I_0)\, I_0+\beta_0-r & H_1(I_1,I_2,I_0)\, I_0 & H_2(I_1,I_2,I_0)\, I_0\\ H_2(I_2,I_0,I_1)\, I_1 &H_3(I_2,I_0,I_1)\, I_1+\beta_1-r& H_1(I_2,I_0,I_1)\, I_1 \\ H_1(I_0,I_1,I_2)\, I_2 & H_2(I_0,I_1,I_2)\, I_2 & H_3(I_0,I_1,I_2)\, I_2+\beta_2-r\\ \end{pmatrix}, \end{align} $$

where $\beta _0=H(I_1,I_2,I_0)$ , $\beta _1=H(I_2,I_0,I_1)$ , and $\beta _2=H(I_0,I_1,I_2)$ . According to (3.4) and Lemma 3.13, we have

(4.14) $$ \begin{align} \begin{cases} H_1(I_{j+1\bmod{3}},I_{j+2 \bmod{3}},I_{j })=\dfrac{ f_2(\beta_{j+1\bmod{3}},I_{j+1\bmod{3}})}{1-\Pi_{i=0}^2f_1(\beta_i,I_i)}=\dfrac{ k_{j}\beta_{j}}{\alpha I_{j}},\\ H_2(I_{j+1\bmod{3}},I_{j+2 \bmod{3}},I_{j})=H_1(I_{j+1\bmod{3}},I_{j+2 \bmod{3}},I_{j })\,k_{j+1\bmod{3}},\\ H_3(I_{j+1\bmod{3}},I_{j+2 \bmod{3}},I_{j })=H_1\,k_{j+1\bmod{3}}\,k_{j+2\bmod{3}},\\ \end{cases}\,\, j=0,1,2, \end{align} $$

where

(4.15) $$ \begin{align} \begin{aligned} k_{i }:=\frac{f_2(\beta_{i+1 \bmod 3},I_{i+1 \bmod 3})f_1(\beta_{i },I_{i})}{f_2(\beta_{i},I_{i})}<0,\qquad i=0,1,2,\\ \end{aligned} \end{align} $$

and

(4.16) $$ \begin{align} \alpha:=1-\Pi_{i=0}^{2}f_1(\beta_i,I_i)=1-k_0k_1k_2>0. \end{align} $$

Denote by $\mu _2(\cdot )$ the Lozinskii measure [Reference Coppel13] with respect to the Euclidean norm $\|\cdot \|_2$ . Let $S_0 := \alpha \, J_3^{[2]}$ and $S_1 := \frac {S_0+S_0^{\top }}{2}.$ Denote by $J_3^{*[2]}$ the matrix $J_3^{[2]}$ evaluated at equilibrium $(I^*,I^*,I^*)$ . Denote by $S^*$ the matrix $S_1$ evaluated at equilibrium $(I^*,I^*,I^*)$ . We have the following result.

The proof can be found in the Appendix.

Different from a corresponding delay differential equation, it seems very difficult (if not impossible, see discussions in the final discussion section) to determine the sign of $\mu _2(J_3^{[2]})$ in the whole $\mathbb {R}_+^3$ , so we estimate a suitable region $\Omega $ , where $\mu _2(J_3^{[2]})<0$ according to Lemma 3.14.

Let $\Delta I_i := I_i - I^*$ for $i=0,1,2$ , and define $ \theta := \|\Delta I\|_\infty = \max _{i=0,1,2} |\Delta I_i|$ . We have the following result.

Lemma 4.5 Suppose that $R_0>2$ (ensuring $k_*\in (1/2,1)$ and $\lambda _{\max }(S^*)<0$ ). If $\theta < \mathcal {G} $ , then $\mu _2(J^{[2]}_3)<0$ (equivalently, $\lambda _{\max }(S_1)<0$ ), where $\mathcal {G}:=\min \{\theta ^*_1,\theta _2^*\}. $ Here, $\theta _1^*$ and $\theta _2^*$ are the unique positive root in the interval $(0,I^*)$ of

$$ \begin{align*}R^*_1(r-\mathcal{B}(x))+C_I(x)R_2^*(r-\mathcal{B}(x)) =-\lambda_{\max}(S^*), \end{align*} $$

and

$$ \begin{align*}R^*_1(\mathcal{B}(-x)-r)+C_I(x)R_2^*(\mathcal{B}(-x)-r) =-\lambda_{\max}(S^*), \end{align*} $$

respectively. The explicit expressions of $C_I(\cdot ),\mathcal {B}(\cdot ), R_1^*(\cdot )$ , and $R_2^*(\cdot )$ are given in (A.8), (A.9), (A.12), and (A.13), respectively.

We defer this important estimation to the Appendix. Based on Lemmas 3.14 and 4.5, we have the following theorem.

As a consequence, we have the following result for system (4.1).

We also need to establish the lower and upper bounds to $\beta (t)$ .

Lemma 4.8 Given that $R_0>1$ and $\tau>\tau ^*$ . If the component of periodic solutions $I(t)$ of system (4.1) is restricted in the interval $(I^*-\mathcal {G},I^*+\mathcal {G})$ for any $t\in R$ , then there exist constants $\beta _{-},\beta _{+}>0$ , given by

$$ \begin{align*}\beta_{-}=\frac{-(1+2\kappa \beta_c \mathcal{G})+\sqrt{(1+2\kappa \beta_c \mathcal{G})^2+4\kappa I_{-}\beta_c}}{2\kappa I_{-}}, \end{align*} $$

and

$$ \begin{align*}\beta_{+}=\frac{-(1-2\kappa \beta_c \mathcal{G})+\sqrt{(1-2\kappa \beta_c \mathcal{G})^2+4\kappa I_{+}\beta_c}}{2\kappa I_{+}}, \end{align*} $$

such that $\beta (t) \in (\beta _{-}, \beta _{+})$ for all $t\in [t_0,\infty )$ , as long as $\beta (s)\in (\beta _{-}, \beta _{+})$ for all $s\in [t_0-\tau ,t_0]$ . Here, $I_{-}:=I^*-\mathcal {G}$ and $I_{+}:=I^*+\mathcal {G}$ . Moreover, as $\mathcal {G}$ decreases, $\beta _{-}$ increases and $\beta _{+}$ decreases, with $\beta ^*\in (\beta _{-},\beta _{+})$ . In particular, as $\mathcal G\downarrow 0$ , we have $\beta _{-}\uparrow r$ , $\beta _{+}\downarrow r$ . Then $|\beta (t)-r|<\max \{\beta _{+}-r,r-\beta _{-}\}$ for all $t\in [t_0-\tau ,\infty )$ .

Proof Applying the relation $I_{+}-I_{-}=2\mathcal {G}$ and solving $F(I_+\beta _+)=\beta _-$ and $F(I_-\beta _-)=\beta _+$ gives

$$ \begin{align*}\beta_{-}:=\frac{-(1+2\kappa \beta_c \mathcal{G})+\sqrt{(1+2\kappa \beta_c \mathcal{G})^2+4\kappa I_{-}\beta_c}}{2\kappa I_{-}}, \end{align*} $$

and

$$ \begin{align*}\beta_{+}:=\frac{-(1-2\kappa \beta_c \mathcal{G})+\sqrt{(1-2\kappa \beta_c \mathcal{G})^2+4\kappa I_{+}\beta_c}}{2\kappa I_{+}}. \end{align*} $$

Define

$$ \begin{align*}U(x):=\frac{-(1+2\kappa \beta_cx)+\sqrt{(1+2\kappa \beta_c x)^2+4\kappa (I^*-x)\beta_c}}{2\kappa (I^*-x)}. \end{align*} $$

Then $U(0)=\beta ^*$ , $\beta _{-}=U(\mathcal {G})$ and $\beta _{+}=U(-\mathcal {G})$ . Moreover,

$$ \begin{align*} U'(x)&=\frac{4 I^*\beta_c^{2} \kappa^{2} x+4 \kappa \beta_c I^* +1 -(2 I^* \beta_c \kappa+1)\sqrt{4 \beta_c^{2} \kappa^{2} x^{2}+4 \kappa \beta_c I^* +1} }{2 \kappa \left( I^* -x \right)^{2}\sqrt{4\beta_c^{2} \kappa^{2} x^{2}+4 k \beta_c I^* +1} }. \end{align*} $$

Since

$$ \begin{align*} &(4 I^*\beta_c^{2} \kappa^{2} x+4 \kappa \beta_c I^* +1)^2-(2 I^* \beta_c \kappa+1)^2(4 \beta_c^{2} \kappa^{2} x^{2}+4 \kappa \beta_c I^* +1)\\ &\quad=-4\beta_c^2\kappa^2(x-I^*)^2(4\kappa\beta_cI^* + 1)<0, \end{align*} $$

we conclude that $U'(x)<0$ for $x\in [0,I^*)$ . Therefore, $\frac {d U(-x)}{dx}>0$ for $x\in [0,I^*)$ . Hence, $\beta _{-}<\beta _{+}$ and $\beta ^*\in (\beta _{-},\beta _{+})$ .

We now prove $\beta (t)\in (\beta _{-},\beta _{+})$ for all $t\geq t_0$ . Fix $t\in [t_0,t_0+\tau ]$ , since $I(t-\tau )\in (I_-,I_+)$ and $\beta (t-\tau )\in [\beta _-,\beta _+]$ , we have

$$\begin{align*}F(I(t-\tau)\beta_+)\ \le\ F(I(t-\tau)\beta(t-\tau))\ =\ \beta(t)\ \le\ F(I(t-\tau)\beta_-). \end{align*}$$

By strict monotonicity of $F(\xi )$ on $[0,\infty )$ , we obtain

$$\begin{align*}\beta_{-}=F(I_+,\beta_+)\ <\ F(I(t-\tau)\beta_+),\qquad F(I(t-\tau)\beta_-)\ <\ F(I_-\beta_-)=\beta_{+}. \end{align*}$$

Hence, $\beta _-< \beta (t) < \beta _+$ , for any $t\in [t_0,t_0+\tau ]$ . Repeating the above on successive intervals $[t_0+n\tau ,t_0+(n+1)\tau ]$ , we obtain $\beta _-\ <\ \beta (t)\ <\ \beta _+$ for all $t\ge t_0$ .

We can now obtain a range of plausible periodic solutions: Suppose $R_0>2$ . Then there exists $\delta _{\mathcal {G}}>0$ such that, for any fixed $\tau \in (\tau ^*,\,\tau ^*+\delta _{\mathcal {G}}]$ with $\tau _M:=\tau ^*+\delta _{\mathcal {G}}$ , for every periodic solution of system (4.1) with period $T$ satisfying $2\tau <T<3\tau $ , denoted as $E(t;\tau ,T):=(I(t;\tau ,T),\beta (t;\tau ,T))$ , $I(t;\tau ,T)\in \left [ I^*-\mathcal {G},\ I^* +\mathcal {G} \right ]$ holds (see Figure 2-(b)). This statement about the range of plausible periodic solutions can be easily shown by way of contradiction. Assuming otherwise, for any $\delta $ , there is some $\tau _{\delta }\in (\tau ^*,\tau ^*+\delta )$ , such that system (4.1) admits a $T$ -periodic solution $I(t;\tau _\delta ,T)$ , for which there is a time $t_s\in R$ such that $I(t_s)=I_{+}$ , $I'(t_s)>0$ and $I(t)\in (I_-,I_+)$ for all $t\in [-\tau ,t_s]$ , then $\beta (t)\in (\beta _{-},\beta _{+})$ for all $t<t_s$ from Lemma 4.8. Hence, $\beta (t_s)>r$ due to $I'(t_s)>0$ . Let $\delta _n:=1/n$ , for every $n$ , there is some $\tau _{n}\in (\tau ^*,\tau ^*+\delta _n)$ , and there exists a $T_n$ periodic solution $I_n(t)$ with $T_n\in (2\tau _n,3\tau _n)$ , there exists $t_{s,n}$ such that $I_n(t_{s,n})=I_{+}$ , $I_n'(t_{s,n})> 0$ , and $ I_n(t)<I_{+}$ for all $s<t_{s,n}$ . Consequently, from $I^{\prime }_n(t)=(\beta _n(t)-r)I_n(t)$ with $I_{+}>0$ , we have $\beta _n(t_{s,n})> r$ . Note that $\beta _n(t_{s,n}+\tau _n)=f(\beta _n(t_{s,n}),I_n(t_{s,n}))=f(\beta _n(t_{s,n}),I_+)\le f(r,I_+). $ Therefore, $ \beta ^*-\beta _n(t_{s,n}+\tau _n)\ \ge F(rI^*)- F(r I_+). $ Using the mean value theorem, there exists $\xi \in [rI^*,rI_+]$ such that

$$\begin{align*}F(rI^*)-F(rI_{+})=\big|F'(\xi)\big|\,r(I_{+}-I^*) \ \ge \big|(F'(rI_{+})\big| r \mathcal{G}, \end{align*}$$

since $-F'(x)$ decreases on $[0,\infty )$ . Hence, $ \big |\beta _n(t_{s,n}+\tau _n)-\beta ^*\big |\ \ge \big |(F'(rI_{+})\big | r \mathcal {G}$ , a contradiction.

Figure 2:

(a) $\lambda _{\max }(S_1)$ for all $(I_0,I_1,I_2)$ in the $\theta $ -neighborhood of equilibrium $I^*$ with $\theta \in [0,I^*)$ . The solid blue curve shows the maximum (“worst-case”) value of $\lambda _{\max }(S_1)$ over the grid for each fixed $\theta \in [0,I^*)$ . This numerical illustration shows that $\lambda _{\max }(S_1)<0$ holds for all $I_0,I_1,I_2\in (I^*-0.951,I^*+0.951)$ . (b) Periodic solution of (4.1) with amplitude in $(I^*- \mathcal {G},I^*+ \mathcal {G})$ ; here, $\tau ^*=3.989327447$ and $\tau _e\approx 4.039$ , $\frac {d\lambda }{d\tau }|_{\lambda =\omega i,\tau =\tau ^*}=0.04807979388-0.1740577i $ . (c) Bifurcation diagram of the amplitude of $I(t)$ with respect to $\tau $ for system (4.1). For panels (a)–(c), the parameter values are $\kappa =1.2$ , $\beta _c=2$ , $r=0.6$ , and $I^*=3.24074$ . The corresponding value $\lambda _{\max }(S^*)=-0.2856$ and $\mathcal {G}=0.2297436746<0.951$ , and panel (a) numerically confirms Lemma. 4.5, showing that $\lambda _{\max }(S_1)<0$ for all $I_0,I_1,I_2\in (I^*-\mathcal {G},\,I^*+\mathcal {G})$ . (d). Periodic solution of (4.1) with amplitude in $(I^*- \mathcal {G},I^*+ \mathcal {G})$ , the parameter values are $\kappa =0.005$ , $\beta _c=2$ , $r=0.6$ , $I^*=777.778$ , $\tau ^*=3.989327447$ , $\mathcal {G}=55.192$ , and $\tau _e\approx 4.0166$ .

Figure 2 Long description

The figure consists of four panels labeled (a) to (d).

* Panel (a): A line graph with theta on the x-axis ranging from 0 to 3 and lambda sub max (S sub 1) on the y-axis ranging from -2 to 18. A solid blue curve increases exponentially. A red horizontal line marks y equals 0. A red circle at (0.951, 0) indicates an intersection point. A gray shaded region exists below the blue curve.

* Panel (b): A line graph titled Solution I(t) with tau equals 4.039. The x-axis is time t (multiplied by 10 super 4) and the y-axis is I(t). The data shows a high-frequency oscillation between approximately 2.6 and 4.2. Two red dashed horizontal lines mark the upper and lower bounds of the oscillation.

* Panel (c): A bifurcation diagram with tau on the x-axis from 0 to 5 and Amplitude on the y-axis from 0 to 45. A red horizontal line remains constant at an amplitude of approximately 3 until tau equals 4, where a blue curve branches upward, showing a sharp increase in amplitude.

* Panel (d): A line graph titled Solution I(t) with tau equals 4.0166. The x-axis is time t (multiplied by 10 super 4) and the y-axis is I(t). It shows a dense, high-frequency oscillation centered around 775, bounded by two red dashed horizontal lines at approximately 725 and 830.

Recall that $C(I^*, \tau ^*, T^*)$ is the connected component through $(I^*, \tau ^*, T^*)$ in $\Sigma (\mathcal {F})$ . We define

$$ \begin{align*}\tau _{e}:=\sup \{\tau> \tau ^*; & \text{ there is a non-constant periodic solution } (\beta (t), I(t)) \text{ of}\ (4.1) \\ & \text{ such that } I(t)\in (I^*-\mathcal{G}, I^*+\mathcal{G}) \text{ for } t\in R\}. \end{align*} $$

The local Hopf bifurcation theorem and the above discussion on the range of plausible periodic solutions ensure that the set defining $\tau _e $ is nonempty, and therefore $\tau _e$ is well defined. The value of $\tau _e$ could be infinity theoretically, but our simulations provided later indicate that after $\tau ^*$ , as $\tau$ increases, the amplitude of the periodic solution becomes so large that the periodic wave presents as a spike. These indicate the time delay is indeed the reason for onset of a large epidemic wave that susceptible populations will be depleted by the infection to a level that our standing assumption that susceptible remains relatively unchanged is no longer valid. Note also that while the expressions for $\tau ^*$ and $\lambda _{\max }(S^*)$ are independent of $\kappa$ , both $I^* \text { and } \mathcal G$ vary with $\kappa$ . Simulations reported in Figure 2-(b) and (d) show that $ \tau _e $ depends on $ \kappa $ , and small variation of $ \kappa $ can lead to larger values of $ I^* $ and $\mathcal {G} $ , resulting in smaller values of $\tau _e $ .

We conclude this section with a note that if $\tau _e>\tau _k$ for some $k\ge 1$ , then a similar argument can be used to show that the projection of $C(I^*, \tau _k, T^*)$ onto the $\tau $ -space contains $(\tau _k, \tau _e)$ , and hence the existence of a fast oscillating periodic solution can also be established for $\tau \in (\tau _k, \tau _e)$ .