Optimal control strategies for a heroin epidemic model with age-dependent susceptibility and recovery-age

1.   Introduction

In this research, we mainly focused on wave equation to study and examine the coupled system. In this system, we assumed a bounded domain $\Omega\in{\bf R}^{N}$ where $\partial\Omega$ indicates sufficiently smooth boundary of $\Omega\in{\bf R}^{N}$ and take the positive constants $\xi_{0}, \xi_{1}, \sigma, \beta_{1}, \beta_{3}$ where $m\geq 1$ for $N = 1, 2$, and $1 < m\leq\frac{N+2}{N-2}$ for $N\geq 3$. The coupled system with these terms is given by

in which $G = \Omega\times(\tau_{1}, \tau_{2})\times(0, \infty)$ and $\tau_{1} < \tau_{2}$ are taken to be non-negative constants in a manner that $\beta _{2}$, $\beta_{4} : [\tau_{1}, \tau_{2}] \rightarrow \mathbb{R}$ indicates distributive time delay while $g_{i}$, $i = 1, 2$ are positive.

The viscoelastic damping term, whose kernel is the function $g$, is a physical term used to describe the link between the strain and stress histories in a beam that was inspired by the Boltzmann theory. There are several publications that discuss this subject and produce a lot of fresh and original findings ^[1,2,3,4,5], particularly the hypotheses regarding the initial condition ^{[6,7,8,9,10,11,12]} and the kernel. See ^{[13,14,15,16,17]}. As it concerns to the plate equation and the span problem, Balakrishnan and Taylor introduced a novel damping model in ^[18] that they dubbed the Balakrishnan-Taylor damping. Here are a few studies that specifically addressed the research of this dampening for further information ^{[18,19,20,21,22,23]}.

Several applications and real-world issues are frequently affected by the delay, which transforms numerous systems into interesting research topics. Numerous writers have recently studied the stability of the evolution systems with time delays, particularly the effect of distributed delay. See ^[24,25,26].

In ^[1], the authors presented the stability result of the system over a considerably broader class of kernels in the absence of delay and Balakrishnan-Taylor damping $\xi_{0} = 1, \xi_{1} = \delta = \beta_{i} = 0, i = 1, \dots, 4$.

Based on everything said above, one specific problem may be solved by combining these damping terms (distributed delay terms, Balakrishnan-Taylor damping and infinite memory), especially when the past history and the distributed delay

are added. We shall attempt to throw light on it since we think it represents a fresh topic that merits investigation and analysis in contrast to the ones mentioned before. Our study is structured into multiple sections: in the second section, we establish the assumptions, notions, and lemmas we require; in the final section, we substantiate our major finding.

2.   Fundamental knowledge

In this section of the paper, we will introduce some basic results related to the theory for the analysis of our problem. Let us take the below:

(G1) $h_{i}:{\bf R}_{+}\rightarrow {\bf R}_{+}$ are a non-increasing $C^{1}$ functions fulfills the following

and

(G2) One can find a function $C^{1}$ functions $G_{i}:{\bf R}_{+}\rightarrow {\bf R}_{+}$ holds true $G_{i}(0) = G_{i}'(0) = 0$.

The functions $G_{i}(t)$ are strictly increasing and convex of class $C^{2}({\bf R}_{+})$ on $(0, \varrho], \quad r\leq g_{i}(0)$ or linear in a manner that

in which $\zeta_{i}(t)$ are a $C^{1}$ functions fulfilling the below

(G3) $\beta _{2}$, $\beta _{4}:[\tau_{1}, \tau_{2}]\rightarrow {\bf R}$ are a bounded function fulfilling the below

(G4) $f_{i}:{\bf R}^{2}\rightarrow {\bf R}$ are $C^{1}$ functions with $f_{i}(0, 0) = 0$, and one can find a function $F$ in a way that

and

Take the below

and

Lemma 2.1. (Sobolev-Poincare inequality ^[27]). Assume that $2\leq q < \infty$ for $n = 1, 2$ and $2\leq q < \frac{2n}{n-2}$ for $n\geq3$. Then, one can find $c_{*} = c(\Omega, q) > 0$ in a manner that

Moreover, choose the below as in ^[26]:

with

Take the auxiliary variable (see ^[28])

Then

Rewrite the problem (1.1) as follows

where

with

In the upcoming Lemma, the energy functional will be introduced.

Lemma 2.2. Let the energy functional is symbolized by $E$, then it is given by

The above fulfills the below

in which $\gamma_{0} = \min\{\beta _{1}-\int_{\tau_{1}}^{\tau_{2}}\vert \beta_{2}(r)\vert dr, \quad \beta _{3}-\int_{\tau_{1}}^{\tau_{2}}\vert \beta_{4}(r)\vert dr \}$.

Proof. To prove the result, we take the inner product of $(2.9)$ with $v_{t}, w_{t}$ and after that integrating over $\Omega$, the following is obtained

in which

Using mathematical skills, the following is obtained

further simplification leads us to the following

The following is obtained after calculation

In the same way, we have

Now, multiplying the equation $(2.9)$ by $- x \vert \beta_{2}(r)\vert, \quad - y\vert \beta_{4}(r)\vert$, and integrating over $\Omega\times(0, 1)\times (\tau_{1}, \tau_{2})$ and utilizing $(2.7)$, the below is obtained

Similarly, we have

Here, we utilize the inequalities of Young as

and

Finally, we have

Thus, after replacement of (2.14)–(2.22) into (2.13), we determined (2.11) and (2.12). As a result, we obtained that $E$ is a non-increasing function by (2.2)–(2.5), which is required.

Theorem 2.3. Take the function ${\bf U} = (v, v_{t}, w, w_{t}, x, y, \eta^{t}, \vartheta^{t})^{T}$ and assume that (2.1)–(2.5) holds true. Then, for any ${\bf U}_{0}\in \mathcal{H}$, then one can find a unique solution ${\bf U}$ of problems (2.9) and (2.10) in a manner that

If ${\bf U}_{0}\in \mathcal{G}_{1}$, then $U$ fulfills the following

in which

3.   Analysis of stability

Here, the stability of the systems (2.9) and (2.10) will be established and investigated. For which the following lemma is needed

Lemma 3.1. Let us suppose that (2.1) and (2.2) fulfills.

where

Proof.

which is obtained through Young's inequality (Eq 3.1).

Lemma 3.2. (Jensens inequality). Let $f : \Omega \rightarrow [c, e]$ and $h : \Omega \rightarrow {\bf R}$ are integrable functions in a manner that for any $z \in \Omega$, $h(z) > 0$ and $\int_{\Omega}h(z)dz = k > 0$. Furthermore, assume a convex function G such that $G : [c, e]\rightarrow {\bf R}$. Then

Lemma 3.3. It is mentioned in ^[12] that one can find a positive constant $\beta$, $\widehat{\beta}$ in a manner that

in which

Proof. As the function $E(t)$ is decreasing and utilizing (2.11), we have the following

for any $t, s \geq 0$. Further, we have

in which $\beta = \max\{\frac{4E(0)}{l_{1}}, 2\}$ and $\mu(t) = \int_{0}^{\infty}g_{1}(t+s)(1+\int_{\Omega}\nabla u_{0}^{2}(z, s) dz)ds$.

In the same way, we can deduce that

in which $\widehat{\beta} = \max\{\frac{4E(0)}{l_{2}}, 2\}$ and $\widehat{\mu}(t) = \int_{0}^{\infty}g_{2}(t+s)(1+\int_{\Omega}\nabla w_{0}^{2}(z, s) dz)ds$. In the upcoming part, we set the following

and

and

Lemma 3.4. In (3.8), the functional $\Psi(t)$ fulfills the following

for any $\varepsilon, \sigma_{1} > 0$ with $l = \min\{l_{1}, l_{2}\}$.

Proof. To prove the result, differentiate (3.8) first and then apply $(2.9)$, we have the following

We estimate the last 6 terms of the RHS of (3.12). The following is obtained by applying Young's, Sobolev-Poincare and Hölder's inequalities on (2.1) and (2.11), we have

In addition to this, for any $\sigma_{1} > 0$, by Lemma 3.1, we have the below

Taking same steps to $I_{12}$, the below is obtained

Same steps for $I_{11}, I_{21}$ and $I_{31}$, we have

Combining (3.13)–(3.21), (3.12) and (2.5), the required (3.11) is obtained.

Lemma 3.5. For any $\sigma, \sigma_{2}, \sigma_{3} > 0$, the functional $\Phi(t)$ introduced in (3.9) holds true

where $\widehat{l} = \max\{l_{1}, l_{2}\}$, $l_{0} = \min\{g_{0}, \widehat{g}_{0}\}$ and $\widehat{l_{0}} = \max\{g_{0}, \widehat{g}_{0}\}$.

Proof. To prove the result, simplification of (3.9) and $(2.9)$ through mathematical skills leads us to the following

Here, we will find our the approximation of the terms of the RHS of (3.18). Using the well-known Young's, Sobolev-Poincare and Hölder's inequalities on (2.1), (2.11) and Lemma 3.1, we proceed as follows

and

In the same, we obtained the following

and to find the approximation of $J_{61}$, we have

the (2.2) implies that

In the same steps, the estimation of $J_{i2}$, $i = 1, .., 6$ are obtained and

Here, put (3.19)–(3.25) into (3.18), the required result is obtained.

Lemma 3.6. The functional $\Theta(t)$ introduced in (3.10) fulfills the below

in which $\beta_{5} = \max\{\beta_{1}, \beta_{3}\}$.

Proof. To prove the result, using $\Theta(t)$, and $(2.9)$, we obtained the following

Utilizing $x(z, 0, r, t) = v_{t}(z, t), y(z, 0, r, t) = w_{t}(z, t)$, and $e^{-r}\leq e^{-r\rho}\leq 1$, for any $0 < \rho < 1$, moreover, select $\gamma_{1} = e^{-\tau_{2}}$, we have

applying (2.4), the required proof is obtained. In the next step, we below functional are introduced

in which $\varphi_{1}(t) = \int_{t}^{\infty}g_{1}(s)ds, \varphi_{2}(t) = \int_{t}^{\infty}g_{2}(s)ds$.

Lemma 3.7. Let us suppose that (2.1) and (2.2) satisfied. Then, the functional $F_{1} = \mathcal{A}_{1}+\mathcal{A}_{2}$ and fulfills the following

Proof. We can easily prove this lemma with the help of Lemma 3.7 in ^[13] and Lemma 3.4 in ^[15].

Now, we have sufficient mathematical tools to prove the below mentioned Theorem.

Theorem 3.8. Take (2.1)–(2.5), then one can find positive constants $\varsigma_{i}, i = 1, 2, 3$ and positive function $\varsigma_{4}(t)$ in a way that the energy functionalmentioned in (2.11) fulfills

in which

and

which are increasing and convex in $(0$, $\varrho]$.

Proof. For the proof, we define the below functional

we determined the positive constants $N, N_{i}, i = 1, 2, 3$. Simplifying (3.36) and utilizing 2.12, the Lemmas 3.4–3.6, we have

We select the various constants at this point such that the values included in parenthesis are positive in this stage. Here, putting

Thus, we arrive at

In the upcoming, we select $\sigma$ in a manner that

After that, we take $N_{2}$ in a way that

and take $N_{3}$ large enough in a way that

As a result, for positive constants $d_{i}, i = 1, 2, 3, 4, 5$, (3.33) can be written as

in which $C_{\kappa} = \max\{C_{\kappa, 1}, C_{\kappa, 2}\}$.

We know that $\frac{\kappa g_{i}^{2}(s)}{\kappa g_{i}(s)-g_{i}(s)}\leq g_{i}(s)$, then from from Lebesgue Dominated Convergence, we have the below

which leads to

As a result of this, one can find $0 < \kappa_{0} < 1$ in a manner that if $\kappa < \kappa_{0}$, then

From (3.8)–(3.10) through mathematical skills, we have the following

By the fact $e^{-\rho r} < 1$ and (2.2), we have the below

that is

Here, set $\kappa = \frac{1}{2N}$ and take $N$ large enough in a manner that

we find

for some $k_{2} > 0$, and

for some $c_{5}, c_{6} > 0$, we have

After that, the below cases are considered:

Case 3.9. $G_{i}, i = 1, 2$ are linear. Multiplying (3.40) by $\zeta_{0}(t) = \min\{\zeta_{1}(t), \zeta_{2}(t)\}$, we find

The last two terms in (3.42), we have

To estimate $I_{1}$, using (2.11),

and by (3.4), we get

In the same way, we obtained

As a result of this, we get

where $\beta_{0} = \max\{\frac{\beta}{4}, \frac{\widehat{\beta}}{4}\}$ and $w(t) = \widehat{\zeta}(t)\mu_{0}(t)$.

Applying $\zeta_{i}'(t)\leq0$, we get

with

we have

then, the following is obtained from (3.48)

Further analysis implies that

From the linearity of $D$, the linearity of the functions $D_{2}, D'_{2}$ and $D_{4}$ can easily be determined. This implies that

which gives (3.29) with $\varsigma_{1} = \lambda_{1}$, $\varsigma_{2} = \frac{\mathcal{H}_{1}(0)}{k_{2}}$, $\varsigma_{3} = \frac{2\beta_{0}}{\lambda_{2}k_{2}}$, and $\varsigma_{4}(t) = Id(t) = t$. Hence, the required proof is completed.

Case 3.10. Let $H_{i}, i = 1, 2$ are nonlinear. Then, with the help of (3.28) and (3.40). Assume the positive functional

then for all $t\geq 0$ and for some $k_{3} > 0$, the following holds true

with the help of (3.4), we have

Therefore

where $k_{6} = \max\{\frac{\mathcal{H}_{2}(0)}{k_{3}}, \frac{\beta_{0}}{k_{3}}\}$ and $\mu_{1}(t) = 1+\int_{0}^{t}\mu_{0}(\varsigma)d\varsigma$.

Corollary 3.11. The following is obtained from (2.11) and (3.53):

Now, we define $\phi_{i}(t), i = 1, 2$ by

where $\mathcal{B}(t) = \frac{\mathcal{B}_{0}}{\mu_{1}(t)}$ and $0 < \mathcal{B}_{0} < \min\{1, \frac{l}{8k_{6}}\}$.

Then, by (3.53), we have

Further, we suppose that $\phi_{i}(t) > 0, \quad \forall t > 0, \quad i = 1, 2$. In addition to this, we define another functional $\Gamma_{1}, \Gamma_{2}$ by

Here, obviously $\Gamma_{i}(t)\leq -cE'(t), \quad i = 1, 2$. As $G_{i}(0) = 0, \quad i = 1, 2$ and $G_{i}(t)$ are convex strictly on $(0$, $\varrho]$, then

Applying (2.3) and (3.56), we get

Taking the same steps, $\overline{G_{i}}, i = 1, 2$ are $C^{2}$-extension of $G_{i}$ that are convex strictly and increasing strictlyon ${\bf R}_{+}$. From (3.59), we have the following

Putting (3.61) and (3.4) into (3.40), we have

Here, introduce $\mathcal{K}_{1}(t)$ for $\varepsilon_{0} < r$ by

in which $D' = \min\{G_{1}, G_{2}\}$ and is equivalent to $E(t)$. Because of this $E^{\prime}(t) \leq 0, \overline{G_{i}}^{\prime} > 0,$ and $\overline{G_{i}}^{\prime \prime} > 0, i = 1, 2$. Also applying (3.62), we obtained that

According to ^[29], we introduce the conjugate function of $\overline{G_{i}}$ by $\overline{G_{i}}^{*},$ which fulfills

For $A = D^{\prime}\left(\varepsilon_{0}(E(t)\mathcal{B}(t)) /(E(0)))\right) \text { and } B_{i} = \overline{G_{i}}^{-1}((\mathcal{B}(t) \Gamma_{i}(t))/(\zeta_{i}(t))), \quad i = 1, 2$ and applying (3.64), we have

Here, we multiply (3.66) by $\zeta_{0}(t)$ and get

where we utilized the following $\varepsilon_{0}(\mathcal{B}(t) E(t) / E(0)) < r$, $D^{\prime} = \min \{G_{1}, G_{2}\}$ and $\Gamma_{i} < -cE'(t), i = 1, 2$, and define the functional $\mathcal{K}_{2}(t)$ as

Effortlessly, one can prove that $\mathcal{K}_{2}(t) \sim E(t)$, i.e., one can find two positive constants $m_{1}$ and $m_{2}$ in a manner that

then, we have

where $\beta_{6} = (k_{2}E(0)-2c\varepsilon_{0})$ and $D_{2}(t) = t D^{\prime}\left(\varepsilon_{0} t\right)$.

Choosing $\varepsilon_{0}$ so small such that $\beta_{6} > 0$, since $D_{2}^{\prime}(t) = D^{\prime}\left(\varepsilon_{0} t\right)+\varepsilon_{0} t D^{\prime \prime}\left(\varepsilon_{0} t\right)$. As $D_{2}^{\prime}(t), D_{2}(t) > 0$ on $(0$, $1]$ and $G_{i}$ on $(0$, $\varrho]$ are strictly increasing. Applying Young's inequality (3.65) on the last term in (3.70)

with $A = D^{\prime}\left(\varepsilon_{0} \frac{ E(t) \mathcal{B}(t)}{E(0)}\right)$ and $B = \frac{k_{6}}{\delta}\mu(t)$, we find

Here, choose $\sigma$ small enough in a manner that $\beta_{6}-\sigma\varepsilon_{0} > 0$ andcombining (3.70) and (3.71), we have

where $\beta_{7} = \beta_{6}-\sigma \varepsilon_{0} > 0$, $D_{3}(t) = t D'^{-1}\left(t\right)$ and $D_{4}(t) = \overline{D}_{3}^{*}\left(t\right)$.

In light of fact $E' < 0$ and $\mathcal{B}' < 0$, then $D_{2}(\frac{E(t) \mathcal{B}(t)}{E(0)})$ is decreasing. As a consequences of this, for $0\leq t\leq T$, we have

In the next step, combine (3.72) with (3.73) and multiply by $\mathcal{B}(t)$, the following is obtained

Since $\mathcal{B}' < 0$, then for any $0 < t < T$

Simplify (3.75) over $[0, T]$ and apply $\mathcal{B}(0) = 1$, the following is obtained

Consequently, we have

As a results of this, we obtain

As a result of this, we get

where, we have (3.29) with $\varsigma_{1} = \frac{ E(0)}{\mathcal{B}(T)}$, $\varsigma_{2} = \frac{\mathcal{K}_{2}(0)}{\beta_{7}}$, $\varsigma_{3} = \frac{\sigma}{\beta_{7}}$, and $\varsigma_{4}(t) = \frac{k_{6}}{\sigma}\mathcal{B}(t)$.

Hence, the required result is obtained 3.8.

4.   Conclusions

The purpose of this work was to study when the coupled system of nonlinear viscoelastic wave equations with distributed delay components, infinite memory and Balakrishnan-Taylor damping. Assume the kernels $g_{i} :{\bf R}_{+}\rightarrow {\bf R}_{+}$ holds true the below

in which $\zeta_{i}$ and $G_{i}$ are functions. We prove the stability of the system under this highly generic assumptions on the behaviour of $g_i$ at infinity and by dropping the boundedness assumptions in the historical data. This type of problem is frequently found in some mathematical models in applied sciences. Especially in the theory of viscoelasticity. What interests us in this current work is the combination of these terms of damping, which dictates the emergence of these terms in the problem. In the next work, we will try to using the same method with same problem. But in added of other dampings.

Acknowledgments

The researchers would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project.

Conflict of interest

The authors declare there is no conflicts of interest.