The dynamics analysis of Gompertz virus disease model under impulsive control

Abstract

The pandemic of Gompertz virus disease remains a pressing issue in agricultural production. Moreover, the dynamics of various infectious diseases is usually investigated by the method of mathematical modelling. A new mathematical model for dynamics on Gompertz virus disease impulsive system is proposed and analyzed in this paper. We prove the dynamic characteristics about the permanence and globally exponential stability of Gompertz virus disease model. Moreover, we also give the sufficient condition that one positive solution which satisfies $E(t)\ge q_2$ if $R_{1\ast }>1$ exists. Eventually, numerical simulations are utilized to validate the validity of the theoretically analyzed conclusion in this paper.

Introduction

It is well-known that a nonlinear system model usually has a rich dynamic behavior which has been payed attention to by many researchers^1–6. Moreover, in recent years, many researchers have paid attention to biological control^7–16.

Due to many cases that we need to describe the biological dynamics with sudden changes as well as other phenomena, impulsive differential equations play important part^17–19. In consideration of hybrid nature, different kinds of evolutionary processes in real-world life are usually described by impulsive systems, and these evolutionary processes usually state a big sudden change at some moments^20–23. Some researchers stated that differential equations with impulse do well in describing biological control, and thought that differential equations with impulsive effect are more practical than ordinary differential equations in stating practical problems^24,25. Liang et al.²⁶ proposed the impulsive control of Leslie predator-prey model, and investigated the dynamic properties of the proposed mathematical biology model. The herbivore-plankton model with cannibalism was proposed by Fang et al.²⁷, and the corresponding heteroclinic bifurcations and three order-1 periodic orbits were analyzed. A new impulsive state feedback model was proposed, and the management strategy with impulsive control was developed²⁸. Wang and Chen²⁹ proposed a new microbial pesticide model and analyzed the condition of the existence of corresponding periodic orbit. Li and Chen³⁰ investigated the dynamics of the impulsive turbidostat model, and also studied the characteristic of the proposed mathematical biology model. Li et al.³¹ proposed a new water hyacinth fish impulsive control model, and investigated the local stability and global dynamics of the proposed ecological system. Li et al.³² proposed a new Filippov predator-prey model, and analyzed the corresponding dynamics using the Filippov theory knowledge. Qin et al.³³ proposed a new non-smooth Filippov ecosystem with group defense and also studied the stability of equilibria and the bifurcation phenomenon numerically. Discontinuous plant disease models with a non-smooth separation line were investigated and discussed by Li et al.³⁴. Khan et al., proposed some new hepatitis B epidemic models and novel corona virus disease models and investigated their dynamic characteristics^35–40.

In fact, exploiting viruses and the release of pest population simultaneously, some research studies have been attached to devote the control of pests^9,41. First of all, introducing pathogens into the pest population is expected to produce an epidemic and then become endemic. Moreover, the infected pests are released to the periodic impulsive pest population. However, the crops will not be affected by the infected pest. The vulnerable pests are infected by direct contact with infected pests or being exposed to an infected environment. Then the pest population and its death will be affected to some extent^42,43. In recent years, the pandemic of Gompertz virus disease remains a pressing issue in epidemiological ecosystem. Mathematical modelling is an important method in investigating the dynamic characteristics of epidemiological ecosystem. Additionally, due to the hybrid nature, in order to well describe different kinds of real-world evolutionary processes, impulsive systems play important role because it states sudden change at some moments. This article will try to develop a new epidemiological ecosystem with impulsive effect and investigate its dynamics for bio-control. Additionally, it is assumed that the pest population will grow according to the logic curve without the disease^9,44,45. Then we investigate the global attractiveness and permanence of the proposed model.

The remainder of this paper is provided as follows. In section “Mathematical model”, the Gompertz virus disease model with impulsive effect is established. In section “Global attractiveness”, utilizing the comparison theorem and Floquet theory of impulsive differential equation, the global attractiveness of the proposed Gompertz virus disease model is strictly proved. In section “Uniform persistence”, we obtain the criteria for the uniform persistence of the proposed Gompertz virus disease model. In section “Numerical simulations”, many numerical results are given to prove the validity of the obtained results. Finally, some conclusions are summarized.

Mathematical model

In this section, we will present the newly developed Gompertz virus disease model with impulsive effect.

The total population of insect pest is denoted by N(t),  and it is divided into five subgroups such as I(t),  E(t), S(t), R(t) and A(t). The susceptible pest is denoted by S(t);  E(t) denotes the exposed pest; the infected (symptomatic) pest is denoted by I(t);  A(t) represents the asymptotically infected pest; the recovered or the removed pest is denoted by R(t). The class M(t) denotes the reservoir or the seafood place or market.

In this paper, the impulsive introduction of disinfectants and the pulse vaccination strategy in the environment are investigated for the control of the Gompertz virus epidemic, and we also assume that the proposed model satisfies:

1. $\mu$ and $\mathrm{\Pi }$ respectively denotes the natural death rate and the birth rate of the pest.

2. The susceptible pest S(t) acquires Gompertz virus infection by ingesting environmental virus from the contaminated reservoir or seafood place or market, through the term given by $\beta \frac{S(t)}{K+S(t)},$ where $\beta$ is the disease transmission coefficient.

3. The proportion of asymptomatic infection is denoted by $\theta$. The transmission rate of the infected is denoted by $\rho .$ $\varpi$ denotes the transmission rate after completing the incubation period. K represents the pathogen concentration.

4. ${\tau }_1$ and ${\tau }_2$ respectively denote the pest in I(t) and A(t) joining R(t) with the removal or recovery rate, and Gompertz virus induced mortality rate of infected individuals is denoted by $d_1$.

5. $\eta M(t)E(t)$ denotes the susceptible pest infected after the interaction with M(t), in which the disease transmission coefficient between S(t) and E(t) is represented by $\eta .$

6. The parameters ${\lambda }_1$ and ${\lambda }_2$ respectively denote the infected symptomatic and the asymptomatically infected contributing the Gompertz virus into the seafood market M(t). The removing rate of the Gompertz virus from the seafood market M(t) is given by the rate d. The natural growth rate of the Gompertz virus is denoted by b. We assume $b<d$ through this paper.

7. The parameter ${\theta }_1(0<{\theta }_1<1)$ denotes the part of susceptible pest inoculated by the vaccine at times $t=nT,$ in which n is a positive integer, and the impulsive period between two consecutive pulse vaccinations is denoted by T. nT represents the pulse vaccination at the time of multiple periods. R(t) represents the temporary immunity which contains the recovered individual and the vaccinated individual.

8. The death of the Gompertz virus dues to the environmental sanitation. Therefore, considering the pulse inoculation of disinfectant in the environment at the time of nT,  the death rate of the virus is denoted by $\xi (0<\xi <1).$ The model is expressed as:

   $$\begin{array}{c}\hfill \left\{\begin{array}{c}\left(\begin{array}{c}\begin{array}{c}\dot{S}(t)=\mathrm{\Pi }-S(t)M(t)E(t)\eta -\frac{S(t)\beta E(t)}{K+S(t)}-\mu S(t)\hfill \\ \dot{E}(t)=\frac{S(t)\beta E(t)}{K+S(t)}-E(t)\mu -E(t)\theta \rho -E(t)(1-\theta )\varpi +\eta E(t)M(t)S(t)\hfill \\ \dot{I}(t)=(1-\theta )\varpi E(t)-({\tau }_1+\mu +d_1)I(t)\hfill \end{array}\hfill \\ \begin{array}{c}\dot{A}(t)=\theta \rho E(t)-({\tau }_2+\mu )A(t)\hfill \\ \dot{R}(t)={\tau }_{1}I(t)+{\tau }_{2}A(t)-\mu R(t)\hfill \\ \dot{M}(t)=M(t)(b-d)+{\lambda }_{1}I(t)+{\lambda }_{2}A(t)\hfill \end{array}\hfill \end{array}\right\}n\in N^{+},t\ne nT.\hfill \\ \hfill \\ \left(\begin{array}{c}\begin{array}{c}S(t^{+})=S(t)(1-{\theta }_1)\hfill \\ E(t^{+})=E(t)\hfill \\ I(t^{+})=I(t)\hfill \end{array}\hfill \\ \begin{array}{c}A(t^{+})=A(t)\hfill \\ R(t^{+})={\theta }_{1}S(t)+R(t)\hfill \\ M(t^{+})=M(t)(1-\xi )\hfill \end{array}\hfill \end{array}\right\}n\in N^{+},t=nT.\hfill \end{array}\right)\end{array}$$ 1

For (1), all the coefficients are greater than 0. Based on

$$\begin{array}{c}\hfill \dot{N}(t)=\mathrm{\Pi }-\mu N(t)-d_{1}I(t),\end{array}$$ 2

then the population size will change with t. If $A(t)=I(t)=E(t)=0$, we have $\underset{t\to \infty }{lim}N(t)=\frac{\mathrm{\Pi }}{\mu }.$

Similarly, we have

$$\begin{array}{c}\hfill \dot{M}(t)={\lambda }_{2}A(t)-(d-b)M(t)+{\lambda }_{1}I(t)\Rightarrow \underset{t\to \infty }{lim}supM(t)\le \frac{\mathrm{\Pi }({\lambda }_1+{\lambda }_2)}{\mu (d-b)}.\end{array}$$

Then we have that when t takes the sufficiently large value,

$$\begin{array}{c}\hfill M(t)\le \frac{\mathrm{\Pi }({\lambda }_1+{\lambda }_2)}{\mu (d-b)},N(t)\le \frac{\mathrm{\Pi }}{\mu }.\end{array}$$

By replacing $N(t)-S(t)-E(t)-I(t)-A(t)$ with R(t),  then

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left(\begin{array}{c}\begin{array}{c}\dot{S}(t)=\mathrm{\Pi }-\mu S(t)-\frac{S(t)\beta E(t)}{K+S(t)}-E(t)S(t)M(t)\eta \hfill \\ \dot{E}(t)=\frac{\beta E(t)S(t)}{S(t)+K}-E(t)\mu -E(t)\theta \rho -(1-\theta )\varpi E(t)+\eta E(t)M(t)S(t)\hfill \\ \dot{I}(t)=-({\tau }_1+\mu +d_1)I(t)+(1-\theta )\varpi E(t)\hfill \end{array}\hfill \\ \begin{array}{c}\dot{A}(t)=-({\tau }_2+\mu )A(t)+\theta \rho E(t)\hfill \\ \dot{N}(t)=-\mu N(t)-d_{1}I(t)+\mathrm{\Pi }\hfill \\ \dot{M}(t)=M(t)(b-d)+I(t){\lambda }_1+A(t){\lambda }_2\hfill \end{array}\hfill \end{array}\right\}n\in N^{+},t\ne nT\hfill \\ \hfill \\ \left(\begin{array}{c}\begin{array}{c}(1-{\theta }_1)S(t)=S(t^{+})\hfill \\ E(t)=E(t^{+})\hfill \\ I(t)=I(t^{+})\hfill \end{array}\hfill \\ \begin{array}{c}A(t)=A(t^{+})\hfill \\ N(t)=N(t^{+})\hfill \\ M(t)(1-\xi )=M(t^{+})\hfill \end{array}\hfill \end{array}\right\}t=nT,n\in N^{+}.\hfill \end{array}\right)\end{array}$$ 3

We provide the initial condition for (3) as:

$$\begin{array}{c}\hfill \varphi (\omega )=({\varphi }_1(\omega ),{\varphi }_2(\omega ),{\varphi }_3(\omega ),{\varphi }_4(\omega ),{\varphi }_5(\omega ),{\varphi }_6(\omega ))\in R_{+}^6,{\varphi }_i(0)>0,i=1,2,\dots ,6.\end{array}$$ *

Next, we will study the dynamic properties of (3). For biological considerations, (3) is investigated in the following closed set and this set is given as

$$\begin{array}{c}\hfill \begin{array}{c}\mathrm{\Omega }=\{\left(S(t),E(t),I(t),A(t),N(t),M(t)\right)\in R_{+}^6:\hfill \\ S(t)\ge 0,E(t)\ge 0,I(t)\ge 0,A(t)\ge 0,N(t)\ge 0,M(t)\ge 0,\hfill \\ 0\le S(t)+E(t)+I(t)+A(t)\le \frac{\mathrm{\Pi }}{\mu },N(t)\le \frac{\mathrm{\Pi }}{\mu },M(t)\le \frac{\mathrm{\Pi }({\lambda }_1+{\lambda }_2)}{\mu (d-b)},d>b\}\begin{array}{cc}& \end{array}\hfill \end{array}\end{array}$$

In fact, it is not difficult to check that $\mathrm{\Omega }$ has a positively invariant property about (3). For further discussion, some important assumptions and definitions are given as follows.

We assume that the fixed solution and the arbitrary solution of (3) are respectively given by

$$\begin{array}{c}\hfill x^{\ast }(t)=(S^{\ast }(t),E^{\ast }(t),I^{\ast }(t),A^{\ast }(t),N^{\ast }(t),M^{\ast }(t)),\end{array}$$

and

$$\begin{array}{c}\hfill x(t)=(S(t),E(t),I(t),A(t),N(t),M(t)).\end{array}$$

Definition 1

$\forall$ $x_0$, $\exists$ $M\ge m>0,$ we have that when t takes the sufficiently large value, $x(t)\in [m,M].$ Then (3) has the uniform persistence.

Definition 2

For arbitrary $x_0$, If $\underset{t\to \infty }{lim}\left|x(t)-x^{\ast }(t)\right|=0$, then $x^{\ast }(t)$ has the global attractivity.

Definition 3

$\forall$ $\eta >0,$ and $\epsilon >0,$ $\left|x^{\ast }(t)-x(t)\right|<\epsilon$ when there is $\delta =\delta (t_0,\epsilon ,\eta )>0.$ Moreover, if $\left|t_0-t\right|>\eta$, and $t\in [t_0,+\infty ],$ then $\left|x_0-x_0^{\ast }\right|<\delta .$ Then $x^{\ast }(t)$ has the characteristic of stability.

Definition 4

When the fixed solution $x^{\ast }(t)$ has the characteristic of stability and global attractivity, then $x^{\ast }(t)$ is called globally asymptotical stable. If there is the disease-free periodic solution for (3) which also has the characteristic of globally asymptotical stability, the Gompertz virus will finally disappear.

Definition 5

(Equicontinuty) For $\forall \epsilon >0,$ if we can find a positive parameter $\delta ,$ which satisfies $|x(t_2)-x(t_1)|<\epsilon$ holds true when $t_i(i=1,2)\in [0,l]$ and $|t_1-t_2|<\delta ,$ x(t) is said to be equicontinuous when $0\le t\le 1.$

For further analysis conveniently, the following lemma is given as follows.

Lemma 1

⁴⁶ We consider

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{u}(t)=-\mu u(t)+\mathrm{\Pi },t\ne nT\hfill \\ u(t^{+})=u(t)(1-{\theta }_1),n\in N^{+},t=nT\hfill \end{array}\right)\end{array}$$ 4

where $0<{\theta }_1<1,$ $\mu >0$ and $\mathrm{\Pi }>0.$

Thus we can obtain that for $t\in [nT,(n+1)T],$ the periodic solution is given as

$$\begin{array}{c}\hfill \begin{array}{c}{u}^{\ast }(t)=\frac{\mathrm{\Pi }}{\mu }+(u^{\ast }-\frac{\mathrm{\Pi }}{\mu })e^{-\mu (t-nT)}\hfill \\ =\frac{\mathrm{\Pi }}{\mu }(1-\frac{{\theta }_1{e}^{-\mu (t-nT)}}{1-(1-{\theta }_1)e^{-\mu T}})\hfill \end{array}\end{array}$$

where

$$\begin{array}{c}\hfill u^{\ast }=\frac{\mathrm{\Pi }}{\mu }\frac{(1-e^{-\mu T})(1-{\theta }_1)}{1-(1-{\theta }_1)e^{-\mu T}}\end{array}$$

exists and is globally asymptotically stable, unique and positive.

Global attractiveness

In this section, we will discuss the global attractiveness of the Gompertz virus disease model with impulsive effect (3).

$\forall$ $t\in [0,+\infty ],$

$$\begin{array}{c}\hfill M(t)=I(t)=A(t)=E(t)=0.\end{array}$$

It means that the asymptotically infectious individuals permanently die out. Based on this prerequisite, we can transform (3) as:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left(\begin{array}{ccc}\hfill \dot{S}(t)& =& \mathrm{\Pi }-\mu S(t),\hfill \\ \hfill \dot{N}(t)& =& \mathrm{\Pi }-\mu N(t),\hfill \end{array}\right\}n\in N^{+},t\ne nT\hfill \\ \hfill \\ \left(\begin{array}{ccc}S(t^{+})\hfill & =\hfill & (1-{\theta }_1)S(t)\hfill \\ N(t^{+})\hfill & =\hfill & N(t),\hfill \end{array}\right\}n\in N^{+},t=nT\hfill \end{array}\right)\end{array}$$ 5

Then we can obtain

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}N(t)=\frac{\mathrm{\Pi }}{\mu }.\end{array}$$

Next, we will prove that for the susceptible S(t),  there is one periodic solution. Firstly, we consider

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{S}(t)=\mathrm{\Pi }-\mu S(t),n\in N^{+},t\ne nT\\ S(t^{+})=(1-{\theta }_1)S(t),n\in N^{+},t=nT.\end{array}\right)\end{array}$$ 6

It follows from Lemma 1 that a unique globally asymptotical periodic solution of (6) exists, and it is given as

$$\begin{array}{c}\hfill S^{\ast }(t)=\frac{\mathrm{\Pi }}{\mu }\left(1-\frac{{\theta }_1{e}^{-\mu (t-nT)}}{1-(1-{\theta }_1)e^{-\mu T}}\right),t\in (nT,(n+1)T].\end{array}$$ 7

Therefore, there exists one infection-free periodic solution which is given by $(S^{\ast }(t),0,0,0,\frac{\mathrm{\Pi }}{\mu },0).$

Theorem 1

For system (3), $\left(S^{\ast },(t),,,0,,,0,,,0,,,\frac{\mathrm{\Pi }}{\mu },,,0\right)$ has global attractivity for $R_1^{\ast }<1$, in which

$$\begin{array}{cc}\hfill R_1^{\ast }=& \frac{\left(\frac{\beta }{K+\zeta }+\frac{\eta \mathrm{\Pi }}{\mu (d-b)}({\lambda }_1+{\lambda }_2)\right)\zeta }{(1-\theta )\varpi +\theta \rho +\mu },\hfill \\ \hfill \zeta =& \frac{\mathrm{\Pi }\left(e^{\mu T}-1\right)}{\mu (e^{\mu T}+{\theta }_1-1)}.\hfill \end{array}$$

Proof

Since $R_1^{\ast }<1$, ${\epsilon }_1^{}>0$ can be chosen such that

$$\begin{array}{c}\hfill \left(\frac{\beta }{K+\zeta +{\epsilon }_1}+\frac{\eta \mathrm{\Pi }({\lambda }_1+{\lambda }_2)}{\mu (d-b)}\right)\left(\zeta +{\epsilon }_1\right)<(1-\theta )\varpi +\theta \rho +\mu .\end{array}$$ 8

According to $\dot{N}(t)\le -\mu N(t)+\mathrm{\Pi },$ then there exists $n_1>0$ satisfying the following inequality

$$\begin{array}{c}\hfill N(t)\le \frac{\mathrm{\Pi }}{\mu },t>n_{1}T.\end{array}$$ 9

By (3), then

$$\begin{array}{c}\hfill \dot{S}(t)<-\mu S(t)+\mathrm{\Pi }.\end{array}$$

When $t>n_{1}T$ and $n>n_1,$ we investigate

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{u}(t)=-\mu u(t)+\mathrm{\Pi },t\ne nT\hfill \\ u(t^{+})=u(t)(1-{\theta }_1),n\in N^{+},t=nT.\hfill \end{array}\right)\end{array}$$ 10

For system (10), we can obtain the globally asymptotical stable periodic solution as $u^{\ast }(t)=S^{\ast }(t),$ utilizing Lemma 1, and also there is $n_2>n_1$ such that when $n>n_2$,

$$\begin{array}{c}\hfill S(t)<S^{\ast }(t)+{\epsilon }_1\le \frac{\mathrm{\Pi }(e^{\mu T}-1)}{\mu (e^{\mu T}+{\theta }_1-1)}+{\epsilon }_1=\delta ,nT<t\le (n+1)T.\end{array}$$ 11

Then

$$\begin{array}{c}\hfill \dot{E}(t)\le \left[\frac{\beta \delta }{\delta +K}+\eta \delta \frac{\mathrm{\Pi }\left({\lambda }_1+{\lambda }_2\right)}{\mu \left(d-b\right)}-(1-\theta )\varpi -\theta \rho -\mu \right]E(t),t>nT,n>n_2.\end{array}$$

Thus

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}E(t)=0.\end{array}$$ 12

Therefore, for $\forall {\epsilon }_2>0$, there is $n_3>n_2$, satisfying $E(t)<{\epsilon }_2$ when $t>n_{3}T$.

Then, according to (3), we can obtain that if $t>nT,n>n_3,$

$$\begin{array}{c}\hfill \dot{I}(t)\le -({\tau }_1+\mu +d_1)I(t)+(1-\theta )\varpi {\epsilon }_2\end{array}$$

i.e.

$$\begin{array}{c}\hfill I(t)<\left[I(nT)-\frac{\varpi {\epsilon }_2\left(1-\theta \right)}{{\tau }_1+\mu +d_1}\right]e^{-(t-nT)(d_1+{\tau }_1+\mu )}+\frac{\left(1-\theta \right)\varpi {\epsilon }_2}{{\tau }_1+\mu +d_1}.\end{array}$$

Hence,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}I(t)=0.\end{array}$$ 13

Similarly, for $\forall {\epsilon }_3>0$, there is $n_4>n_3$, satisfying

$$\begin{array}{c}\hfill I(t)<{\epsilon }_3\end{array}$$

when $t>n_{4}T$.

Exploiting the fourth one of (3), the following inequality can be obtained as

$$\begin{array}{c}\hfill \dot{A}(t)\le \theta \rho {\epsilon }_3-({\tau }_2+{\mu }_1)A(t)\end{array}$$

for all $t>nT(n>n_4).$

Obviously

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}A(t)=0.\end{array}$$ 14

Therefore, for arbitrary $\forall {\epsilon }_4>0$, there is $n_5>n_4$ satisfying $A(t)<{\epsilon }_4$ when $t>n_{5}T.$

Next, according to the sixth one of (3), we get

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{M}(t)\le {\lambda }_1{\epsilon }_3+{\lambda }_2{\epsilon }_4-dM(t),t\ne nT\\ M(t^{+})=M(t)(1-\xi ),n>n_5,t=nT.\end{array}\right)\end{array}$$

Obviously,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}M(t)=0.\end{array}$$ 15

Therefore, for $\forall {\epsilon }_5>0$, there is $n_6>n_5$ satisfying $M(t)<{\epsilon }_5$ at the case $t>n_{6}T.$

Eventually, we can obtain

$$\begin{array}{c}\hfill \dot{N}(t)\ge \mathrm{\Pi }-\mu N(t)-d_1{\epsilon }_3.\end{array}$$

when $t>n_{6}T.$

When $t>n_{6}T$, the following comparative model will be investigated.

$$\begin{array}{c}\hfill \dot{Z}(t)=\mathrm{\Pi }-\mu Z(t)-d_1{\epsilon }_3,\end{array}$$ 16

Obviously,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}Z(t)=\frac{\mathrm{\Pi }-d_1{\epsilon }_3}{\mu }.\end{array}$$

Exploiting the corresponding lemma of²⁵, we can obtain that if $t>n_{7}T$, there is $n_7>n_6$ such that

$$\begin{array}{c}\hfill N(t)\ge \frac{\mathrm{\Pi }-d_1{\epsilon }_3}{\mu }-{\epsilon }_6.\end{array}$$ 17

Because ${\epsilon }_3>0$ and ${\epsilon }_6>0,$ and they are extremely small, we can obtain by (9) and (17) that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}N(t)=\frac{\mathrm{\Pi }}{\mu }.\end{array}$$ 18

From (9) and (12)–(15), we can get the following conclusion:

For $\forall {\epsilon }_7>0$, there is $n_8>n_7$ satisfying

$$\begin{array}{c}\hfill M(t)<{\epsilon }_7,A(t)<{\epsilon }_7,I(t)<{\epsilon }_7,E(t)<{\epsilon }_7,N(t)>\frac{\mathrm{\Pi }}{\mu },\end{array}$$

when $t>n_{8}T.$

When $t>n_{8}T$, it follows from (3) that

$$\begin{array}{c}\hfill \dot{S}(t)\ge \mathrm{\Pi }-\frac{\beta \delta }{\delta +K}{\epsilon }_7-\left(\mu ,+,\eta ,{\epsilon }_7^2\right)S(t),\end{array}$$

when $t>nT$ and $n>n_8.$

Considering

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{u}(t)\ge \mathrm{\Pi }-\frac{\beta \delta }{\delta +K}{\epsilon }_7-\left(\mu ,+,\eta ,{\epsilon }_7^2\right)S(t),t\ne nT\hfill \\ u\left(t^{+}\right)=u(t)(1-{\theta }_1),n\in N^{+},t=nT,n>n_8,\hfill \end{array}\right)\end{array}$$ 19

based on Lemma 1, its unique asymptotical stable periodic solution can be computed as

$$\begin{array}{c}\hfill \stackrel{-}{u}(t)=\frac{\mathrm{\Pi }-\frac{\beta \delta }{\delta +K}{\epsilon }_7}{\mu +\eta {\epsilon }_7^2}+\left(u^{\ast },-,{\frac{\mathrm{\Pi }-\frac{\beta \delta }{\delta +K}{\epsilon }_7}{\mu +\eta {\epsilon }_7^2}}_{}\right)e^{-(t-nT)\left(\eta ,{\epsilon }_7^2,+,\mu \right)}\end{array}$$

when $nT<t\le (n+1)T,$ in which

$$\begin{array}{c}\hfill u^{\ast }=\frac{\mathrm{\Pi }-\frac{\beta \delta }{\delta +K}{\epsilon }_7}{\mu +\eta {\epsilon }_7^2}\frac{\left(1,-,e^{-\left(\mu ,+,\eta ,{\epsilon }_7^2\right)T}\right)(1-{\theta }_1)}{1-e^{-\left(\mu ,+,\eta ,{\epsilon }_7^2\right)T}(1-{\theta }_1)}.\end{array}$$

According to²⁵, there exists $n_9>n_8$ satisfying

$$\begin{array}{c}\hfill S(t)>\stackrel{-}{u}(t)-{\epsilon }_7\end{array}$$ 20

for

$$\begin{array}{c}\hfill n>n_9,nT<t\le (n+1)T.\end{array}$$

Because ${\epsilon }_7$ and ${\epsilon }_1$ take the very small value, then we have

$$\begin{array}{c}\hfill \underset{t\leftarrow \infty }{lim}S(t)=S^{\ast }(t)=\left(1-\frac{{\theta }_1\left(1,-,e^{-\mu (t-nT)}\right)}{1-\left(1,-,{\theta }_1\right)e^{-\mu T}}\right)\frac{\mathrm{\Pi }}{\mu }\end{array}$$ 21

when $nT<t\le (n+1)T.$

By (12)–(15), (18) and (21), the infection-free periodic solution $(S^{\ast }(t),0,0,0,\frac{\mathrm{\Pi }}{\mu },0)$ is the globally attractive. Based on Theorem 1, we can obtain some assertions as follows.

Proposition 1

1. The infection-free periodic solution which is given by $(S^{\ast }(t),0,0,0,\frac{\mathrm{\Pi }}{\mu },0)$ is globally attractive, when

   $$\begin{array}{c}\hfill \mathrm{\Pi }\left(\frac{\beta \mu }{\mathrm{\Pi }+\mu K}+\frac{\eta \mathrm{\Pi }({\lambda }_1+{\lambda }_2)}{\mu (d-b)}\right)<\mu ((1-\theta )+\theta \rho +\mu );\end{array}$$
2. If $T<T^{\ast }$ and

   $$\begin{array}{c}\hfill \mathrm{\Pi }\left(\frac{\beta \mu }{\mathrm{\Pi }+\mu K}+\frac{\eta \mathrm{\Pi }({\lambda }_1+{\lambda }_2)}{\mu (d-b)}\right)>\mu ((1-\theta )+\theta \rho +\mu ),\end{array}$$

   $\left(S^{\ast },(t),,,0,,,0,,,0,,,\frac{\mathrm{\Pi }}{\mu },,,0\right)$ is globally attractive, where

   $$\begin{array}{c}\hfill \begin{array}{c}\begin{array}{c}{T}^{\ast }=\frac{1}{\mu }ln\left(1+\frac{\upsilon {\theta }_1\mu }{\mathrm{\Pi }-\upsilon \mu }\right),\hfill \\ A=\eta \pi ({\lambda }_1+{\lambda }_1),\hfill \\ B=\mu (d-b),\hfill \\ C=(1-\theta )\varpi +\theta \rho +\mu ,\hfill \end{array}\\ \upsilon =\frac{-\left(B+KA-BC\right)+\sqrt{{\left(B+KA-BC\right)}^2+4KABC}}{2A}.\end{array}\end{array}$$

Proposition 2

If

$$\begin{array}{c}\hfill {\theta }_1>{\theta }_1^{\ast }=\frac{\left(\mathrm{\Pi },-,\mu ,\upsilon \right)\left(e^{\mu T},-,1\right)}{\mu \upsilon },\end{array}$$

$\left(S^{\ast },(t),,,0,,,0,,,0,,,\frac{\mathrm{\Pi }}{\mu },,,0\right)$ is globally attractive.

Uniform persistence

In this section, we will discuss the sufficient conditions for the Gompertz virus disease model with impulsive effect. In order to obtain uniform persistence, the following lemma needs to be proved first.

Lemma 2

If $R_{1\ast }>1,$ then for arbitrary $t_0>0,E(t)<\stackrel{\_}{E}$ doesn’t hold true for all $t>t_0,$ where

$$\begin{array}{cc}\hfill R_{1\ast }^{}=& \frac{\frac{\beta \zeta }{K+\zeta }}{(1-\theta )\varpi +\theta \rho +\mu },\hfill \\ \hfill \zeta =& \frac{\mathrm{\Pi }\left(e^{\mu T}-1\right)(1-{\theta }_1)}{\mu (e^{\mu T}+{\theta }_1-1)},\hfill \\ \hfill \stackrel{\_}{E}=& \frac{\mathrm{\Pi }\left(e^{\mu T}-1\right)(1-{\theta }_1)}{(e^{\mu T}+{\theta }_1-1)\left((1-\theta )\varpi +\theta \rho +\mu \right)\left(K+\zeta \right)}-\frac{\mu }{\beta }.\hfill \end{array}$$

Proof

Herein we use the method of contradiction. We assume that there is $t_0>0,$ which satisfies when $t>t_0,$

$$\begin{array}{c}\hfill E(t)<\stackrel{\_}{E}.\end{array}$$ 22

When $t_0<t,$

$$\begin{array}{c}\hfill \dot{I}(t)\le \stackrel{\_}{E}(1-\theta )\varpi -I(t)({\tau }_1+\mu +d_1)\end{array}$$ 23

and

$$\begin{array}{c}\hfill \dot{A}(t)\le \theta \rho \stackrel{\_}{E}-({\tau }_2+{\mu }_1)A(t).\end{array}$$ 24

By (23) and (24), then $\exists {\epsilon }^{\ast }>0$ and when $t_1>t_0,$ then

$$\begin{array}{cc}\hfill I(t)& \le \frac{(1-\theta )\varpi \stackrel{\_}{E}}{{\tau }_1+\mu +d_1}+{\epsilon }^{}\ast ,\hfill \\ \hfill & \frac{\frac{\beta (\zeta -{\epsilon }^{\ast })}{k+\zeta -{\epsilon }^{\ast }}}{(1-\theta )\varpi +\theta \rho +\mu }>1,\hfill \end{array}$$

and

$$\begin{array}{c}\hfill A(t)\le \frac{\theta \rho \stackrel{\_}{E}}{{\tau }_2+\mu }+{\epsilon }^{}\ast \end{array}$$

for arbitrary $t>t_1.$

Thus, when $t>t_1,$

$$\begin{array}{c}\hfill \dot{M}(t)\le {\lambda }_1\frac{(1-\theta )\varpi \stackrel{\_}{E}}{{\tau }_1+\mu +d_1}+{\lambda }_2\frac{\theta \rho \stackrel{\_}{E}}{{\tau }_2+\mu }-(d-b)M(t).\end{array}$$

It is not difficult to check that there is $t_2>t_1,$ such that

$$\begin{array}{c}\hfill M(t)\le \frac{{\lambda }_1\left(\frac{(1-\theta )\varpi \stackrel{\_}{E}}{{\tau }_1+\mu +d_1}+{\epsilon }^{\ast }\right)+{\lambda }_2\left(\frac{\theta \rho \stackrel{\_}{E}}{{\tau }_2+\mu }+\epsilon \ast \right)}{d-b}+{\epsilon }^{\ast }.\end{array}$$

Hence, when $t\ne nT(n\in N),$ and $t>t_2,$ we can obtain

$$\begin{array}{cc}\hfill & \dot{S}(t)\ge \mathrm{\Pi }-g_{1}S(t),\hfill \\ \hfill & g_1=\mu +\frac{\beta \stackrel{\_}{E}}{K}+\eta \stackrel{\_}{E}\left({\epsilon }^{\ast },+,\frac{{\lambda }_1\left(\frac{(1-\theta )\varpi \stackrel{\_}{E}}{{\tau }_1+\mu +d_1}+{\epsilon }^{\ast }\right)+{\lambda }_2\left(\frac{\theta \rho \stackrel{\_}{E}}{{\tau }_2+\mu }+{\epsilon }^{\ast }\right)}{d-b}\right).\hfill \end{array}$$

For the case of $n_{10}T>t_2$ and $t>n_{10}T,$

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{v}(t)\ge \mathrm{\Pi }-g_{1}v(t),t\ne nT\\ v(t^{+})=v(t)(1-{\theta }_1),t=nT,\end{array}\right)\begin{array}{cc}& n>n_{10}\end{array},n\in N^{+}.\end{array}$$ 25

It follows from Lemma 1 that

$$\begin{array}{c}\hfill \overline{v}(t)=-e^{-g_1(t-nT)}(\frac{\mathrm{\Pi }}{g_1}-v^{\ast })+\frac{\mathrm{\Pi }}{g_1},t\in (nT,(n+1)T]\end{array}$$

is globally attractive, in which

$$\begin{array}{c}\hfill v^{\ast }=\frac{\mathrm{\Pi }(1-e^{-g_{1}T})(1-{\theta }_1)}{(1-(1-{\theta }_1)e^{-g_{1}T})g_1}.\end{array}$$

Thus, there exists a constant $t_3>t_2,$ satisfying that when $t>t_3,$

$$\begin{array}{c}\hfill S(t)\ge \stackrel{\_}{v}(t)-{\epsilon }^{\ast }\ge v^{\ast }-{\epsilon }^{\ast }\stackrel{\mathrm{\Delta }}{=}{S}^L.\end{array}$$ 26

According to (26) and (3), we can obtain

$$\begin{array}{c}\hfill \dot{E}(t)\ge \left(\beta \frac{S^L}{S^L+K}-((1-\theta )\varpi )+\theta \rho +\mu \right)E(t).\end{array}$$ 27

Integrating (27) from nT to $(n+1)T,$ considering that ${\epsilon }^{\ast }$ is arbitrary small and $\overline{E}>0$, then we can get

$$\begin{array}{c}\hfill \begin{array}{c}E((n+1)T)\ge \begin{array}{cc}E(nT)e^{\left(\beta \frac{S^L}{S^L+K}-((1-\theta )\varpi +\theta \rho +\mu )\right)}& \end{array}\hfill \\ \begin{array}{ccc}\begin{array}{cccc}& & \begin{array}{cc}& \end{array}& \ge \end{array}& E(nT)e^{{}^{\left(\beta \frac{\zeta }{\zeta +K}-((1-\theta )\varpi +\theta \rho +\mu )\right)}}& .\end{array}\hfill \end{array}\end{array}$$

Using $R_{1\ast }>1,$ we get

$$\begin{array}{c}\hfill \beta \frac{\zeta }{\zeta +K}-((1-\theta )\varpi +\theta \rho +\mu )>0.\end{array}$$

Thus

$$\begin{array}{c}\hfill \begin{array}{c}E((n_0+k)T)\hfill \\ \ge \begin{array}{c}{\left(e^{\beta \frac{\zeta }{\zeta +K}-((1-\theta )\varpi +\theta \rho +\mu )}\right)}^{k}E(n_{0}T)\to \infty ,k\to \infty ,n_{0}T>t_3.\end{array}\hfill \end{array}\end{array}$$

This is the contradiction with that E(t) is bounded. Thus, there is $t^{\ast }>t_0,$ such that $E(t^{\ast })\ge \stackrel{\_}{E},$ and the above prediction is proved.

Theorem 2

When $R_{1\ast }>1,$ for system (3), there will be $q_2$ satisfying that there exists some positive solution which satisfies $q_2\le E(t).$

Proof

Define

$$\begin{array}{c}\hfill q_2=\stackrel{\_}{E}{e}^{-((1-\theta )\varpi +\theta \rho +\mu )T}.\end{array}$$

According to Lemma 2, $\left(S(t),E(t),I(t),A(t),N(t),M(t)\right)$ is discussed based on the following cases.

Firstly, $E(t)\ge \stackrel{\_}{E}$ when t takes sufficiently large value.

Secondly, E(t) vibrates about $\stackrel{\_}{E}$ when t takes sufficiently large value.

For case 1, we have $\underset{t\to \infty }{lim}infE(t)>q_2.$ Immediately, we have the conclusion of Theorem 2.

For case 2, let $t_4>t_3$ and $t_5$ be sufficiently large satisfying

$$\begin{array}{c}\hfill E(t_4)=E(t_5)=\overline{E},E(t)<\overline{E},\forall t\in (t_4,t_5).\end{array}$$

If $t_5-t_4\le T$, since

$$\begin{array}{c}\hfill E(t_4)=\stackrel{\_}{E},\end{array}$$

and

$$\begin{array}{c}\hfill -E(t)(\mu +\theta \rho +(1-\theta )\varpi )\le \dot{E}(t).\end{array}$$

we have $E(t)\ge q_2$ ($t\in [t_4,t_5]$).

When $t_5-t_4>T,$ then

$$\begin{array}{c}\hfill E(t)\ge q_2\end{array}$$

if $t\in [t_4,t_5]$.

Similarly, we can prove

$$\begin{array}{c}\hfill E(t)\ge q_2\end{array}$$

when $t\in [t_4,t_4+T]$ and

$$\begin{array}{c}\hfill S^L<S(t)\end{array}$$

when $t\in [t_4+T,t_5]$.

In the following, the conclusion that $E(t)\ge q_2$ holds true when $t_4+T\le t\le t_5,$ will be proved.

Herein we use the method of contradiction, and it means there is a positive constant $T^{\prime }$ satisfying $q_2\le E(t)$ if $t_4\le t\le T^{\prime }+T+t_4,$

$$\begin{array}{c}\hfill E(T^{\prime }+T+t_4)=q_2\end{array}$$

and

$$\begin{array}{c}\hfill \dot{E}(t_4+T+T^{\prime })\le 0.\end{array}$$

Exploiting (3), and $t=T+T^{\prime }+t_4$, then

$$\begin{array}{c}\hfill \begin{array}{c}\dot{E}(t)=\eta E(t)M(t)S(t)-\varpi (1-\theta )E(t)+\frac{\beta S(t)}{S(t)+K}-\theta \rho E(t)-\mu E(t)\hfill \\ \ge q_2\left(\beta \frac{S^L}{S^L+K}-(1-\theta )\varpi -\theta \rho -\mu \right)>0.\hfill \end{array}\end{array}$$

This is a contradiction. So $E(t)\ge q_2$ when $t\in [t_4,t_5].$

Theorem 3

If $R_{1\ast }>1,$ we have the conclusion that model (3) is permanent.

Proof

We assume the arbitrary solution of (3) is denoted by $\left(S(t),E(t),I(t),A(t),N(t),M(t)\right).$

If t takes a sufficiently large value, we can obtain

$$\begin{array}{c}\hfill \dot{S}(t)\ge \mathrm{\Pi }-S(t)\left(\mu ,+,\frac{\mathrm{\Pi }\beta }{\mu K},+,\eta ,\frac{{\mathrm{\Pi }}^2}{{\mu }^2}\right).\end{array}$$

Using the same method, we have

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}S(t)\ge q_1,\end{array}$$

where

$$\begin{array}{c}\hfill q_1=\frac{\mathrm{\Pi }}{\beta \frac{\mathrm{\Pi }}{\mu K}+\eta \frac{{\mathrm{\Pi }}^2}{{\mu }^2}+\mu }\frac{(1-{\theta }_1)\left(e^{\left(\beta \frac{\mathrm{\Pi }}{\mu K}+\eta \frac{{\mathrm{\Pi }}^2}{{\mu }^2}+\mu \right)T}-1\right)}{e^{\left(\beta \frac{\mathrm{\Pi }}{\mu K}+\eta \frac{{\mathrm{\Pi }}^2}{{\mu }^2}+\mu \right)T}+{\theta }_1-1}-{\epsilon }^{\ast }\end{array}$$

(${\epsilon }^{\ast }$ is sufficiently small).

When t takes a sufficiently large value, we derive

$$\begin{array}{c}\hfill \dot{I}(t)\ge (1-\theta )\varpi q_1-({\tau }_1+\mu +d_1)I(t).\end{array}$$

Therefore

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}I(t)\ge q_3,\end{array}$$

where

$$\begin{array}{c}\hfill q_3=\frac{(1-\theta )\varpi q_1}{({\tau }_1+\mu +d_1)}.\end{array}$$

When t is sufficiently large, we can obtain

$$\begin{array}{c}\hfill \dot{A}(t)\ge \theta \rho q_1-A(t)({\tau }_2+{\mu }_1).\end{array}$$

So

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}A(t)\ge q_4,\end{array}$$

where

$$\begin{array}{c}\hfill q_4=\frac{\theta \rho q_1}{{\tau }_2+{\mu }_1}.\end{array}$$

According to Theorem 2, when t takes sufficiently large value, then

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{M}(t)\ge -(d-b)M(t)+{\lambda }_2{q}_4+q_3{\lambda }_1\hfill \\ M(t^{+})=M(t)(1-\xi ).\hfill \end{array}\right)\end{array}$$

Therefore

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}M(t)\ge q_5,\end{array}$$

where

$$\begin{array}{c}\hfill q_5=\frac{{\lambda }_1{q}_3+{\lambda }_2{q}_4}{b-d}\frac{\left(1-e^{T\left(d-b\right)}\right)(1-\xi )}{e^{\left(d-b\right)T}+\xi -1}-{\epsilon }^{\ast }\end{array}$$

( ${\epsilon }^{\ast }$ is sufficiently small).

Set

$$\begin{array}{c}\hfill \begin{array}{c}{\mathrm{\Omega }}_0=\{\left(S(t),E(t),I(t),A(t),N(t),M(t)\right)\in R_{+}^6:I(t)\ge q_3,A(t)\ge q_4,N(t)\ge 0,\hfill \\ E(t)\ge q_2,S(t)\ge q_1,M(t)\le \frac{\mathrm{\Pi }({\lambda }_1+{\lambda }_2)}{\mu (d-b)},d>b,\hfill \\ \frac{\mathrm{\Pi }}{\mu }\ge N(t)\ge A(t)+I(t)+E(t)+S(t),M(t)\ge q_5\}.\hfill \end{array}\end{array}$$

It follows from Theorem 3 and the analysis above that ${\mathrm{\Omega }}_0$ belongs to the globally attractive region of $\mathrm{\Omega }.$ For the system (3), the positive solution satisfying the condition (*) will finally enter and remain in ${\mathrm{\Omega }}_0,$ i.e., the system will be permanent.

According to Theorem 2 and Theorem 3, the following assertions can be easily obtained.

Proposition 3

If ${\theta }_1<{\theta }_1^{\ast }$, system (3) will be permanent, i.e., the Gompertz virus will be an endemic disease, in which

$$\begin{array}{c}\hfill \begin{array}{c}{\theta }_1^{\ast }=\frac{\left\{\mathrm{\Pi }\beta -\left(\mathrm{\Pi }+K\mu \right)[(1-\theta )\varpi +\theta \rho +\mu ]\right\}(e^{\mu T}-1)}{K\mu [(1-\theta )\varpi +\theta \rho +\mu ]+\mathrm{\Pi }[\beta -(1-\theta )\varpi +\theta \rho +\mu ](e^{\mu T}-1)}\\ \left(\mathrm{\Pi }\beta >\left(\mathrm{\Pi }+K\mu \right)[(1-\theta )\varpi +\theta \rho +\mu ]\right)\end{array}.\end{array}$$

When $T>\stackrel{\_}{T}$, system (3) will be permanent, and the Gompertz virus will be an endemic disease, in which

$$\begin{array}{c}\hfill \overline{T}=\frac{1}{\mu }ln\frac{(1-{\theta }_1)[k\alpha \mu -(\beta -\alpha )\mathrm{\Pi }]}{k\alpha \mu -\mathrm{\Pi }(\beta -\alpha )(1-{\theta }_1)},\alpha =(1-\theta )\varpi +\theta \rho +\mu .\end{array}$$

It follows from Theorem 1 and Theorem 3 that the Gompertz virus will not appear when $R_1^{\ast }<1$, and when $R_{1\ast }>1,$ the Gompertz virus will be uniformly persistent.

Numerical simulations

In this section, the Runge-Kutta method is exploited to perform numerical simulations of Gompertz virus disease model with impulsive effect. Numerical simulations are presented to validate the analyzed conclusion from the view of impulsive differential equation theory.

Firstly, when we choose $\mathrm{\Pi }=5.1,{\lambda }_1=0.1,\eta =0.2,\beta =0.1,\mu =1,b=0.3,{\lambda }_2=0.15,\rho =1,T=1,d=1,K=2,\varpi =1,{\tau }_1=1.5,{\tau }_2=1,\xi =0.02,{\theta }_1=0.1,\theta =0.5,d_1=0.2,$ we compute $R_1^{\ast }=0.6498<1,$ which satisfies the condition of Theorem 1. By Theorem 1, the infection-free periodic solution of (3) is globally asymptotically stable. Figure 1 also demonstrates this conclusion: The periodic solution has the characteristic of global attractivity. It can be shown from Fig. 1 that S(t) goes to oscillatory; E(t),  I(t),  A(t) and M(t) go to extinction; N(t) goes to some stable value. It implies that the susceptible pest population oscillate with a positive amplitude, i.e., they will not go extinct. Additionally, the infective pests almost will go extinct and have little effect on the crop, which can be shown in Fig. 1.

Figure 1.

Dynamical behavior of system (3) with $\beta =0.1,\eta =0.2,{\lambda }_1=0.1,\mathrm{\Pi }=5.1,\mu =1,b=0.3,{\lambda }_2=0.15,\rho =1,T=1,d=1,K=2,\varpi =1,{\tau }_1=1.5,{\tau }_2=1,\xi =0.02,{\theta }_1=0.1,\theta =0.5,d_1=0.2,$ and all the parameters satisfy $R_1^{\ast }=0.6498<1.$ (a) (I(t), E(t), S(t));  (b) trajectory of the susceptible pest S(t);  (c) trajectory of the exposed pest E(t);  (d) trajectory of the infected pest I(t);  (e) trajectory of the asymptotically infected pest A(t);  (f) trajectory of the total population of pest N(t);  (g) trajectory of the reservoir or the seafood place or market M(t).

Moreover, when we choose $\mathrm{\Pi }=5.1,{\lambda }_1=0.1,\eta =2,\beta =5,\mu =0.3,b=0.3,{\lambda }_2=0.15,\rho =1.3,T=1,d=1,K=3,\varpi =2,{\tau }_1=25,{\tau }_2=0.2,\xi =0.02,{\theta }_1=0.13,\theta =0.5,d_1=0.2,$ we compute $R_{1\ast }=2.01>1,$ which satisfies the condition of Theorem 3. By Theorem 3, we can obtain that the proposed model is permanent, which can be shown in Fig. 2. Figure 2 also demonstrates this conclusion: The periodic solution has the characteristic of permanence. It can be shown in Fig. 2 that M(t),  A(t),  I(t),  E(t) and S(t) go to oscillatory; N(t) goes to some stable value. It means that all the species will not go extinct and will almost maintain a stable value, which can be shown in Fig. 2.

Figure 2.

Dynamical behavior of system (3) with $\beta =5,\eta =2,{\lambda }_1=0.1,\mathrm{\Pi }=5.1,\mu =0.3,b=0.3,{\lambda }_2=0.15,\rho =1.3,T=1,d=1,K=3,\varpi =2,{\tau }_1=25,{\tau }_2=0.2,\xi =0.02,{\theta }_1=0.13,\theta =0.5,d_1=0.2,$ and all the parameters satisfy $R_{1\ast }=2.0644>1.$ (a) (I(t), E(t), S(t));  (b) trajectory of the susceptible pest S(t);  (c) trajectory of the exposed pest E(t);  (d) trajectory of the infected pest I(t);  (e) trajectory of the asymptotically infected pest A(t);  (f) trajectory of the total population of pest N(t);  (g) trajectory of the reservoir or the seafood place or market M(t).

From the view of ecology, infective pests almost have no effect on crops, but the pests that have a significant impact on the crops will become extinct. This will also demonstrate the effectiveness of scientific control. Actually, the purpose of scientific control is not to eradicate pests, but to control the number of pests to a certain level.

Conclusions

A new mathematical biological model on the Gompertz virus impulsive system is proposed in this paper. This proposed model considers the factors of the recovered or the removed population, asymptotically infected, infected (symptomatic), exposed and susceptible population. Utilizing the Floquet theory, we strictly prove the infection-free periodic solution of the proposed model is globally attractive if $R_1^{\ast }<1.$ It means that the infective pests almost will go extinct and have little effect on the crop. Moreover, we also obtain that the system (3) is permanent if $R_{1\ast }>1.$ Hence, we exploit the impulsive control method based on the effect of the Gompertz virus on pest such that $R_1^{\ast }<1,$ and then drive the virus to extinction. In addition, we can also control that the susceptible, exposed, infected (symptomatic), asymptotically infected population and the Gompertz virus in the market or seafood place or the reservoir oscillate with a positive amplitude. Actually, the scientific impulsive control is not to eradicate pests, but to control the number of pests to a certain level. Therefore, the proposed impulsive control strategy in this paper is very effective in impulsive control for the Gompertz virus model. Considering that time delays unavoidably exist in the transmission of the impulsive information and sampling in lots of practical cases, the proposed system will be extended to the delayed impulsive control for the Gompertz virus disease model in the future.

Author contributions

L.W. contributed to the conception of the study. A.S. contributed significantly to data analysis and manuscript preparation; Y.X. helped to perform the analysis with constructive discussions.

Data availibility

Data will be made available on request from the corresponding author (Youxiang Xie).

Competing interests

The authors declare no competing interests.