In this section, we will study the condition for rumor eradication and system permanence, respectively. Firstly, we show that all solutions of system (2) ultimately are bounded from Theorem 4.1. By using Floquet theory and the small amplitude perturbation method [28], we obtain the conditions for the local stability of a rumor eradication periodic solution \(( 0,y^{*} ( t ) )\) from Theorem 4.2. We named it the rumor eradication theorem. Based on Definition 3.2, we will show the conditions for the permanence of system (2) from Theorem 4.3. We named it the system permanence theorem.

### Theorem 4.1

*There exists a constant*
\(M > 0\)
*such that*
\(x ( t ) \le M\), \(y ( t ) \le M\)
*for each solution*
\(( x ( t ),y ( t ) )\)
*of system* (2) *with all*
*t*
*large enough*.

### Proof

Suppose \(( x ( t ),y ( t ) )\) is any solution of system (2). Define

$$ V ( t ) = \gamma x ( t ) + y ( t ). $$

(7)

Then \(V \in V_{0}\),

$$ \textstyle\begin{cases} D^{ +} V ( t ) + \lambda V ( t ) = \gamma ( r + \lambda )x ( t ) - \frac{\gamma r}{k}x ^{2} ( t ) + ( \lambda - d )y ( t ), \quad t \ne nT, \\ V ( nt^{ +} ) = V ( nt ) + p, \quad t = nT. \end{cases} $$

(8)

Clearly, the right hand side of system (8) is bounded when \(0 < \lambda < d\). Select a suitable \(\lambda _{0}\) and let \(M_{0}\) be the bound. Thus, system (8) leads to

$$ \textstyle\begin{cases} D^{ +} V ( t ) \le M_{0} - \lambda _{0}V ( t ),\quad t \ne nT, \\ V ( nt^{ +} ) = V ( nt ) + p, \quad t = nT. \end{cases} $$

(9)

By Lemma 3.2, if \(t \in ( nT, ( n + 1 )T ]\), we can get

$$\begin{aligned} V ( t ) \le& \biggl( V \bigl( 0^{ +} \bigr) - \frac{M _{0}}{\lambda _{0}} \biggr)\exp ( - \lambda _{0}t ) \\ &{}+ \frac{p ( 1 - \exp ( - n\lambda _{0}T ) )}{\exp ( \lambda _{0}T ) - 1}\exp ( \lambda _{0}T ) \exp \bigl( - \lambda _{0} ( t - nT ) \bigr) + \frac{M _{0}}{\lambda _{0}}. \end{aligned}$$

Hence

$$ \lim_{t \to \infty } V ( t ) \le \frac{M_{0}}{\lambda _{0}} + \frac{p\exp ( \lambda _{0}T )}{ \exp ( \lambda _{0}T ) - 1}. $$

Therefore \(V ( t )\) is ultimately bounded by a constant and there exists a constant \(M > 0\), such that \(x ( t ) \le M\), \(y ( t ) \le M\) for each solution \(( x ( t ),y ( t ) )\) of system (2) with all *t* large enough. The proof is completed. □

### Theorem 4.2

*Let*
\(( x ( t ),y ( t ) )\)
*be any solution of system* (2), *if*
\(p > \frac{d}{ \beta } [ rT - \ln \frac{1}{ ( 1 - q )} ]\)
*then*
\(( 0,y^{*} ( t ) )\)
*is said to be locally asymptotically stable*.

### Proof

The locally asymptotical stability of the periodic solution \(( 0,y^{*} ( t ) )\) may be determined by considering the behavior of a small amplitude perturbation of the solution. Define \(x ( t ) = u ( t )\), \(y ( t ) = y^{*} ( t ) + v ( t )\). Here \(v ( t )\) is a small perturbation from system (2), and we have

\left(\begin{array}{c}u(t)\\ v(t)\end{array}\right)=\Phi (t)\left(\begin{array}{c}u(0)\\ v(0)\end{array}\right),\phantom{\rule{1em}{0ex}}0\le t<T.

The matrix solution matrix \(\varPhi ( t )\) satisfies

\Phi (t)=\left(\begin{array}{cc}exp(t(r-\beta {y}^{\ast}))& 0\\ \frac{exp(-dt)[exp(t(d+r-\beta {y}^{\ast}))-1]}{d+r-\beta {y}^{\ast}}\gamma \beta {y}^{\ast}& exp(-dt)\end{array}\right).

(10)

After adding impulses, system (2) becomes

\left(\begin{array}{c}u(n{T}^{+})\\ v(n{T}^{+})\end{array}\right)=\left(\begin{array}{cc}1-q& 0\\ 0& 1\end{array}\right)\left(\begin{array}{c}u(nT)\\ v(nT)\end{array}\right).

(11)

By the Floquet theory of impulsive differential equations [28], \(( 0,y^{*} ( t ) )\) is locally asymptotically stable if \(\vert p_{1} \vert = \exp ( - d \times T )\) and \(\vert p_{2} \vert = ( 1 - q )\exp \int _{0}^{T} ( r - \beta y^{*} ) \,dt\) the absolute values of both eigenvalues are less than one, thus we obtain the conditions \(p > \frac{d}{\beta } [ rT - \ln \frac{1}{ ( 1 - q )} ]\) for the local stability of the rumor eradication periodic solution. □

### Theorem 4.3

*If*
\(p < \frac{d}{\beta } [ rT - \ln \frac{1}{ ( 1 - q )} ]\)
*then system* (2) *is persistent*.

### Proof

Suppose that \(( x ( t ),y ( t ) )\) is any solution of system (2) with \(x ( 0 ) > 0\). By Theorem 4.1, we have proved that there exists a constant \(M > 0\), such that \(x ( t ) \le M\), \(y ( t ) \le M\) and \(M > k\), for *t* large enough. Then we will prove that the solution of system (2) has lower bounds to complete the proof of system permanence.

Define

$$ m_{1} = \frac{p\exp ( - dT )}{1 - \exp ( - dT )} - \sigma , \quad \sigma > 0. $$

(12)

From Lemma 3.2 we can get \(\lim_{t \to \infty } y ( t ) > m_{1}\), which is to say \(y ( t )\) has lower bounds. Next we shall find a \(m_{2} > 0\) so that \(x ( t ) \ge m_{2}\) for *t* large enough. We will prove this in the following two steps.

Firstly, since \(p < \frac{d}{\beta } [ rT - \ln \frac{1}{ ( 1 - q )} ]\), the solution of system (2) is unstable. We select \(0 < m_{3} < k\) and \(\varepsilon > 0\) small enough such that \(\delta = \frac{\gamma \beta m_{3}}{1 + \alpha m_{3}^{2}} < d\), \(\sigma = rT - \frac{r}{k}m_{3}T - \beta \frac{p}{d - \delta } - \varepsilon _{1}T > 0\). Now we assume that there exists \(t_{1} \in ( 0, \infty )\) such that \(x ( t_{1} ) < m_{3}\). We call this assumption one.

Thus, we can get \(\dot{y} ( t ) \le y ( t ) ( \delta - d )\). Considering the impulsive differential equation.

$$ \textstyle\begin{cases} \dot{u} ( t ) = u ( t ) ( \delta - d ),\quad t \ne nT, \\ u ( t^{ +} ) = u ( t ) + p,\quad t = nT, \\ u ( 0^{ +} ) = y_{0} > 0. \end{cases} $$

(13)

By Lemmas 3.2 and 3.3, we have \(y ( t ) \le u ( t )\) and \(u ( t ) \to \overline{u} ( t )\), \(t \to \infty \). Here

$$ \overline{u} ( t ) = \frac{p\exp [ ( \delta - d ) ( t - nT ) ]}{1 - \exp [ ( \delta - d )T ]},\quad t \in \bigl( nT, ( n + 1 )T \bigr] $$

(14)

exists \(T_{1} > 0\), when \(t > T_{1}\),

$$ y ( t ) \le u ( t ) \le \overline{u} ( t ) + \varepsilon $$

(15)

and

$$ \dot{x} ( t ) \ge x ( t ) \biggl[ r - \frac{r}{k}m_{3} - \beta \bigl( \overline{u} ( t ) + \varepsilon _{1} \bigr) \biggr] $$

(16)

let \(N_{1} \in N\) and \(N_{1}T > T_{1}\), \(n \le N_{1}\), we will have

$$ \textstyle\begin{cases} \dot{x} ( t ) \ge x ( t ) [ r - \frac{r}{k}m_{3} - \beta ( \overline{u} ( t ) + \varepsilon _{1} ) ], \quad t \ne nT, \\ x ( t^{ +} ) = ( 1 - q )x ( t ),\quad t = nT \end{cases} $$

(17)

moreover, we can obtain the following result:

$$\begin{aligned} x \bigl( ( n + 1 )T \bigr) \ge& x ( nT ) \exp \biggl( \int _{nT}^{ ( n + 1 )T} \biggl[ r - \frac{r}{k}m_{3} - \beta \bigl( \overline{u} ( t ) + \varepsilon _{1} \bigr) \biggr]\,dt \biggr) \\ =& x ( nT ) \exp ( \sigma ). \end{aligned}$$

(18)

Then \(x ( ( N_{1} + l )T ) \ge x ( N_{1}T ^{ +} )\exp ( l\sigma ) \to \infty \) as \(l \to \infty \), which is a contradiction to Theorem 4.1. Therefore, the first assumption is not true. Hence there exists a \(t_{1} > 0\) such that \(x ( t_{1} ) \ge m_{3}\).

Secondly, we only need to consider those solutions that leave the region \(R = \{ X ( t ) \in R_{ +}^{2}:x ( t ) < m_{3} \} \) moreover, reenter again. Let \(t^{*} = \inf_{t \ge t_{1}} \{ x ( t ) < m_{3} \}\), then we have two cases for discussion.

Case 1: if the \(t^{*}\) is an impulsive point, there exists \(n_{0} \in N\), \(t^{*} = n_{0}T\), \(\varepsilon ^{*} > 0\) small enough. We have \(t_{*} = t^{*} - \varepsilon ^{*}\), so \(t_{*}\) is a non-impulsive point and \(x ( t_{*} ) \ge m_{3}\).

Case 2: if the \(t^{*}\) is a non-impulsive point, \(x ( t ) \ge m_{3}\) on \([ t_{1},t^{*} )\) and \(x ( t^{*} ) = m_{3}\).

Let us assume \(t^{*} \in [ n_{1}, ( n_{1} + 1 )T )\), \(n_{1} \in N\), is a non-impulsive point. We call this assumption two. Choose \(n_{2},n_{3} \in N\), such that

$$ n_{2}T > T_{2} = \frac{\ln ( \frac{\varepsilon _{1}}{M + p} )}{ - d + \delta },\qquad \exp \bigl( ( n_{2} + 1 ) \sigma _{1}T \bigr)\exp ( n_{3}\sigma ) > 1. $$

(19)

Here \(\sigma _{1} = r - \frac{r}{k}m_{3} - \beta M < 0\). Let \(\overline{T} = ( n_{2} + n_{3} )T\), there exists \(t_{2} \in ( ( n_{1} + 1 )T, ( n_{1} + 1 )T + \overline{T} )\), such that \(x ( t_{2} ) \ge m_{3}\), now suppose that \(t \in ( ( n_{1} + 1 )T, ( n _{1} + 1 )T + \overline{T} )\), satisfy \(x(t) < m_{3}\) we call this assumption three. Considering

$$ \textstyle\begin{cases} \dot{u} ( t ) = u ( t ) ( \delta - d ),\quad t \ne nT, \\ u ( t^{ +} ) = u ( t ) + p,\quad t = nT, \\ u ( 0^{ +} ) = y_{0} > 0 \end{cases} $$

(20)

let \(y ( ( n_{1} + 1 )T^{ +} ) = u ( ( n_{1} + 1 )T^{ +} )\) and we have

$$ u ( t ) = \biggl( u \bigl( ( n_{1} + 1 )T ^{ +} \bigr) - \frac{p}{1 - \exp ( ( - d + \delta )T )} \biggr)\exp ( - d + \delta ) \bigl( t - ( n_{1} + 1 )T \bigr) + \overline{u} ( t ), $$

(21)

where \(t \in ( nT, ( n + 1 )T )\), \(n_{1} + 1 \le n \le n_{1} + 1 + n_{2} + n_{3}\).

So, when \(( n_{1} + 1 + n_{2} )T \le t \le ( n_{1} + 1 )T + \overline{T}\), is satisfied then

$$ \bigl\vert u ( t ) - \overline{u} ( t ) \bigr\vert < ( M + q )\exp ( - d + \delta ) < \varepsilon _{1} $$

(22)

and

$$ y ( t ) \le u ( t ) \le \overline{u} ( t ) + \varepsilon . $$

(23)

as in the first step, when \(t \in [ ( n_{1} + 1 + n_{2} )T, ( n_{1} + 1 )T + \overline{T} ]\) we have

$$ x \bigl[ ( n_{1} + 1 + n_{2} + n_{3} )T \bigr] \ge x \bigl[ ( n_{1} + 1 + n_{2} )T \bigr]\exp ( n _{3}\sigma ) $$

(24)

the first equation of system (2) gives

$$ \dot{x} ( t ) \ge x ( t ) \biggl[ r - \frac{r}{k}m_{3} - \beta M \biggr] = \sigma _{1}x ( t ) $$

(25)

integrating Eq. (25) on \([ t^{*}, ( n_{1} + 1 + n_{2} )T ]\), we can get

$$ x \bigl[ ( n_{1} + 1 + n_{2} )T \bigr] \ge m_{3} \exp \bigl( \sigma _{1} ( n_{2} + 1 )T \bigr) $$

(26)

thus, from Eqs. (24) and (26), we have

$$ x \bigl[ ( n_{1} + 1 + n_{2} + n_{3} )T \bigr] \ge m _{3}\exp \bigl( \sigma _{1} ( n_{2} + 1 )T \bigr)\exp ( n_{3}\sigma ) > m_{3} $$

(27)

which is a contradiction to assumption three. Therefore, assumption three is not true. Thus for \(t \in ( n_{1}T, ( n_{1} + 1 )T + \overline{T} ]\) we get \(x ( t ) \ge m _{3}\). And assumption two is really true, \(t^{*}\) is a non-impulsive point, \(x ( t^{*} ) = m_{3}\).

Let \(\overline{t} = \inf_{t \ge t^{*}} \{ x ( t ) \ge m_{3} \}\), then \(x ( \overline{t} ) \ge m_{3}\), for \(t \in [ t^{*},\overline{t} )\), we have

$$ x ( t ) \ge x \bigl( t^{*} \bigr)\exp \bigl( \sigma _{1} \bigl( t - t^{*} \bigr) \bigr) \ge m_{3}\exp \bigl( \sigma _{1} ( 1 + n_{2} + n_{3} )T \bigr) \triangleq m_{2}. $$

(28)

For \(t > \overline{t}\), the same arguments hold since \(x ( \overline{t} ) \ge m_{3}\). Hence \(x ( t ) \ge m _{2}\), for all \(t > t_{1}\). That is, for \(t \to \infty \), \(x ( t )\) has lower bounds. The proof is completed. In summary, for \(p < \frac{d}{\beta } [ rT - \ln \frac{1}{ ( 1 - q )} ]\) system (2) is persistent. □