1 Introduction

The human immunodeficiency virus (HIV) predominantly targets CD4 + T cells, the immune system's biggest white blood cells [1, 2]. HIV infection attacks all cells, but it is CD4 + T cells that are the most damaging and impair the immune system by eliminating them [3]. When the number of CD4 + T cells falls below a particular threshold, the cell-mediated immune system vanishes, the immune system weakens, and the body becomes vulnerable to infection [3]. Pearson [4] provided a straightforward mathematical model for HIV infection. This model has served as an inspiration for scientists working on HIV modeling [4,5,6]. The mathematical models for HIV described here are extremely helpful in understanding the dynamics of HIV infection [7,8,9,10]. A group of scientists led by Agosto et al. [11] constructed an HIV model that included HIV cell-to-cell transmission and studied the characteristics of CD4 + T cells in depth. HIV-1 latency and processes have been described by Ruelas et al. [12] in order to better understand this deadly retrovirus. Sun et al. [13] indicated the c-myc proto-oncogene in the context of HIV-1 infection. The literature has supplied several mathematical models that have applications in varied fields akin to engineering, physics, etc., to the last of those sciences [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50]. In this regard, the 2019-nCoV pandemic that has afflicted the world has been thoroughly explored and studied [51,52,53,54,55,56,57]. Furthermore, COVID has impacted nearly every country on the globe in terms of economic, social, and psychological issues. Some key features of DNA have been observed by Cattani et al. [58] utilizing a mathematical model to describe them, and Owolabi et al. [59, 60] examined epidemiological models that included fractional order.

This study is arranged as follows: Sect. 2 offers an overview of the research model. The analytical solutions to this model and some diagrams for some of the analytical solutions we shall get are in Sect. 3. In the final part, we present the conclusion.

2 New algorithm scheme of the HIV-1 infection of CD4+ T cell

Let us first consider the new biomathematical model as follows [61,62,63]

$$K_{t}^{\nu } f\left( t \right) = \beta_{1} - \beta_{2} f\left( t \right) - \beta_{3} f\left( t \right)h\left( t \right)$$
$$K_{t}^{\nu } g\left( t \right) = \beta_{3} f\left( t \right)h\left( t \right) - \beta_{4} g\left( t \right)$$
(1)
$$K_{t}^{\nu } h\left( t \right) = \beta_{5} g\left( t \right) - \beta_{6} h\left( t \right)$$

This model with f(t), g(t), h(t), \(\nu \in \left( {0,\;1} \right]\) and \(\beta_{i}\), (i = 1, …, 6) is arbitrary constants that indicate the rate of production of CD4 + T cells, the rate of natural death rate, infected CD4 + cells from uninfected CD4 + cells, virus-producing cells' death, creation of vision viruses by infected cells, and virus particle death.

The conformable derivative (CD) of order \(\nu\) can be computed using the following formula:

$$M_{t}^{\nu } f\left( t \right) = \mathop {\lim }\limits_{\varepsilon \to 0} \left( {{{f\left( {t + \varepsilon t^{1 - \nu } } \right) - f\left( t \right)} \mathord{\left/ {\vphantom {{f\left( {t + \varepsilon t^{1 - \nu } } \right) - f\left( t \right)} \varepsilon }} \right. \kern-0pt} \varepsilon }} \right)$$
(2)

for all t > 0, \(\nu \in \left( {0,\;1} \right]\).

Using the transformation described below

$$\zeta = c\frac{{t^{\nu } }}{\nu }$$
(3)

Then Eq. (1), becomes

$$cf^{\prime}\left( \zeta \right) = \beta_{1} - \beta_{2} f\left( \zeta \right) - \beta_{3} f\left( \zeta \right)h\left( \zeta \right)$$
$$cg^{\prime}\left( \zeta \right) = \beta_{3} f\left( \zeta \right)h\left( \zeta \right) - \beta_{4} g\left( \zeta \right)$$
(4)
$$ch^{\prime}\left( \zeta \right) = \beta_{5} g\left( \zeta \right) - \beta_{6} h\left( \zeta \right)$$

In the following, the analytical scheme is used for generating new exact traveling wave solutions of the reduced Eq. (4).

3 New analytical solutions HIV-1 infection of CD4+ T cell

Here, the extended analytical method has been used for getting new exact solutions for fractional Eq. (4).

3.1 Improved \(\left( {{{\Upsilon^{\prime}} \mathord{\left/ {\vphantom {{\Upsilon^{\prime}} \Upsilon }} \right. \kern-0pt} \Upsilon }} \right)\) expansion method

In what’s follows [64,65,66,67,68,69,70], let the polynomial form of solution for Eq. (4) can be presented as follows:

$$f\left( \zeta \right) = \sum\limits_{n = 0}^{N} {A_{n} } \, F^{n} \left( \zeta \right)$$
$$g\left( \zeta \right) = \sum\limits_{n = 0}^{U} {B_{n} } \, F^{n} \left( \zeta \right)$$
(5)
$$h\left( \zeta \right) = \sum\limits_{n = 0}^{O} {M_{n} } \, F^{n} \left( \zeta \right)$$

where \(A_{0}\), \(A_{1}\), \(A_{2}\)…, \(A_{N}\), \(B_{0}\), \(B_{1}\), \(B_{2}\)…, \(B_{U}\),and \(M_{0}\), \(M_{1}\), \(M_{2}\),… \(M_{O}\) are constants, which can be determined by considering the derivative term of highest order with comparison of nonlinear terms of the governing equation, while \(F\left( \varsigma \right)\) is introduced by

$$F\left( \zeta \right) = \frac{{\Upsilon^{\prime}\left( \zeta \right)}}{\Upsilon \left( \zeta \right)}$$
(6)

where \(\Upsilon = \Upsilon \left( \zeta \right)\) follows the ODE in the following form:

$$\Upsilon \Upsilon^{\prime\prime} = \alpha \Upsilon^{2} + \beta \Upsilon \Upsilon^{\prime} + \delta \left( {\Upsilon^{\prime}} \right)^{2}$$
(7)

where \(\alpha\), \(\beta\), and \(\delta\) are constants.

Equation (7) can be rewritten as

$$\frac{d}{d\zeta }\left( {\frac{{\Upsilon^{\prime}}}{\Upsilon }} \right) = \alpha + \beta \left( {\frac{{\Upsilon^{\prime}}}{\Upsilon }} \right) + \left( {\delta - 1} \right)\left( {\frac{{\Upsilon^{\prime}}}{\Upsilon }} \right)^{2}$$
(8)

The generalized solutions of Eq. (7) are the four types of solutions as follows:

  • Case (1) if \(\Lambda = \beta^{2} + 4\alpha - 4\alpha \delta \ge 0\) and \(\beta \ne 0\), then

$$F\left( \zeta \right) = \frac{\beta }{{2\left( {1 - \delta } \right)}} + \frac{\beta \sqrt \Lambda }{{2\left( {1 - \delta } \right)}}\left( {\frac{{\delta_{1} \exp \left( {\frac{\sqrt \Delta }{2}\zeta } \right) + \delta_{2} \exp \left( {\frac{ - \sqrt \Delta }{2}\zeta } \right)}}{{\delta_{1} \exp \left( {\frac{\sqrt \Delta }{2}\zeta } \right) - \delta_{2} \exp \left( {\frac{ - \sqrt \Delta }{2}\zeta } \right)}}} \right)$$
(9)

  • Case (2) If \(\Lambda = \beta^{2} + 4\alpha - 4\alpha \delta < 0\) and \(\beta \ne 0\), then

    $$F\left( \zeta \right) = \frac{\beta }{{2\left( {1 - \delta } \right)}} + \frac{{\beta \sqrt { - \Lambda } }}{{2\left( {1 - \delta } \right)}}\left( {\frac{{i\delta_{1} \cos \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right) - \delta_{2} \sin \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right)}}{{i\delta_{1} \sin \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right) + \delta_{2} \cos \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right)}}} \right)$$
    (10)

  • Case (3) If \(\Lambda = \alpha \left( {1 - \delta } \right) \ge 0\) and \(\beta = 0\), then

    $$F\left( \zeta \right) = \frac{\sqrt \Lambda }{{\left( {1 - \delta } \right)}}\left( {\frac{{\delta_{1} \cos \left( {\sqrt \Lambda \,\zeta } \right) + \delta_{2} \sin \left( {\sqrt {\Lambda \,} \zeta } \right)}}{{\delta_{1} \sin \left( {\sqrt \Lambda \,\zeta } \right) - \delta_{2} \cos \left( {\sqrt \Lambda \,\zeta } \right)}}} \right)$$
    (11)

  • Case (4) If \(\Lambda = \alpha \left( {1 - \delta } \right) < 0\) and \(\beta = 0\), then

    $$F\left( \zeta \right) = \frac{{\sqrt { - \Lambda } }}{{\left( {1 - \delta } \right)}}\left( {\frac{{i\delta_{1} \cosh \left( {\sqrt { - \Lambda \,} \zeta } \right) - \delta_{2} \sinh \left( {\sqrt { - \Lambda \,} \zeta } \right)}}{{i\delta_{1} \sinh \left( {\sqrt { - \Lambda \,} \zeta } \right) - \delta_{2} \cosh \left( {\sqrt { - \Lambda \,} \zeta } \right)}}} \right)$$
    (12)

    where, \(\zeta = c\frac{{t^{\nu } }}{\nu }\) and \(\alpha ,\beta ,\delta ,\delta_{1} ,\delta_{2}\) are real parameters.

3.2 Application of improved \(\left( {{{\Upsilon^{\prime}} \mathord{\left/ {\vphantom {{\Upsilon^{\prime}} \Upsilon }} \right. \kern-0pt} \Upsilon }} \right)\) expansion method

In order to use this approach on our model, we must strike a balance between \(f^{\prime}\left( \zeta \right)\) with \(f\left( \zeta \right)h\left( \zeta \right)\), \(g^{\prime}\left( \zeta \right)\) with \(f\left( \zeta \right)h\left( \zeta \right)\) and \(h^{\prime}\left( \zeta \right)\) with \(h\left( \zeta \right)\) in Eq. (4), and we obtain N = U = 2, O = 1. Then, the solutions of Eq. (4) yields

$$f\left( \zeta \right) = A_{0} + A_{1} F\left( \zeta \right) + A_{2} F^{2} \left( \zeta \right)$$
$$g\left( \zeta \right) = B_{0} + B_{1} F\left( \zeta \right) + B_{2} F^{2} \left( \zeta \right)$$
(13)
$$h\left( \zeta \right) = M_{0} + M_{1} F\left( \zeta \right)$$

Inserting Eq. (13) into Eq. (4) with the aid of (8) yields a system of algebraic equations. By solving them, it gains.

  • Set (1)

$$\left\{ \begin{gathered} A_{0} = \mp \frac{{A_{1} \alpha }}{\ell },A_{2} = \mp \frac{1}{4}\frac{{\ell A_{1} }}{\alpha },\ell = \left( { \mp \beta + \sqrt {\beta^{2} - 4\alpha \delta + 4\alpha } } \right), \hfill \\ A_{1} = A_{1} ,c = c,B_{0} = B_{0} ,B_{1} = - A_{1} ,M_{1} = - \frac{{2c\left( {\delta - 1} \right)}}{{\beta_{3} }},\beta_{1} = 0,\beta_{2} = 0,\beta_{4} = 0,\beta_{6} = \beta_{6} , \hfill \\ B_{2} = \frac{1}{4}\frac{{\ell A_{1} }}{\alpha },M_{0} = \mp \frac{{4c\left( { - \delta \mp \frac{\ell \beta }{{2\alpha }} + 1} \right)\alpha }}{{\ell \beta_{3} }} \hfill \\ \end{gathered} \right\}$$
(14)

  • Set (2)

    $$\left\{ \begin{gathered} c = c,A_{0} = - B_{0} ,A_{1} = \frac{1}{2}\frac{{\ell B_{0} }}{\alpha },A_{2} = 0,B_{0} = B_{0} ,,\beta_{6} = \beta_{6} ,\ell = \left( { \mp \beta + \sqrt {\beta^{2} - 4\alpha \delta + 4\alpha } } \right), \hfill \\ B_{2} = 0,M_{0} = M_{0} ,M_{1} = - \frac{{c\left( {\delta - 1} \right)}}{{\beta_{3} }},\beta_{1} = 0,B_{1} = \frac{1}{2}\frac{{\ell B_{0} }}{\alpha }, \hfill \\ \beta_{2} = \mp \frac{{2\alpha \left( {c\delta - c \pm \frac{1}{2}\frac{{\beta_{3} \ell M_{0} }}{\alpha } \pm \frac{c\beta \ell }{{2\alpha }}} \right)}}{\ell },\beta_{4} = \mp \frac{{2\alpha \left( {c\delta - c \pm \frac{1}{2}\frac{{\beta_{3} \ell M_{0} }}{\alpha } \pm \frac{c\beta \ell }{{2\alpha }}} \right)}}{\ell } \hfill \\ \end{gathered} \right\}$$
    (15)

  • Set (3)

    $$\left\{ \begin{gathered} c = c,A_{0} = A_{0} ,A_{1} = A_{1} ,A_{2} = 0,B_{0} = B_{0} ,B_{1} = - A_{1} , \hfill \\ B_{2} = 0,M_{0} = \frac{{c\left( {B_{0} A_{0} \delta - B_{0} A_{0} - B_{0} A_{1} \beta - A_{1}^{2} \alpha } \right)}}{{\beta_{3} A_{1} \left( {A_{0} + B_{0} } \right)}},M_{1} = - \frac{{c\left( {\delta - 1} \right)}}{{\beta_{3} }},\beta_{1} = \frac{{c\left( {A_{0}^{2} \delta - A_{0}^{2} - A_{1}^{2} \alpha - A_{1} \beta A_{0} } \right)}}{{A_{1} }}, \hfill \\ \beta_{2} = \frac{{c\left( {A_{0}^{2} \delta - A_{0}^{2} - A_{1}^{2} \alpha - A_{1} \beta A_{0} } \right)}}{{A_{1} \left( {A_{0} + B_{0} } \right)}},\beta_{4} = \frac{{c\left( {A_{0}^{2} \delta - A_{0}^{2} - A_{1}^{2} \alpha - A_{1} \beta A_{0} } \right)}}{{A_{1} \left( {A_{0} + B_{0} } \right)}},\beta_{6} = \beta_{6} \hfill \\ \end{gathered} \right\}$$
    (16)

  • Set (4)

    $$\left\{ \begin{gathered} c = c,A_{0} = \frac{{A_{1} \alpha }}{\beta },A_{1} = A_{1} ,A_{2} = \frac{{A_{1} \left( {\delta - 1} \right)}}{\beta },B_{0} = B_{0} ,B_{1} = B_{1} ,\beta_{2} = - \frac{{\beta c\left( {A_{1} + B_{1} } \right)}}{{A_{1} }}, \hfill \\ B_{2} = - \frac{{A_{1} \left( {\delta - 1} \right)}}{\beta },M_{0} = \frac{{cB_{1} \beta }}{{\beta_{3} A_{1} }},M_{1} = - \frac{{2c\left( {\delta - 1} \right)}}{{\beta_{3} }},\beta_{1} = 0,\beta_{4} = 0,\beta_{6} = \beta_{6} \hfill \\ \end{gathered} \right\}$$
    (17)

  • Set (5)

    $$\left\{ \begin{gathered} c = c,A_{0} = \frac{{A_{1} \alpha }}{\beta },A_{1} = A_{1} ,A_{2} = \frac{{A_{1} \left( {\delta - 1} \right)}}{\beta },B_{0} = B_{0} ,B_{1} = B_{1} ,\beta_{4} = 0,\beta_{6} = \beta_{6} , \hfill \\ B_{2} = - \frac{{A_{1} \left( {\delta - 1} \right)}}{\beta },M_{0} = \frac{{cB_{1} \beta }}{{\beta_{3} A_{1} }},M_{1} = - \frac{{2c\left( {\delta - 1} \right)}}{{\beta_{3} }},\beta_{1} = 0,\beta_{2} = - \frac{{\beta c\left( {A_{1} + B_{1} } \right)}}{{A_{1} }} \hfill \\ \end{gathered} \right\}$$
    (18)

  • Set (6)

    $$\left\{ \begin{gathered} c = 0,A_{0} = \frac{1}{2}\frac{{A_{1}^{2} \delta - A_{1} \beta A_{2} - A_{1}^{2} + 2A_{2}^{2} \alpha }}{{A_{2} \left( {\delta - 1} \right)}},A_{1} = A_{1} ,A_{2} = A_{2} , \hfill \\ B_{0} = \frac{1}{2}\left( {\frac{{A_{1}^{2} \delta - A_{1}^{2} - A_{1} B_{1} + A_{1} B_{1} \delta - A_{1} \beta A_{2} + 2A_{2}^{2} \alpha }}{{A_{2} \left( {\delta - 1} \right)}}} \right), \hfill \\ B_{1} = B_{1} ,B_{2} = - A_{2} ,M_{0} = \frac{{c\left( {B_{1} \delta - A_{2} \beta + A_{1} \delta - A_{1} - B_{1} } \right)}}{{A_{2} \beta_{3} }},M_{1} = - \frac{{2c\left( {\delta - 1} \right)}}{{\beta_{3} }}, \hfill \\ \beta_{1} = \frac{1}{2}c\left( {\frac{{ - 2A_{1}^{3} \delta + 2A_{2}^{2} \beta^{2} A_{1} + 3A_{2} \beta A_{1}^{2} - 4A_{2}^{3} \beta \alpha + A_{1}^{3} + 4A_{1} \delta A_{2}^{2} \alpha - 4A_{1} A_{2}^{2} \alpha + A_{1}^{3} \delta^{2} - 3A_{2} \beta A_{1}^{2} \delta }}{{A_{2}^{2} \left( {\delta - 1} \right)}}} \right), \hfill \\ \beta_{2} = - \frac{{c\left( {A_{2} \beta + B_{1} \delta - B_{1} } \right)}}{{A_{2} }},\beta_{4} = \frac{{c\left( {A_{1} \delta - A_{1} - A_{2} \beta } \right)}}{{A_{2} }},\beta_{6} = \beta_{6} \hfill \\ \end{gathered} \right\}$$
    (19)

  • Set (7)

    $$\left\{ \begin{gathered} c = 0,A_{0} = \frac{1}{2}\frac{{A_{1}^{2} \delta - A_{1} \beta A_{2} - A_{1}^{2} + 2A_{2}^{2} \alpha }}{{A_{2} \left( {\delta - 1} \right)}},A_{1} = A_{1} ,A_{2} = A_{2} ,\beta_{2} = \beta_{4} , \hfill \\ B_{0} = \frac{1}{2}\left( {\frac{\begin{gathered} cA_{1}^{3} \delta^{2} - 3cA_{2} \beta A_{1}^{2} \delta - 2cA_{1}^{3} \delta + 2cA_{2}^{2} \beta^{2} A_{1} + 3cA_{2} \beta A_{1}^{2} - 4cA_{2}^{3} \beta \alpha \hfill \\ - \beta_{4} A_{2} A_{1}^{2} \delta + \beta_{4} A_{2}^{2} A_{1} \beta + \beta_{4} A_{2} A_{1}^{2} - 2\beta_{4} A_{2}^{3} \alpha + cA_{1}^{3} + 4cA_{1} \delta A_{2}^{2} \alpha - 4cA_{1} A_{2}^{2} \alpha \hfill \\ \end{gathered} }{{\beta_{4} A_{2}^{2} \left( {\delta - 1} \right)}}} \right), \hfill \\ B_{1} = - A_{1} ,B_{2} = - A_{2} ,M_{0} = \frac{{cA_{1} \delta - 2cA_{2} \beta - \beta_{4} A_{2} - cA_{1} }}{{A_{2} \beta_{3} }},M_{1} = - \frac{{2c\left( {\delta - 1} \right)}}{{\beta_{3} }}, \hfill \\ \beta_{1} = \frac{1}{2}c\left( {\frac{\begin{gathered} - 2A_{1}^{3} \delta + 2A_{2}^{2} \beta^{2} A_{1} + 3A_{2} \beta A_{1}^{2} - 4A_{2}^{3} \beta \alpha + A_{1}^{3} \hfill \\ + 4A_{1} \delta A_{2}^{2} \alpha - 4A_{1} A_{2}^{2} \alpha + A_{1}^{3} \delta^{2} - 3A_{2} \beta A_{1}^{2} \delta \hfill \\ \end{gathered} }{{A_{2}^{2} \left( {\delta - 1} \right)}}} \right),\beta_{4} = \beta_{4} ,\beta_{6} = \beta_{6} \hfill \\ \end{gathered} \right\}$$
    (20)

  • Set (8)

    $$\left\{ \begin{gathered} c = 0,A_{0} = A_{0} ,A_{1} = A_{1} ,A_{2} = A_{2} ,B_{0} = \frac{{A_{0} B_{2} }}{{A_{2} }},B_{1} = \frac{{A_{1} B_{2} }}{{A_{2} }},B_{2} = B_{2} ,M_{0} = \frac{{\beta_{4} B_{2} }}{{A_{2} \beta_{3} }},M_{1} = 0,\beta_{1} = 0, \hfill \\ \beta_{2} = - \frac{{\beta_{4} B_{2} }}{{A_{2} }},\beta_{4} = \beta_{4} ,\beta_{6} = \beta_{6} \hfill \\ \end{gathered} \right\}$$
    (21)

  • Set (9)

    $$\left\{ \begin{gathered} c = c,A_{0} = A_{0} ,A_{1} = A_{1} ,A_{2} = A_{2} ,B_{0} = B_{0} ,B_{1} = B_{1} ,B_{2} = B_{2} ,M_{0} = 0,M_{1} = 0,\beta_{1} = 0, \hfill \\ \beta_{2} = 0,\beta_{4} = 0,\beta_{6} = \beta_{6} \hfill \\ \end{gathered} \right\}$$
    (22)

In view of set [1], Eq. (13) yields

$$f\left( \zeta \right) = \mp \frac{{A_{1} \alpha }}{\ell } + A_{1} F\left( \zeta \right) \mp \frac{1}{4}\frac{{\ell A_{1} }}{\alpha }F^{2} \left( \zeta \right)$$
$$g\left( \zeta \right) = B_{0} - A_{1} F\left( \zeta \right) + \frac{1}{4}\frac{{\ell A_{1} }}{\alpha }F^{2} \left( \zeta \right)$$
(23)
$$h\left( \zeta \right) = \mp \frac{{4c\left( { - \delta \mp \frac{\ell \beta }{{2\alpha }} + 1} \right)\alpha }}{{\ell \beta_{3} }} - \frac{{2c\left( {\delta - 1} \right)}}{{\beta_{3} }}F\left( \zeta \right)$$

Substituting Eqs. (23) in Eqs. (9–12), we have.

Case (1) If \(\Lambda = \beta^{2} + 4\alpha - 4\alpha \delta \ge 0\) and \(\beta \ne 0\) (Fig. 1), then

$$\begin{gathered} f_{1,1} \left( t \right) = \mp \frac{{A_{1} \alpha }}{\ell } + A_{1} \left\{ {\frac{\beta }{{2\left( {1 - \delta } \right)}} + \frac{\beta \sqrt \Lambda }{{2\left( {1 - \delta } \right)}}\left( {\frac{{\delta_{1} \exp \left( {\frac{\sqrt \Delta }{2}\zeta } \right) + \delta_{2} \exp \left( {\frac{ - \sqrt \Delta }{2}\zeta } \right)}}{{\delta_{1} \exp \left( {\frac{\sqrt \Delta }{2}\zeta } \right) - \delta_{2} \exp \left( {\frac{ - \sqrt \Delta }{2}\zeta } \right)}}} \right)} \right\} \hfill \\ \, \mp \frac{1}{4}\frac{{\ell A_{1} }}{\alpha }\left\{ {\frac{\beta }{{2\left( {1 - \delta } \right)}} + \frac{\beta \sqrt \Lambda }{{2\left( {1 - \delta } \right)}}\left( {\frac{{\delta_{1} \exp \left( {\frac{\sqrt \Delta }{2}\zeta } \right) + \delta_{2} \exp \left( {\frac{ - \sqrt \Delta }{2}\zeta } \right)}}{{\delta_{1} \exp \left( {\frac{\sqrt \Delta }{2}\zeta } \right) - \delta_{2} \exp \left( {\frac{ - \sqrt \Delta }{2}\zeta } \right)}}} \right)} \right\}^{2} \hfill \\ \end{gathered}$$
$$\begin{gathered} g_{1,1} \left( t \right) = B_{0} - A_{1} \left\{ {\frac{\beta }{{2\left( {1 - \delta } \right)}} + \frac{\beta \sqrt \Lambda }{{2\left( {1 - \delta } \right)}}\left( {\frac{{\delta_{1} \exp \left( {\frac{\sqrt \Delta }{2}\zeta } \right) + \delta_{2} \exp \left( {\frac{ - \sqrt \Delta }{2}\zeta } \right)}}{{\delta_{1} \exp \left( {\frac{\sqrt \Delta }{2}\zeta } \right) - \delta_{2} \exp \left( {\frac{ - \sqrt \Delta }{2}\zeta } \right)}}} \right)} \right\} \hfill \\ \, + \frac{1}{4}\frac{{\ell A_{1} }}{\alpha }\left\{ {\frac{\beta }{{2\left( {1 - \delta } \right)}} + \frac{\beta \sqrt \Lambda }{{2\left( {1 - \delta } \right)}}\left( {\frac{{\delta_{1} \exp \left( {\frac{\sqrt \Delta }{2}\zeta } \right) + \delta_{2} \exp \left( {\frac{ - \sqrt \Delta }{2}\zeta } \right)}}{{\delta_{1} \exp \left( {\frac{\sqrt \Delta }{2}\zeta } \right) - \delta_{2} \exp \left( {\frac{ - \sqrt \Delta }{2}\zeta } \right)}}} \right)} \right\}^{2} \hfill \\ \end{gathered}$$
(24)
$$\begin{gathered} h_{1,1} \left( t \right) = \mp \frac{{4c\left( { - \delta \mp \frac{\ell \beta }{{2\alpha }} + 1} \right)\alpha }}{{\ell \beta_{3} }} \hfill \\ \, - \frac{{2c\left( {\delta - 1} \right)}}{{\beta_{3} }}\left\{ {\frac{\beta }{{2\left( {1 - \delta } \right)}} + \frac{\beta \sqrt \Lambda }{{2\left( {1 - \delta } \right)}}\left( {\frac{{\delta_{1} \exp \left( {\frac{\sqrt \Delta }{2}\zeta } \right) + \delta_{2} \exp \left( {\frac{ - \sqrt \Delta }{2}\zeta } \right)}}{{\delta_{1} \exp \left( {\frac{\sqrt \Delta }{2}\zeta } \right) - \delta_{2} \exp \left( {\frac{ - \sqrt \Delta }{2}\zeta } \right)}}} \right)} \right\} \hfill \\ \end{gathered}$$
Fig. 1
figure 1

Analytical solution of Eqs. (24) with different \(\nu\) at \(\alpha = 0.5\), \(\delta_{2} = 0.02\), \(c = 0.1\),\(\delta = \delta_{1} = \beta = A_{1} = \beta_{3} = B_{0} = 2\)

Full size image

Case (2) If \(\Lambda = \beta^{2} + 4\alpha - 4\alpha \delta < 0\) and \(\beta \ne 0\) (Fig. 2), then

$$\begin{gathered} f_{1,2} \left( t \right) = \mp \frac{{A_{1} \alpha }}{\ell } + A_{1} \left\{ {\frac{\beta }{{2\left( {1 - \delta } \right)}} + \frac{{\beta \sqrt { - \Lambda } }}{{2\left( {1 - \delta } \right)}}\left( {\frac{{i\delta_{1} \cos \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right) - \delta_{2} \sin \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right)}}{{i\delta_{1} \sin \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right) + \delta_{2} \cos \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right)}}} \right)} \right\} \hfill \\ \, \mp \frac{1}{4}\frac{{\ell A_{1} }}{\alpha }\left\{ {\frac{\beta }{{2\left( {1 - \delta } \right)}} + \frac{{\beta \sqrt { - \Lambda } }}{{2\left( {1 - \delta } \right)}}\left( {\frac{{i\delta_{1} \cos \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right) - \delta_{2} \sin \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right)}}{{i\delta_{1} \sin \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right) + \delta_{2} \cos \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right)}}} \right)} \right\}^{2} \hfill \\ \end{gathered}$$
$$\begin{gathered} g_{1,2} \left( t \right) = B_{0} - A_{1} \left\{ {\frac{\beta }{{2\left( {1 - \delta } \right)}} + \frac{{\beta \sqrt { - \Lambda } }}{{2\left( {1 - \delta } \right)}}\left( {\frac{{i\delta_{1} \cos \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right) - \delta_{2} \sin \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right)}}{{i\delta_{1} \sin \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right) + \delta_{2} \cos \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right)}}} \right)} \right\} \hfill \\ \, + \frac{1}{4}\frac{{\ell A_{1} }}{\alpha }\left\{ {\frac{\beta }{{2\left( {1 - \delta } \right)}} + \frac{{\beta \sqrt { - \Lambda } }}{{2\left( {1 - \delta } \right)}}\left( {\frac{{i\delta_{1} \cos \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right) - \delta_{2} \sin \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right)}}{{i\delta_{1} \sin \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right) + \delta_{2} \cos \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right)}}} \right)} \right\}^{2} \hfill \\ \end{gathered}$$
(25)
$$h_{1,2} \left( t \right) = \mp \frac{{4c\left( { - \delta \mp \frac{\ell \beta }{{2\alpha }} + 1} \right)\alpha }}{{\ell \beta_{3} }} - \frac{{2c\left( {\delta - 1} \right)}}{{\beta_{3} }}\left\{ \begin{gathered} \frac{\beta }{{2\left( {1 - \delta } \right)}} + \hfill \\ \frac{{\beta \sqrt { - \Lambda } }}{{2\left( {1 - \delta } \right)}}\left( {\frac{{i\delta_{1} \cos \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right) - \delta_{2} \sin \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right)}}{{i\delta_{1} \sin \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right) + \delta_{2} \cos \left( {\frac{{\sqrt { - \Lambda } }}{2}\zeta } \right)}}} \right) \hfill \\ \end{gathered} \right\}$$
Fig. 2 
figure 2

Analytical solution of Eqs. (25) with different \(\nu\) at \(\alpha = 3\), \(\delta_{2} = 0.8\), \(c = 0.1\),\(\delta = \delta_{1} = \beta = A_{1} = \beta_{3} = B_{0} = 2\)

Full size image

Case (3) If \(\Lambda = \alpha \left( {1 - \delta } \right) \ge 0\) and \(\beta = 0\) (Fig. 3), then

$$\begin{gathered} f_{1,3} \left( t \right) = \mp \frac{{A_{1} \alpha }}{\hbar } + A_{1} \left\{ {\frac{\sqrt \Lambda }{{\left( {1 - \delta } \right)}}\left( {\frac{{\delta_{1} \cos \left( {\sqrt \Lambda \,\zeta } \right) + \delta_{2} \sin \left( {\sqrt {\Lambda \,} \zeta } \right)}}{{\delta_{1} \sin \left( {\sqrt \Lambda \,\zeta } \right) - \delta_{2} \cos \left( {\sqrt \Lambda \,\zeta } \right)}}} \right)} \right\} \hfill \\ \, \mp \frac{1}{4}\frac{{\hbar A_{1} }}{\alpha }\left\{ {\frac{\sqrt \Lambda }{{\left( {1 - \delta } \right)}}\left( {\frac{{\delta_{1} \cos \left( {\sqrt \Lambda \,\zeta } \right) + \delta_{2} \sin \left( {\sqrt {\Lambda \,} \zeta } \right)}}{{\delta_{1} \sin \left( {\sqrt \Lambda \,\zeta } \right) - \delta_{2} \cos \left( {\sqrt \Lambda \,\zeta } \right)}}} \right)} \right\}^{2} \hfill \\ \end{gathered}$$
$$\begin{gathered} g_{1,3} \left( t \right) = B_{0} - A_{1} \left\{ {\frac{\sqrt \Lambda }{{\left( {1 - \delta } \right)}}\left( {\frac{{\delta_{1} \cos \left( {\sqrt \Lambda \,\zeta } \right) + \delta_{2} \sin \left( {\sqrt {\Lambda \,} \zeta } \right)}}{{\delta_{1} \sin \left( {\sqrt \Lambda \,\zeta } \right) - \delta_{2} \cos \left( {\sqrt \Lambda \,\zeta } \right)}}} \right)} \right\} \hfill \\ \, + \frac{1}{4}\frac{{\hbar A_{1} }}{\alpha }\left\{ {\frac{\sqrt \Lambda }{{\left( {1 - \delta } \right)}}\left( {\frac{{\delta_{1} \cos \left( {\sqrt \Lambda \,\zeta } \right) + \delta_{2} \sin \left( {\sqrt {\Lambda \,} \zeta } \right)}}{{\delta_{1} \sin \left( {\sqrt \Lambda \,\zeta } \right) - \delta_{2} \cos \left( {\sqrt \Lambda \,\zeta } \right)}}} \right)} \right\}^{2} \hfill \\ \end{gathered}$$
(26)
$$h_{1,3} \left( t \right) = \mp \frac{{4c\left( { - \delta + 1} \right)\alpha }}{{\hbar \beta_{3} }} - \frac{{2c\left( {\delta - 1} \right)}}{{\beta_{3} }}\left\{ {\frac{\sqrt \Lambda }{{\left( {1 - \delta } \right)}}\left( {\frac{{\delta_{1} \cos \left( {\sqrt \Lambda \,\zeta } \right) + \delta_{2} \sin \left( {\sqrt {\Lambda \,} \zeta } \right)}}{{\delta_{1} \sin \left( {\sqrt \Lambda \,\zeta } \right) - \delta_{2} \cos \left( {\sqrt \Lambda \,\zeta } \right)}}} \right)} \right\}$$

where, \(\hbar = \sqrt { - 4\alpha \delta + 4\alpha }\).

Fig. 3
figure 3

Analytical solution of Eqs. (26) with different \(\nu\) at \(\alpha = 3\), \(\delta_{2} = 0.8\), \(c = \delta = \delta_{1} = 0.02\),\(A_{1} = \beta_{3} = B_{0} = 2\)

Full size image

Case (4) If \(\Lambda = \alpha \left( {1 - \delta } \right) < 0\) and \(\beta = 0\), then

$$\begin{gathered} f_{1,4} \left( t \right) = \mp \frac{{A_{1} \alpha }}{\hbar } + A_{1} \left\{ {\frac{{\sqrt { - \Lambda } }}{{\left( {1 - \delta } \right)}}\left( {\frac{{i\delta_{1} \cosh \left( {\sqrt { - \Lambda \,} \zeta } \right) - \delta_{2} \sinh \left( {\sqrt { - \Lambda \,} \zeta } \right)}}{{i\delta_{1} \sinh \left( {\sqrt { - \Lambda \,} \zeta } \right) - \delta_{2} \cosh \left( {\sqrt { - \Lambda \,} \zeta } \right)}}} \right)} \right\} \hfill \\ \, \mp \frac{1}{4}\frac{{\hbar A_{1} }}{\alpha }\left\{ {\frac{{\sqrt { - \Lambda } }}{{\left( {1 - \delta } \right)}}\left( {\frac{{i\delta_{1} \cosh \left( {\sqrt { - \Lambda \,} \zeta } \right) - \delta_{2} \sinh \left( {\sqrt { - \Lambda \,} \zeta } \right)}}{{i\delta_{1} \sinh \left( {\sqrt { - \Lambda \,} \zeta } \right) - \delta_{2} \cosh \left( {\sqrt { - \Lambda \,} \zeta } \right)}}} \right)} \right\}^{2} \hfill \\ \end{gathered}$$
$$\begin{gathered} g_{1,4} \left( t \right) = B_{0} - A_{1} \left\{ {\frac{{\sqrt { - \Lambda } }}{{\left( {1 - \delta } \right)}}\left( {\frac{{i\delta_{1} \cosh \left( {\sqrt { - \Lambda \,} \zeta } \right) - \delta_{2} \sinh \left( {\sqrt { - \Lambda \,} \zeta } \right)}}{{i\delta_{1} \sinh \left( {\sqrt { - \Lambda \,} \zeta } \right) - \delta_{2} \cosh \left( {\sqrt { - \Lambda \,} \zeta } \right)}}} \right)} \right\} \hfill \\ \, + \frac{1}{4}\frac{{\hbar A_{1} }}{\alpha }\left\{ {\frac{{\sqrt { - \Lambda } }}{{\left( {1 - \delta } \right)}}\left( {\frac{{i\delta_{1} \cosh \left( {\sqrt { - \Lambda \,} \zeta } \right) - \delta_{2} \sinh \left( {\sqrt { - \Lambda \,} \zeta } \right)}}{{i\delta_{1} \sinh \left( {\sqrt { - \Lambda \,} \zeta } \right) - \delta_{2} \cosh \left( {\sqrt { - \Lambda \,} \zeta } \right)}}} \right)} \right\}^{2} \hfill \\ \end{gathered}$$
(27)
$$h_{1,4} \left( t \right) = \mp \frac{{4c\left( { - \delta + 1} \right)\alpha }}{{\hbar \beta_{3} }} - \frac{{2c\left( {\delta - 1} \right)}}{{\beta_{3} }}\left\{ {\frac{{\sqrt { - \Lambda } }}{{\left( {1 - \delta } \right)}}\left( {\frac{{i\delta_{1} \cosh \left( {\sqrt { - \Lambda \,} \zeta } \right) - \delta_{2} \sinh \left( {\sqrt { - \Lambda \,} \zeta } \right)}}{{i\delta_{1} \sinh \left( {\sqrt { - \Lambda \,} \zeta } \right) - \delta_{2} \cosh \left( {\sqrt { - \Lambda \,} \zeta } \right)}}} \right)} \right\}$$

where \(\zeta = c\frac{{t^{\nu } }}{\nu }\) and \(\alpha ,\beta ,\delta ,\delta_{1} ,\delta_{2}\) are real parameters. For simplicity, the sets [2,3,4,5,6,7,8,9] should be omitted here (Fig. 4).

Fig. 4 
figure 4

Analytical solution of Eqs. (26) with different \(\nu\) at \(\alpha = 3\), \(\delta_{2} = 0.8\),\(c = 0.1\), \(A_{1} = \beta_{3} = B_{0} = \delta = \delta_{1} = 2\)

Full size image

4 Conclusions

In conclusion, we introduced a novel fractional model in biology, namely HIV-1 infection of CD4 + T cells. Here, an extended \(\left( {{{\Upsilon^{\prime}} \mathord{\left/ {\vphantom {{\Upsilon^{\prime}} \Upsilon }} \right. \kern-0pt} \Upsilon }} \right)\) method has been studied for constructing new exact traveling wave solutions such as exponential function, trigonometric function and hyperbolic function which are shown graphically in 2D plots to show the dynamical behavior of the proposed model for a different fractal order to see how unique our solutions are, as they are all fresh and different. Therefore, we came to the conclusion that the analytical findings presented here are both useful and fascinating. We want to propose a simple and trustworthy way to the research that will be conducted for the future of human beings.