1 Introduction

Comparatively, HIV like most viruses is very minute, unadorned organism which cannot reproduce unaided. It remains the most deadly disease [1] which has ever hit the planet since the last three decades. However, as the fight for survival between the virus and the human immune system persist [1–7], CD4+ T-lymphocytes and macrophage remediate the cellular defence mechanism of the body. The immune system [1, 7–14], congregates to filter diverse information from invading pathogens and perceives them through the T cell receptors (TCRs) (Fig. 1).

Figure 1
figure 1

Specialized Macrophage cell which ingest foreign antigens invading the host cell

Full size image

1.1 Modelling of intracellular delay

Delay differential equations (DDEs) have been valuable for innumerable years in control theory and recently interrelated to biological and mathematical models. Most biological frames [10] have time delay embedded in them, yet few scientist collaborate them due to the intricacies they stimulate. The principal intricacies in studying DDEs lie in their solitary transcendental character.

In this paper, we first formulate a system of ordinary differential equations expounded as follows

$$\begin{aligned} &\frac{dx}{dt} = a_{1} (1-xy-x), \\ &\frac{dy}{dt} = a_{2} ( a_{3} xy-y ) - a_{4} yz, \\ &\frac{dz}{dt} = a_{5} (y-z), \end{aligned}$$
(1.1)

where CD4+ T-cell is symbolized as (x), tainted cells as (y) and CTL as (z).

Next, we define a translation operator \(T_{\tau }\) where \(\tau \geq 0 \), and \(T_{\tau } ( T ) =s ( t- \tau ) \). Therefore by nominal convenience \(x_{1}:= T_{\tau } x, y_{1}:= T_{\tau } y\), and \(z_{1}: =T_{\tau } z\), hence we express the system of equations with delay as follows

$$\begin{aligned} \textstyle\begin{cases} \frac{dx}{dt} = a_{1} (1-xy-x),\\ \frac{dy}{dt} = a_{2} ( a_{3} x_{1} y_{1} -y ) - a_{4} yz,\\ \frac{dz}{dt} = a_{3} (y-z). \end{cases}\displaystyle \end{aligned}$$
(1.2)

The term \(a_{2} a_{3} x_{1} y_{1}\), represents the rate equation and reflects a boundless time lag between the infection of CD4+ T-cell and the staging of new virions. Next we consider the well posedness and existence of equilibrium points.

2 Well posedness and existence of equilibrium points

We define equation (1.2) in terms of F, G and H as shown below;

$$\begin{aligned} &\frac{dx}{dt} =F(x,y,z, x_{1}, y_{1},z_{1} ), \\ &\frac{dy}{dt} =G(x,y,z, x_{1}, y_{1},z_{1} ), \\ &\frac{dz}{dt} =H(x,y,z, x_{1}, y_{1},z_{1} ). \end{aligned}$$
(2.1)

Equating (1.2) and (2.1), yields

$$\begin{aligned} &F ( x,y,z,x_{1}, y_{1},z_{1} ):= a_{1} ( 1-xy-x ), \\ &G ( x,y,z, x_{1}, y_{1},z_{1} ):= a_{2} ( a_{3}, x_{1},y_{1} -y ) - a_{4} yz, \\ &H ( x,y,z, x_{1}, y_{1},z_{1} ):= a_{5} (y-z). \end{aligned}$$
(2.2)

Further, we define a steady state for equation (2.2) in terms of \(({\overline{x}}, {\overline{y}}, {\overline{z}})\) and procure the equations below:

$$\begin{aligned} &F ( {\overline{x}}, {\overline{y}}, {\overline{z}}, {\overline{x}}_{1}, {\overline{y}}_{1}, {\overline{z}}_{1}) =0, \\ &G ( {\overline{x}}, {\overline{y}}, {\overline{z}}, {\overline{x}}_{1}, {\overline{y}}_{1}, {\overline{z}}_{1}) =0, \\ &H ( {\overline{x}}, {\overline{y}}, {\overline{z}}, {\overline{x}}_{1}, {\overline{y}}_{1}, {\overline{z}}_{1}) =0. \end{aligned}$$
(2.3)

Proposition 2.1

If\(a_{2} a_{3} \)and\(z_{1}: =T_{\tau } z\), then the steady state solution of equation (2.1) is in unison with the steady state solution of equation (2.3).

Proof

Consider an immutable function \(S(t)\) and its derivative \(\frac{ds}{dt}\), assuming that the derivative of the function is \(\frac{ds}{dt} = \frac{d s_{1}}{dt}\), then:

$$\begin{aligned} &F ( {\overline{x}}, {\overline{y}}, {\overline{z}}, {\overline{x}}_{1}, {\overline{y}}_{1}, {\overline{z}}_{1}) =0, \qquad G ( {\overline{x}}, {\overline{y}}, {\overline{z}}, {\overline{x}}_{1}, {\overline{y}}_{1}, {\overline{z}}_{1}) =0 \quad \text{and} \quad H ( {\overline{x}}, {\overline{y}}, {\overline{z}}, {\overline{x}}_{1}, {\overline{y}}_{1}, {\overline{z}}_{1}) =0 , \end{aligned}$$

hence equations (2.1) and (2.3) have the same unvarying states.

Further, by codifying a neighborhood close to the unvarying state solution, we let

$$\begin{aligned} &x= \overline{x} +X,\qquad y= \overline{y} +Y,\qquad z= \overline{z} +Z, \qquad x_{1} = \overline{x} + X_{1},\\ & y_{1} = \overline{y} + Y_{1} \quad\text{and}\quad z_{1} = \overline{z} + Z_{1}. \end{aligned}$$

Next, we exploit the Taylor series about the point (\(\overline{x}, \overline{y}, \overline{z} \)) and withholding only the linear terms yields

$$\begin{aligned} &\frac{dX}{dT} =- a_{1} ( \overline{y} +1 ) X- a_{1} \overline{x} Y, \\ &\frac{dY}{dt} = a_{2} a_{3} \overline{y} X_{1} + \bigl[ a_{2} ( a_{3} \overline{x} Y_{1} -Y ) - a_{4} \overline{z} Y \bigr] - a_{4} \overline{y} Z \quad \text{and} \\ &\frac{dZ}{dt} = a_{5} Y- a_{5} Z. \end{aligned}$$
(2.4)

Further, we linearized equation (2.4) as:

⌊ d X d t d Y d t d Z d t ⌋ = ⌊ − a 1 ( y ‾ + 1 ) − a 1 x ‾ 0 a 2 a 3 y ‾ e − λ t a 2 ( a 3 x ‾ e − λ t − 1 ) − a 4 z ‾ − a 4 y ‾ 0 a 5 − a 5 ⌋ ⌊ X Y Z ⌋ .
(2.5)

Next we let

B= ⌊ − a 1 ( y ‾ + 1 ) − a 1 x ‾ 0 a 2 a 3 y ‾ e − λ t a 2 ( a 3 x ‾ e − λ t − 1 ) − a 4 z ‾ − a 4 y ‾ 0 a 5 − a 5 ⌋ .

Where B is the coefficient matrix of the linearized system and λ is the eigenvalue of matrix B. Hence the unvarying state of the delay system can be explored in terms of the eigenvalues of matrix B. □

3 Global stability of equilibrium \(E_{O}\)

In this section, we proceed by analyzing the universal stability of the disease-free equilibrium of the model. Hence we let (\(x_{o}, y_{o}, z_{o} \)) be an unvarying state solution of the following equations: \(\frac{dx}{dt} =F ( x,y,z, x_{1}, y_{1}, z_{1} ), \frac{dy}{dt} =G(x,y,z, x_{1}, y_{1}, z_{1} )\) and \(\frac{dz}{dt} =H(x,y,z, x_{1}, y_{1}, z_{1} )\), then (\(x_{o}, y_{o}, z_{o} \)) is stable if \(\epsilon >0 \) and \(\delta >0\) for (\(x,y,z\)) when

$$\begin{aligned} \bigl[x ( t_{o} ) - x_{o} \bigr]^{2} + \bigl[y ( t_{o} ) - y_{o} \bigr]^{2} +\bigl[z ( t_{o} ) - z_{o} \bigr]^{2} < \delta ^{2}. \end{aligned}$$
(3.1)

Given that \(t_{o} \in [ t_{1} -\tau, t_{1} ]\), then \([x ( t ) - x_{o} ]^{2} +[y ( t ) - y_{o} ]^{2} + [z ( t ) - z_{o} ]^{2} < \varepsilon ^{2} \) for all \(t> t_{1}\).

On the other hand if (\(x_{o}, y_{o}, z_{o} \)) is stable and δ is chosen so that equation (3.1) is elucidated as \([ x ( t ), y ( t ),z ( t ) \rightarrow ( x_{o}, y_{o}, z_{o} ) ] \). Then as \(t \rightarrow \infty \), (\(x_{o}, y_{o}, z_{o}\)) is globally asymptotically stable.

Proposition 3.1

Suppose\(a_{1},a_{2}, a_{3}, a_{5} >0\):

  1. (i)

    Then if\(a_{3} <1 \), then the unvarying state of equation (1.2) of the system (2.1) is universally asymptotically stable for\(\tau \geq 0\). Hence\(E_{O}\)is stable.

  2. (ii)

    If\(a_{3} >1 \),then the unvarying state of equation (1.2) of the system (2.1) is globally unstable, hence\(E_{O} \)is unstable.

Proof

Suppose the characteristic equation of matrix B is written as

$$\begin{aligned} U ( \lambda ) = \lambda ^{3} + b_{1} \lambda ^{2} + b_{2} \lambda + b_{3} +\bigl( d_{1} \lambda ^{2} + d_{2} \lambda + d_{3} \bigr)e^{-\tau \lambda } =0. \end{aligned}$$
(3.2)

Now if \(\tau =0 \), then there is no delay in equation (3.2), hence Proposition 3.1 reduces to Proposition 2.1.

Conversely, when \(\tau >0\), and we surrogate the unvarying state of equations (1.2) and (2.2) and solve, we generate the characteristic equation

$$( \lambda + a_{1} ) \bigl( \lambda + a_{2} - a_{2} a_{3} e^{-\lambda \tau } \bigr) ( \lambda + a_{5} ) =0. $$

Next, by considering the unvarying state of equation (1.2), we ascertain that matrix B has three eigenvalues: \(\lambda _{1} =- a_{1} <0, \lambda _{2} =- a_{5} <0 \) and \(\lambda _{3}\) in the form

$$\lambda _{3} + a_{2} \bigl( a_{3} e^{-\tau \lambda _{3}} -1 \bigr) =0. $$

Therefore, to find the location of \(\lambda _{3}\), we propose a function of the form

$$u ( t ):=t+ a_{2} - a_{2} a_{3} e^{-\lambda \tau },\quad \text{where }t\in R. $$

By differentiating, we obtain \(u ' ( t ) =1+ a_{2} a_{3} \tau e^{-\lambda \tau }\), which is always positive, hence the limit of the original equation is written as \(\lim_{t\rightarrow \infty } u ( t ) =-\infty \) and \(\lim_{t\rightarrow \infty } u ( t ) =\infty \). Therefore u(t) has a distinctive zero revealed as \(u ( 0 ) = a_{2} (1- a_{3} )\). Hence we conclude that if \(a_{3} <1, u(0)>0\) and \(\lambda _{3} <0\). Then the unvarying state of equation (2.5) is globally asymptotically stable for \(\tau >0 \).

Conversely if \(a_{3} >1 \),then we conclude that the unvarying state of equation (1.2) is globally asymptotically unstable for \(a_{3} <0\), \(\lambda _{3} >0\) and \(\tau >0\). □

4 The endemic equilibrium

The endemic equilibrium \(E^{*}\) is obtained by filtering the real and imaginary parts from the equation below

$$\begin{aligned} & [b_{3} + b_{2} \alpha + b_{1} \alpha ^{2} + \alpha ^{3} - b_{1} \omega ^{2} -3\alpha \omega ^{2} +i \bigl( b_{2} \omega +2 b_{1} \alpha \omega +3 \alpha ^{2} \omega - \omega ^{3} \bigr) \\ &\quad{}+ e^{-\tau \alpha } [ ( d_{2} \omega +2 d_{1} \alpha \omega ) \sin \omega \tau + \bigl( d_{3} + d_{2} \alpha + d_{1} \alpha ^{2} - d_{1} \omega ^{2} \bigr) \cos \omega \tau \\ &\quad{}+i e^{-\tau \alpha } \bigl[ \bigl( d_{1} \omega ^{2} - d_{3} - d_{2} \alpha - d_{1} \alpha ^{2} \bigr) \sin \omega \tau + ( d_{2} \omega +2 d_{1} \omega ) \cos \omega \tau \bigr] =0, \end{aligned}$$
(4.1)

which yields

$$\begin{aligned} &d_{2} \omega \sin \omega \tau + \bigl( d_{3} - d_{1} \omega ^{2} \bigr) \cos \omega \tau = b_{1} \omega ^{2} - b_{3}, \end{aligned}$$
(4.2)
$$\begin{aligned} &d_{2} \omega \cos \omega \tau - \bigl( d_{3} - d_{1} \omega ^{2} \bigr) \sin \omega \tau = \omega ^{3} - b_{2} \omega. \end{aligned}$$
(4.3)

Summing the squares of equations (4.2) and (4.3)

$$\begin{aligned} u ( \omega ):= \omega ^{6} + \bigl( b_{1}^{2} -2 b_{2} - d_{1}^{2} \bigr) \omega ^{4} + \bigl( b_{2}^{2} -2 b_{1} b_{3} +2 d_{1} d_{3} - d_{2}^{2} \bigr) \omega ^{2} + b_{3}^{2} - d_{3}^{2} =0. \end{aligned}$$
(4.4)

We further exploit equation (4.4) by surrogating the parameters below

$$m:= \omega ^{2},\qquad p:= b_{1}^{2} -2 b_{2} - d_{1}^{2},\qquad q:= b_{2}^{2} -2 b_{1} b_{3} +2 d_{1} d_{3} - d_{2}^{2}\quad \text{and}\quad r:= b_{3}^{2} - d_{3}^{2}. $$

Hence, the new equation reads

$$\begin{aligned} k ( m ) = m^{3} +p m^{2} +qm+r=0. \end{aligned}$$
(4.5)

Lemma 4.1

Suppose\(\tau >0\)then equation (4.5) has no positive real root, hence all the roots have negative real parts.

Proof

Since equation (4.5) has no positive root, then ω is not the root of equation (4.4). Therefore for any real number ω, the value of iω is also not the root of equation (4.4), hence \(\lambda ( \tau _{c}) =i\omega ( \tau _{c} )\) is not the root of equation (4.4). This implies that only one positive real root exist for equation (4.5). Hence differentiating equation (4.5) yields

$$\begin{aligned} k ' ( m ) =3 m^{2} +2pm+q. \end{aligned}$$
(4.6)

Therefore the roots of equation (4.5) are

$$ m_{o} = \frac{-p+ \sqrt{p^{2} -3q}}{3}\quad \text{and}\quad \frac{-p- \sqrt{p^{2} -3q}}{3}. $$

 □

Lemma 4.2

  1. (i)

    If\(r<0 \)then it follows that\(p^{2} -3q>0 \)has a positive root, provided\(p<0 \).

  2. (ii)

    On the other hand if\(r>0\)then\(p^{2} -3q<0\), has no positive real roots.

Proof

(i). Suppose condition (i) holds and \(r<0\), then \(k(0)= r<0\), hence \(\lim_{m\rightarrow \infty } k ( m ) =\infty \). Exploiting the intermediate value theorem, equation (4.6) yields \(t_{o}\) as a positive root and therefore \(k(t_{o} )=0\).

Conversely, if condition (ii) holds for which \(r>0\) and \(p^{2} -3q<0\), then \(m_{o} \) is real for which \(m_{o} >0\). Since \(k ( 0 ) =r>0\) and \(k( m_{o} )<0\), then by the intermediate value theorem, k has zero between the origin and \(m_{o}\)

(ii). conversely, if \(q> \frac{1}{3} p^{2} \) then the zeros of \(m_{o}\) and \(m_{1}\) of \(K ' (m)\) are not real. Henceforth \(K ' ( m ) =0\) has no real root.

Consequently \(q> \frac{1}{3} p^{2} \geq 0\), has no real roots, given that K is an increasing function where \(K ( 0 ) =r\geq 0\). We conclude that the model has a unique endemic equilibrium which is asymptotically stable if and only if \(R_{o} >1\), otherwise unstable. This unique endemic equilibrium occurs at \(\tau \geq 0\). □

5 Existence of Hopf bifurcation

We introduce the Hopf bifurcation and therefore appraise the 3-dimensional system of differential equations as follows

$$\begin{aligned} &\frac{dx}{dt} =F(x,y,z,\tau ), \\ &\frac{dy}{dt} =G(x,y,z,\tau ), \\ &\frac{dy}{dt} =H(x,y,z,\tau ). \end{aligned}$$
(5.1)

The following conditions holds if:

  1. (i)

    \(F ( \overline{x}, \overline{y}, \overline{z},\tau ) =G ( \overline{x}, \overline{y}, \overline{z},\tau ) =H ( \overline{x}, \overline{y}, \overline{z},\tau ) =0\), then \(\tau _{c}\) and \(( \overline{x}, \overline{y}, \overline{z}, ) \) remains as a steady state solution for equation (4.5).

  2. (ii)

    F, G and H are analytic in terms of (\(x, t, z\)), then they are within the neighborhood of \(( \overline{x}, \overline{y}, \overline{z}, \tau _{c} )\).

  3. (iii)

    the Jacobian matrix of equation (2.5) at \(( \overline{x}, \overline{y}, \overline{z},\tau )\) has a pair of complex conjugate eigenvalues, λ and λ̅, then

    $$\begin{aligned} \begin{aligned} &\lambda ( t ) =\alpha ( \tau ) +i\omega ( \tau ),\qquad \omega ( \tau _{c} ) = \omega _{c} >0, \qquad\alpha ( \tau _{c} ) =0\quad \text{and}\\ & \frac{d\alpha (\tau )}{dr} I_{\tau } = \tau _{C} \neq 0. \end{aligned} \end{aligned}$$
    (5.2)
  4. (iv)

    the remaining eigenvalues of the Jacobian matrix at \(( \overline{x}, \overline{y}, \overline{z}, \tau _{c} )\) have strictly negative real part. Then the system (4.1) has a family of periodic solution: \(\varepsilon _{H} >0\) and analytic function \(\tau ^{H} ( \epsilon ) = \sum_{2} \infty T_{i}^{H} \varepsilon ^{i}\), where \(0<\varepsilon < \varepsilon _{H}\). Hence, \(T^{H} ( \varepsilon )\) of \(p_{\varepsilon } (t)\) is analytic and of the form

    $$\begin{aligned} T^{H} ( \varepsilon ) = \frac{2\pi }{\omega _{c}} \Biggl(1+ \sum _{2}^{\infty } T_{i}^{H} \varepsilon ^{i} \Biggr) \quad (0< \varepsilon < \varepsilon _{H} ). \end{aligned}$$
    (5.3)

Next, we exploit the parameters of the analytic function to ensure the occurrence of Hopf bifurcation. Hence we denote the positive roots of equation (4.1) by \(m_{j}, j\), where \(j\in [0,1,2]\), depending on the number of positive roots in the equation. Therefore equation (4.1) has six positive roots such as \(\omega _{j}, \lambda =i\omega \) and \(\pm \sqrt{m_{j}}\) where \(j=0,1,2\). Consequently, substituting \(\omega _{j}\) into equation (4.2) and (4.3) and solving for τ produces

$$\begin{aligned} \tau _{j}^{(n)} = \frac{1}{\omega _{j}} \arcsin \biggl\lceil \frac{d_{1} \omega _{j}^{5} + ( b_{1} d_{2} - d_{3} - d_{1} b_{2} ) \omega _{j}^{3} + ( d_{3} b_{2} - b_{3} d_{2} ) \omega _{j}}{d_{2}^{2} \omega _{j}^{2} +( d_{3} - d_{1} \omega _{j}^{2} )^{2}} \biggr\rceil + \frac{2\pi (n-1}{\omega _{j}}. \end{aligned}$$
(5.4)

Where \(j=0, 1,2\) and \(n=1,2,\dots \).

Again, when we consider \(\tau _{c} >0\) to be a smaller value, where \(\alpha ( \tau _{c} ) =0\), then

$$\begin{aligned} \tau _{c} T_{jc}^{(nc)} = \min \bigl\{ \tau _{j}^{ ( \tau )} >0, 0\leq j\leq 2, n\geq 1\bigr\} . \end{aligned}$$
(5.5)

Where \(\omega _{c} = \omega _{jc}\).

Theorem 5.1

Consider the time lagτand the critical time lag\(\tau _{c}\)and\(\omega _{c} \)as defined in equation (4.5) and suppose\(3 \omega _{c}^{6} +2p \omega _{c}^{4} +q \omega _{c}^{2} \neq 0\)then the system of delay differential equation in terms of equation (2.1), portrays the Hopf bifurcation at the steady state in equation (1.2).

Proof

We show that

$$\begin{aligned} \frac{d\alpha (\tau )}{d\tau } I_{\tau } = \tau _{C} \neq 0. \end{aligned}$$
(5.6)

The existence of Equation (4.6) personifies the occurrence of Hopf bifurcation. Hence equating the real and imaginary parts of equations (4.2) and (4.3) to zero, we obtain the following

$$\begin{aligned} &e^{-\tau \alpha } \bigl[ (d_{2} +2 d_{1} \alpha ) \omega \sin \omega \tau + \bigl[ (d_{3} + d_{2} \alpha + d_{1} \bigl( \alpha ^{2} - \omega ^{2} \bigr) \bigr] \cos \omega \tau \bigr] \\ &\quad = (b_{1} +3\alpha ) \omega ^{2} - b_{3} - b_{2} \alpha - b_{1} \alpha ^{2} - \alpha ^{3} ] \end{aligned}$$
(5.7)

and

$$\begin{aligned} & e^{-\tau \alpha } \bigl[ \bigl(d_{1} \omega ^{2} + d_{3} - d_{2} \alpha - d_{1} \alpha ^{2} \bigr) \sin \omega \tau + ( d_{2} \omega +2 d_{1} \alpha \omega ) \cos \omega \bigr] \\ &\quad =\omega ^{3} - b_{2} \omega -2 b_{1} \alpha \omega -3 \alpha ^{2} \omega. \end{aligned}$$
(5.8)

Further, we differentiate equations (5.7) and (5.8) with respect to τ and evaluate at \(\tau = \tau _{c}\) where \(\alpha \tau _{c} =0\) and \(\omega ( \tau _{c} ) = \omega _{c}\) and obtains the following

$$\begin{aligned} &E_{1} \frac{d\omega }{d\tau }\bigg\vert \tau = \tau _{c} - E_{2} \frac{d\alpha }{d\tau } \bigg\vert \tau = \tau _{c} = E_{3} \cos \omega _{c} \tau _{c} + E_{4} \sin \omega _{c} \tau, _{c}, \end{aligned}$$
(5.9)
$$\begin{aligned} &E_{2} \frac{d\omega }{d\tau } \bigg\vert \tau = \tau _{c} - E_{1} \frac{d\alpha }{d\tau } \bigg\vert \tau = \tau _{c} = E_{3} \sin \omega _{c} \tau _{c} + E_{4} \cos \omega _{c} \tau _{c}. \end{aligned}$$
(5.10)

Where \(E_{1}:=2 b_{1} \omega _{c} + ( 2 d_{1} \omega _{c} - \tau _{c} d_{2} \omega _{c} ) \cos \omega _{c} \tau _{c} + ( \tau _{c} d_{3} - d_{2} - \tau _{c} d_{1} \omega _{c}^{2} ) \sin \omega _{c} \tau _{c}\), \(E_{2}:= b_{2} -3 \omega _{c}^{2} + ( d_{2} + \tau _{c} d_{1} \omega _{c}^{2} - \tau _{c} d_{3} ) \cos \omega _{c} \tau _{c} +(2 d_{1} \omega _{c} - \tau _{c} d_{2} \omega _{c} )\sin \omega _{c} \omega _{c}\), \(E_{3}:= d_{2} \omega _{c}^{2}\) and \(E_{4}:= d_{1} \omega _{c}^{3} - d_{3} \omega _{c}\).

Now by solving equations (5.9) and (5.10) together we have

$$\begin{aligned} \frac{d\alpha }{d\tau } \bigg\vert \tau = \tau _{c} = \frac{ ( E_{1} E_{4} - E_{2} E_{4} ) \sin \omega _{c} \tau _{c} - ( E_{1} E_{4} + E_{2} E_{3} ) \cos \tau _{c}}{E_{1}^{2} + E_{2}^{2}}. \end{aligned}$$
(5.11)

Next we substitute equations (4.2) and (4.3) into equations (5.9) and (5.10) and yields

$$\begin{aligned} &\cos \omega _{c} \tau _{c} = \frac{( b_{1} \omega _{c}^{2} - b_{3} )( d_{3} - d_{1} \omega _{c}^{2} + d_{2} \omega _{c} ( \omega _{c}^{3} - b_{2} \omega _{c} )}{d_{2}^{2} +( d_{3} - d_{1} \omega _{c}^{2} )^{2}}, \end{aligned}$$
(5.12)
$$\begin{aligned} &\sin \omega _{c} \tau _{c} = \frac{d_{2} \omega _{c} ( b_{1} \omega _{c}^{2} - b_{3} ) -( d_{3} - d_{1} \omega _{c}^{2} ) (\omega _{c}^{3} - b_{2} \omega _{c} )}{d_{2}^{2} \omega _{c}^{2} +( d_{3} - d_{1} \omega _{c}^{2} )^{2}}. \end{aligned}$$
(5.13)

Finally, we incorporate equations (5.12) and (5.13) into equation (5.11) and obtains

$$\begin{aligned} \frac{d\alpha }{d\tau } \bigg\vert \tau = \tau _{c} = \frac{3 w_{c}^{6} +2p \omega _{c}^{4} +q \omega _{c}^{2}}{E_{1}^{2} + E_{2}^{2}} \neq 0. \end{aligned}$$
(5.14)

Consequently we conclude based on equations (5.12), (5.13) and (5.14), the existence of Hopf bifurcation when, τ passes through the critical value \(\tau _{c}\). □

5.1 Proof for the existence of Hopf bifurication

In this section we prove the existence of Hopf Bifurcation by assuming that \(a_{1}, a_{2}, a_{5} >0\) and \(a_{3} >1\), for \(\tau _{c}\) and \(\omega _{c}\) as defined in equation (4.6). Therefore by Lemma 4.2(i) if \(p\geq 0, q\geq 0\), then we conclude that the system of delay differential equations proposed in equation (2.1) for the whole delay process has Hopf bifurcation. Consequently, the Hopf bifurcation occurs at \(p\geq 0, q\geq 0\), when \(3 w_{c}^{6} +2p \omega _{c}^{4} +q \omega _{c}^{2} \neq 0\). This is further supported by Lemma 4.1 and Theorem 5.1. The forward bifurcation analysis is portrayed by Fig. 2, where \(\tau =q \omega _{c}\), \(\omega _{c} =0.024, P=1000, q=2200, E_{1} =0.029\) and \(E_{2} =0.06\). The conditions for the existence of Hopf bifurcation have been extrapolated by the above analysis. Hence, Fig. 2 portrays the conditions for the model to have a forward bifurcation under a single endemic equilibrium.

Figure 2
figure 2

Demonstration of the forward bifurcation process of the model with τ against \(\tau _{c}\): where \(\tau _{c} =q \omega _{c}\), \(\omega _{c} =0.024, P=1000, q=2200, E_{1} =0.029\) and \(E_{2} =0.06\)

Full size image

6 Numerical simulation

The numerical simulations are engaging around the convergence of orbits of the global stability and the existence of Hopf bifurcation for the system (5.14). The results are enjoined to flaunt the effects of intracellular delay on the qualitative behavior of the three variables: CD4+ T-cell, symbolized (x), cells tainted with HIV, symbolized (y) and CTL, symbolized (z) [7, 14].

Firstly, the model was analyzed without the delay component embedded (ODE), under the following conditions: affirmation of the qualitative conduct of the three variables such as CD4+ T-cell (x), HIV (y), and CTL (z). The affirmation of the three variables was done alongside with controlling the fundamental reproductive rate of the virus. Therefore when equation (1.1) was solved numerically as per the 6th order Runge–Kutta strategy with \(a_{1} =0.224, a_{2} =0.941, a_{3} =0.369, a_{4} =4.651, a_{5} =1.311\). It was ascertained that the quantum of CD4+ T-cells were consistent over time, whilst CTL and the virus level in the blood approaches zero (Fig. 3). In relation to Proposition 3.1(i) and equation (1.1), when \(a_{3} <1\), a steady state was attained at \((1, 0, 0)\). Therefore the conceptive pace of the infection happens to be under control and hence the decline rate of viral production is higher than the production rate. The virus at this stage cannot spread and hence eliminated from the blood as the quantum of CD4+ Tcells remains constant at \(\delta _{1} / \delta _{3}\) (Fig. 3).

Figure 3
figure 3

(a), (b), (c), (d) shows the Numerical simulation of the model without the delay component. Figure (a) shows how the quantum of CD4+ T-cells approaches zero, while (b) and (c) shows the amount of HIV and CTL in the blood respectively. Figure (d) shows how the virus component decreases over time as CTL approaches zero

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On the contrary, the model was further analyzed with the delay component (DDE), under the following conditions: affirmation of the qualitative conduct of the three variables such as CD4+ T-cell (x), HIV (y), and CTL (z). The affirmation of the three variables was done alongside with controlling the fundamental reproductive rate of the virus. Next equation (1.2) was solved numerically using the 6th order Runge–Kutta technique where \(a_{1} =0.224, a_{2} =0.941, a_{3} =0.369, a_{4} =4.651, \tau =0.191\). It was ascertained that the quantum of CD4+ T-cells were consistent over time, whilst CTL and the virus level in the blood approaches zero (Fig. 4). In relation to Proposition 3.1(i) and equation (1.2), when \(a_{3} <1\), a steady state was attained at (1, 0, 0). Therefore the conceptive pace of the infection happens to be under control and hence the decline rate of viral production is higher than the production rate. The virus at this stage cannot spread, hence eliminated from the blood as the quantum of CD4+ T-cells remains constant at \(\delta _{1} / \delta _{3}\) (Fig. 4).

Figure 4
figure 4

(a), (b), (c), (d) shows the Numerical simulation of the model with the delay component system. Figure (a) shows how the quantum of CD4+ T-cells approaches a constant value whilst (b) and (c) shows how the quantum of HIV decreases whilst CTL remains constant over time. Figure (d) shows how the virus component decreases over time as CTL in the blood weakens and approaches zero

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The model with delay was more advantageous over the non-delay model, because it was more stable at the trajectory. It supports viral peak postponement and virological suppression better than the non-delay (Figs. 3 and 4).

Another dimension to the analysis was the removal of the delay component under the following conditions: affirmation of the qualitative conduct of the three variables such as CD4+ T-cell (x), HIV (y), and CTL (z). This time the fundamental reproductive rate of the virus was uncontrolled. Now when equation (1.1) was solved numerically as per the 6th order Runge–Kutta strategy with now \(a_{1} =0.1038, a_{2} =3.22, a_{3} =2.64, a_{4} =7.41, a_{5} =4.17\). The density of HIV and CTL approaches the vanishing point to confirm the connection between the virus and the immune response, as depicted in Fig. 5. This implies that the essential conceptive pace of the infection is not under control and hence the decline rate of viral production is lower than the production rate. Infection rate surges and more virions are produced in the blood (Fig. 5).

Figure 5
figure 5

(a), (b), (c), (d) shows the numerical simulation of the model without the delay component and the reproductive number is not constant. Figure (a), the quantum of CD4+ T cells approaches a constant value: (b) and (c) shows how HIV and CTL component in the blood fluctuates before assuming a common point, with time. Figure (d) depicts an escalation of the virus as CTL level decreases drastically

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Contrary to the above, the model was analyzed further with the delay component (DDE), under the following conditions: affirmation of the qualitative conduct of the three variables such as CD4+ T-cell (x), HIV (y), and CTL (z). The fundamental reproductive rate of the virus was uncontrolled and equation (1.2) was solved numerically using the 6th order Runge–Kutta technique where \(a_{1} =0.1038, a_{2} =3.22, a_{3} =2.64, a_{4} =7.41, \tau =0.19\). It was observed that the essential conceptive pace of the infection was not under control and hence the decline rate of viral production was lower than the production rate (Fig. 6). Therefore the elongation of the infection [2, 3, 13] on an individual leads to the inception of AIDS.

Figure 6
figure 6

(a), (b), (c), (d) shows the numerical simulation of the model with delay whilst the reproductive number is not constant. Figure (a) shows how the quantum of CD4+ T cells decreases over time and assume a constant value: Figure (b) and (c) shows fluctuations of HIV and CTL in the blood. The Virus has reached an uncontrollable state as CTL decreases. Figure (d) shows an escalation of the virus component whilst CTL diminishes drastically for the interception of HIV

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7 Structured treatment interruption

The quest for legit structured treatment interruptions has necessitated the essence of intracellular delay in STI models. Currently, the adopted STI used by many HIV individuals does not incorporate days-off for drug assimilation by the body. Arguably therapy interruptions should align across days and weeks, in order to allow for drug assimilation. Find below some of the STI systems embedded to validate the model.

7.1 Two-weeks-on two-weeks-off strategy (14/14)

The scenario above relates to 14 days on treatment and 14 days off regime therapy, designed for HIV patients. However, adherence to the regime yields an auto viral control compared to the primitive STI system [11], which aligns with lengthy periods of therapy. In reference to Fig. 7, this strategy yields a momentous average viral load of 586 virions/ml and 3371 virions/m at peak level. On the contrary, a continuous therapy without any interruption increases the viral haul to 633 virions/ml on the average. This implies that the shorter the period for a particular therapy, the lower the peak viral haul. Therefore the lower the peak viral haul, the less the augmentation of viral strains, responsible for drug resistance mutations (Fig. 7).

Figure 7
figure 7

Graphs of STI which shows 14 days on and 14 days off treatment

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Further, the graphs also deal with non-delay and therefore an average viral haul of 8446 virions/ml was produced (Fig. 7(b)). The amount of virions produced by the non-delay therapy was higher and therefore signals the essence of delay. Hence, with reference to the ongoing discussion, a short cycle therapy is recommended on the basis of controlling viral rebound.

7.2 Five-days-on two-days-off strategy (5/2)

The five days on and two days off STI strategy allows treatment from Monday to Friday, with Saturday and Sunday as weekends off treatment. This strategy, also inferred as weekend off strategy is vital in subduing long term viral replication [1, 11]. It enhances a decrease in drug residual level, since its squatty intermission period allows for the assimilation of DRM. Hence in curbing down monotherapy situations, the usage of particular drugs should be disengaged for suitable metabolism rate.

In addition, the five days on and two days off regime, supports a decline of the maximum viral haul. It is worth to note that escalations in viral haul have been a headache for most scientists. The peak viral haul is crucial and personifies a larger viral pool which begets mutations. Based on the 5/2 strategy a cost saving of 29% is marginalized, compared to the continual therapy. In addition, the 5/2 strategy produces low viral haul due to limited viral rebound. Furthermore, the strategy is useful to places where therapy is not consistent and hence reliable to stop therapy for two days, than to abort the whole process due to limited drug availability. In accordance with the above discussions, the 5/2 strategy appears to be reasonably successful in producing positive results [1, 11], as depicted by Fig. 8.

Figure 8
figure 8

Demonstrates the 5 days on and 2 days off therapy

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From Fig. 8, the 5 days on and two days off therapy induces a mean viral haul of 331 virions/ml with zero delay and 412 virions/ml within 24 hours.

7.3 Impact of varying the on and off period

The idea of varying treatment on and off has conceited imperative due to less access to ARV, cost concerns and ceaseless treatment. However off-treatment should not be prolonged due to high viral rebound and viral mutation to drugs.

This chapter contrasts the 5/2 system, the 20/8 system and the 26/2 system and suggest the way forward. The ceaseless treatment period was used as reference and therapy was tempered as per the three systems above (Fig. 9). It was observed that tempering with therapy for a short period of time was essential due to the following reasons: accessibility to ARV, cost concerns and discontinuity of therapy due to ARV shortage. Therefore the shorter the time frames for off treatment, the better the results and the lesser the viral rebound [11–14].

Figure 9
figure 9

Correlation of the 5/2, 20/8 and 26/2 STI Regime

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In accordance with the ongoing analysis, it was ascertained that the 26/2 system (Table 1) stimulates a reliable outcome, however the system is not decisive due to its little cost sparing of about 7%. On the other side, the 20/8 system appears to have a lower viral haul than the 5/2 system. The respective average and peak viral hauls for the 20/8 system were 352 virions/ml and 751 virions/ml respectively. The 5/2 system also articulated 437 virions/ml and 756 virions/ml respectively. In view of the above outcome a trade- off is simulated to enhance the choice and adoption of a suitable treatment interruption (Fig. 9). Therefore the next section enhances the selection of the overall regime for compliance.

Table 1 Demonstration of treatment strategies, viral load level and corresponding cost
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7.4 Overall STI regime comparison

The above examination has conceited essential in adjusting treatment on and off, thereby diminishing the pinnacle viral haul. Therefore, based on the above analysis the 18/3 and 24/4 systems have been recommended for compliance and adoption due to low viral haul production (Fig. 10). The 18/3 system produces an average and peak virions of 120 and 285 virions/ml respectively, whilst the 24/4 system produces 118 and 338 virions/ml respectively (Fig. 10).

Figure 10
figure 10

Comparison of the STI systems; (24/4, 21/3, 12/2)

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8 Conclusion

The aim of the paper is to unveil the niche of delay differential equation in harmonizing low level HIV viral haul and thereby articulating the adopted model to delve into structured treatment interruptions.

Hence, the sturdiness of the model with delay (equation (1.2)) and without delay (equation (1.1)) was assessed. Numerical simulations were used to consolidate the results.

The demands for the stability of Hopf bifurcation were authenticated in itemizing the initial conditions of the model (Fig. 2). The existence of Hopf bifurcation [3, 4] has been proved and hence occurs when, τ passes through the critical value \(\tau _{c}\).

The analysis of the results indicated that when the basic reproductive rate of the virus was under control and the delay component τ were embedded in the model to verify the qualitative behavior of the three variables, such CD4+ Tcells (x), HIV cells (y) and CTL (z). It was concluded that when \(a_{3} <1\), then by Proposition 3.1(i): a steady state exists at (1,0, 0). Therefore the pace of infection by the virus is under control and the decline rate of viral production is higher than the production rate at \(\delta _{1} / \delta _{3}\)

Conversely when \(a_{3} >1\), then as per Proposition 3.1(ii): a non-existence steady state occurs. Therefore the pace of the contagion is higher than the decline rate at \(\delta _{1} / \delta _{3}\) and AIDS intercepts. [2]

Adherence to the conditions imposed on Proposition 3.1(i) when \(a_{3} <1\), intimates that the reproductive ratio of the virus is under control. This signifies a stable CD4+ T cells, hence adherence to therapy could delay AIDS interception. Further, by the conditions of Proposition 3.1(i) we have revealed an abortive attempt by the virus due to the consistent increase in CD4+ T cells (Fig. 4). Hence, CTL consequently eliminates the virus from the body. This is made possible when CD4+ T cells converges at \(\delta _{1} / \delta _{3}\)

Again sustenance of the imposed conditions on Proposition 3.1(i), is central to virological suppression and increased life expectancy under HIV [9, 11, 14]. Therefore it is imperative to ascertain the time frame, for the adaptive immune response of the body to emerge in regulating viral replication. This is supported when \(a_{3} <1\).

Further, in compliance with the rigorous analysis imposed on treatment options, the study hereby recommends for a short period off medication. This certifies that treatment should be expelled for only few days to allow for drug assimilation [8]. Therefore in reference to the imposed interactions on STI systems, the study recommends for the 24 days on treatment and 4 days off treatment for compliance. 18 days on treatment and 3 days off treatment is also supported by the study. [8, 12]

Notwithstanding, due to increased imposition on treatment interruptions the study recommends for the application of controlled techniques to enhance the adopted model.

Finally, the use of optimal control with distributed delay could also be exploited due to increase in target cells complexities [6].