1 Introduction

Till today, the Aedes aegypti, Aedes albopictus, and yellow fever mosquitoes still present a danger as they are the main vectors of some viral diseases to humans, such as the Chikungunya virus which is transmitted to the human population through the bites of infected mosquitoes. Therefore, it is still imperative to research how to study this disease to overcome it before it spreads. A remarkable effort in recent years has been undertaken to model this Chikungunya virus (CHIKV) transmitted to the human population through the Egyptian mosquito [1,2,3]. Among the symptoms of CHIKV, we cite headaches, fever, muscle pain, and joint disorders.

In this paper, we propose to model and numerically study the problem of Chikungunya virus propagation, which may help in controlling the spread of the disease if the community is exposed to it in the future. The approved modeling is based on the differential equations system, it is mainly found in applications in biology [4] and in engineering sciences [5]. Taking into account the difficulty of solving these equations by analytical methods, a certain number of numerical methods were used, which worked well [6, 7]. For example, stability analyses and semi-analytical solutions for a Zika virus dynamics model, SVEIR epidemic model for the measles transmission and SIR epidemic model were studied, respectively, in [8,9,10]. Without forgetting, of course, these methods are the basis of the numerical simulations. Despite these convincing successes, numerical methods only give solutions in the form of curves or tables. For example, the researcher in biology may sometimes not need equations per se, but he may need non-implicit solutions to cope with studies of status according to different and shifting parameters. Therefore, many scientific researchers have been drawn to researching semi-analytical methods because exact and closed-form solutions of equations can be analytically difficult to obtain, or a method exists, yet it is complex. In literature, many numerical methods have been proposed such as the Adomian decomposition method [11], the variable frequency method [12, 13] and the symmetric perturbation method [14]. Here, we propose the differential transformation method (DTM) which was formulated by Zhou (1986) in [15] to solve differential equations in electric fields. Since the DTM has been used to solve a variety of equations. Relevant examples include differential and integral systems, DDEs, and differential-difference equations. In this work, we extend the use of DTM to more generalized nonlinear dynamical systems and improve the efficiency of this method by a multi-step procedure in time, which can widen the maximum time span to observe and study well the behavior of the solutions obtained.

In this article, we provide an easy method that does not require much prior programming knowledge, unlike many researchers who have used the RK4 method for the numerical treatment of this topic as a ready-made method (see, as an example [19]). We will show that there is a number of positive results that can be achieved using this proposed technique, such as its greater flexibility and ease in overcoming nonlinearity while respecting the conditions and characteristics of the problem. The objective is to propose an efficient modified differential method for solving some more generalized dynamical systems and, at the same time, with high accuracy and efficiency of solutions obtained. Through the numerical examples, the DTM demonstrates the advantages by the comparison of its results with another numerical method.

We will consider the nonlinear differential compact system

$$\begin{aligned} \dot{{\textbf {v}}}(t)={\varvec{\varphi }}\big (t,{\textbf {v}}(t)\big ), \quad ~~ ~~~ t\in [0,T]. \end{aligned}$$
(1)

The \(\dot{{\textbf {v}}}\) is the time derivative of the vector function \({\textbf {v}}=(\mathrm v_1,\mathrm v_2,\ldots ,\mathrm v_n)\), \(n\in \mathbb {N}^{*}\), and \({\varvec{\varphi }}\) is a vector quadratic form ( special nonlinear vector function having only the square of a variable and/or the product of two variables). In [7], the authors consider (1) as a nonlinear differential equations (\(n=1\)) with a more time delays in a polynomial type.

We define the differential transform of the ith derivative of a vector function \({\textbf {v}}\) in a multi-dimensional case by:

$$\begin{aligned} {\textbf {V}}_k(i)=\frac{1}{i!}\bigg [\frac{d^i{\textbf {v}}(t)}{d t^i}\bigg ]_{t=t_k}. \end{aligned}$$
(2)

where \({\textbf {v}}\) is the original vector function and \({\textbf {V}}_k\) is the overall transformed function. As in scientific literature, differential inverse transform of \({\textbf {V}}_k\) is defined as

$$\begin{aligned} {\textbf {v}}(t)=\sum _{i=0}^{\infty }{\textbf {V}}_k(i)(t-t_k)^i. \end{aligned}$$
(3)

The basic concept of differential transform is derived from Taylor series expansion such we notice that by substituting Eq. (2) in Eq. (3) and with taking into account the particular case when the Taylor series expansion is originally based we get

$$\begin{aligned} {\textbf {v}}(t)=\sum _{i=0}^{\infty }\frac{(t-t_k)^i}{i!}\bigg [\frac{d^i{\textbf {v}}(t)}{d t^i}\bigg ]_{t=t_k}. \end{aligned}$$
(4)

In the Table 1, we can deduce from Eqs. (2) and (3) some necessary results already proved in [16, 17].

Table 1 Fundamental DT operations for the one-variable equations
Full size table

2 CHIKV model with latency

CHIKV is a viral illness transmitted to human population by the bites of an infected mosquitoes of the genus Aedes or yellow fever mosquitoes that, in turn, become infected by feeding on infected persons. That is, a person-to-mosquito-to-person cycle. Several mathematical models have been presented in the CHIKV infection literature to describe the evolution of disease transmission in a population over time or to describe the population growth of chikungunya virus within-host at each stage, see for example [1, 3, 18]. Modelling, mathematical analysis and optimal control for this virus have been studied by El Hajji et al. in [19, 20].

In this section, we study the following CHIKV model with CHIKV-to-cell transmission:

$$\begin{aligned} \left\{ \begin{array}{llll} \dot{S}(t) &{} =\mu -a S(t)- \alpha S(t) V(t) , \\ \dot{L}(t) &{} =(1-\rho )\alpha S(t) V(t) -(h+\nu )L(t), \\ \dot{I}(t) &{} =\rho \alpha S(t) V(t) +\nu L(t)-p I(t), \\ \dot{V}(t) &{} =m I(t)-rV(t)-q B(t) V(t), \\ \dot{B}(t) &{} =\eta +c B(t) V(t) -d B(t). \end{array} \right. \end{aligned}$$
(5)

We assume that all uppercase Latin letters (S, L, I, V, and B) are interpreted as follows:

  • S Concentrations of uninfected monocytes.

  • L Concentrations of latently infected monocytes.

  • I Concentrations of actively infected monocytes.

  • V Concentrations of CHIKV-particles.

  • B Concentrations of B cells.

The uninfected susceptible host cells are created at \(\mu \)-rate and die at aS-rate. The uninfected susceptible host cells are attacked by the CHIKV−particles at \(\alpha SV\)-rate, where the constant \(\alpha \) is the contact rate between susceptible host cells and free CHIKV−particles. The parameter \(\nu \) represents the transmission rate from a latent to active infected host cell. The constant \(0<\rho <1\) represents a part of infected cells that are assumed to be actively infected cells, so that \(1-\rho \) is a part of infected cells that are assumed to be latently infected cells, where the sum is 1. The multiplication of CHIKV−particles is assumed at a constant m-rate. The CHIKV−particles are attacked by the B cells at qBV-rate. The B cells are created at \(\eta \)-rate, proliferated at cBV-rate. Infected host cells, free CHIKV−particles and B cells die at pI-rate, rV-rate and dB-rate, respectively.

2.1 Basic properties

Lemma 1

There exist constants \(M_{1},M_{2},M_{3}>0,\) such that the following compact subset is positively invariant for system (5)

$$\begin{aligned} \varGamma & = \Bigg\{ (S,~L,~I,~V,~B)\in \mathbb {R}_{+}^{5} : 0 \le S, L, I \le M_{1}, \\ & \quad 0 \le V \le M_{2}, 0 \le B \le M_{3} \Bigg\} . \end{aligned}$$

Proof

We have

$$\begin{aligned} \dot{S}\mid _{S=0}&=\mu>0,\\ \dot{L}\mid _{L=0}&=(1-\rho )\alpha S V \ge 0, \quad \forall \; S, V \ge 0,\\ \dot{I}\mid _{I=0}&=\rho \alpha S V +\nu L \ge 0, \quad \forall \; S, L, V \ge 0,\\ \dot{V}\mid _{V=0}&=m I \ge 0, \quad \forall \; I\ge 0,\\ \dot{B}\mid _{B=0}&=\eta >0. \end{aligned}$$

Then, \(\mathbb {R}_{+}^{5}\) is positively invariant for the system (5). We let \(H_{1}(t)=S(t)+L(t)+I(t)\), and \(H_{2}(t)=V(t)+\dfrac{q}{c}B(t)\), then we obtain

$$\begin{aligned} \dot{H}_{1}(t) = \mu -a S(t)-h L(t)-p I(t) \le \mu -\sigma _{1} (S(t)+ L(t)+ I(t)) = \mu -\sigma _{1} H_{1}(t), \end{aligned}$$

where, \(\sigma _{1}=\min \{a, h, p\}\). Hence \(H_{1}(t)\le M_{1}\) if \(H_{1}(0)\le M_{1}\) where \(M_{1}=\dfrac{\mu }{\sigma _{1}}\). Hence, \(0 \le S(t), L(t), I(t) \le M_{1}\) if \(0\le S(0)+L(0)+I(0)\le M_{1}\). Moreover, we have

$$\begin{aligned} \dot{H}_{2}(t)& = m I(t)-r V(t)+\dfrac{q}{c }\eta -\dfrac{qd}{c}B(t)\\\le & {} m M_{1}+\dfrac{q}{c }\eta -\sigma _{2}\bigg (V(t)+\dfrac{q}{c }B(t)\bigg )\\& = m M_{1} +\dfrac{q}{c }\eta -\sigma _{2} H_{2}(t), \end{aligned}$$

where, \(\sigma _{2}=min\{r,d\}\). Hence \(H_{2}(t)\le M_{2}\) if \(H_{2}(0)\le M_{2}\) where \(M_{2}=\dfrac{m M_{1} +\dfrac{q}{c }\eta }{\sigma _{2}}\). Thus, \(0 \le V(t)\le M_{2}\) and \(B(t)\le M_{3}\) if \(0 \le V(0)+\dfrac{q}{c }B(0)\le M_{2}\) where \(M_{3}=\dfrac{c M_{2}}{q}\). \(\square \)

2.2 Steady states

Lemma 2

Let \(S_{0}=\dfrac{\mu }{a}\) and \(B_{0}=\dfrac{\eta }{d}\), then for the system (5) there exists a strictly positive threshold parameter \(\mathscr {R}_{0}\) given by \(\mathscr {R}_{0}=\dfrac{(\rho h+\nu )}{p(h+\nu )} \dfrac{m\alpha S_0}{r+ qB_0} \) such that

  • if \(\mathscr {R}_{0} \le 1,\) there exists only the disease-free equilibrium \(E_{0}\),

  • if \(\mathscr {R}_{0} >1,\) there exist two steady states: the disease-free equilibrium \(E_{0}\) and the endemic equilibrium \(E_{1}\).

Proof

Let (S,  L,  I,  V,  B) be any steady state satisfying

$$\begin{aligned} \begin{array}{llll} 0 &{} =\mu -a S- \alpha S V , \\ 0 &{} =(1-\rho )\alpha S V -(h+\nu )L, \\ 0 &{} =\rho \alpha S V +\nu L-p I, \\ 0 &{} =m I-rV-q B V, \\ 0 &{} =\eta +c B V -d B, \end{array} \end{aligned}$$
(6)

Solving Eqs. (6) there exists a CHIKV−free steady state \(E_{0}=(S_{0},~0,~0,~0,~B_{0})\).

Now, let \(E_1=(S,L,I,V,B)\) be any steady state, then from Eq. (6) we have

$$\begin{aligned} \begin{array}{llll} 0&{}=\mu -\left[ a + \alpha V \right] \dfrac{p(h+\nu )}{\alpha (\nu +\rho h)} \left[ \dfrac{r}{m}+\dfrac{q\eta }{m(d-cV)}\right] \\ S&{}=\dfrac{(h+\nu )L}{(1-\rho ) \alpha V }=\dfrac{p(h+\nu )}{\alpha (\nu +\rho h)} \left[ \dfrac{r}{m}+\dfrac{q\eta }{m(d-cV)}\right] \\ L &{} =\dfrac{p(1-\rho )}{\nu +\rho h}\left[ \dfrac{rV}{m}+\dfrac{q\eta V}{m(d-cV)}\right] \\ I &{} =\dfrac{V(r+qB)}{m}=\dfrac{rV}{m}+\dfrac{q\eta V}{m(d-c V)}, \\ B &{} =\dfrac{\eta }{d-c V}. \end{array} \end{aligned}$$
(7)

Define the function

$$\begin{aligned} g(V)=\mu -\left[ a + \alpha V \right] \dfrac{p(h+\nu )}{\alpha (\nu +\rho h)} \left[ \dfrac{r}{m}+\dfrac{q\eta }{m(d-cV)}\right] . \end{aligned}$$

Then we obatin

$$\begin{aligned} \lim _{V\rightarrow 0^+} g(V)& = \lim _{V\rightarrow 0^+} \left( \mu -\left[ a + \alpha V \right] \dfrac{p(h+\nu )}{\alpha (\nu +\rho h)} \left[ \dfrac{r}{m}+\dfrac{q\eta }{m(d-cV)}\right] \right) \\& = \mu -\dfrac{a p(h+\nu )}{\alpha md(\nu +\rho h)} (rd+q\eta ) \end{aligned}$$

Since \(a = \displaystyle \dfrac{\mu }{S_0}\) and \(B_0 = \displaystyle \dfrac{\eta }{d}\), we obtain

$$\begin{aligned} \lim _{V\rightarrow 0^+} g(V)& = \mu -\dfrac{a p(h+\nu )}{\alpha m (\nu +\rho h)} (r +q B_0)\\& = \mu \left( 1-\dfrac{1}{\mathscr {R}_0}\right) \\& = \dfrac{\mu }{\mathscr {R}_{0}}\left( \mathscr {R}_{0}-1\right)>0\quad \text {if}\quad \mathscr {R}_{0} > 1. \end{aligned}$$

Now, we have

$$\begin{aligned} \lim _{V \rightarrow \left( \dfrac{d}{c}\right) ^-}g(V)& = \lim _{y\rightarrow +\infty }\left[ \mu -\left( a + \dfrac{\alpha d}{c} \right) \dfrac{p(h+\nu )}{\alpha (\nu +\rho h)} \left( \dfrac{r}{m}+\dfrac{q\eta }{m}y\right) \right] = -\infty . \end{aligned}$$

Furthermore, we have

$$\begin{aligned} g'(V)& = -\alpha \dfrac{p(h+\nu )}{\alpha (\nu +\rho h)} \left[ \dfrac{r}{m}+\dfrac{q\eta }{m(d-cV)}\right] -\left[ a + \alpha V \right] \dfrac{p(h+\nu )}{\alpha (\nu +\rho h)} \left[ \dfrac{c q\eta }{m(d-cV)^2}\right] . \end{aligned}$$

Therefore \(g'(V) < 0\) for all \(V\in \left( 0,\dfrac{d}{c}\right) \). Finally, we conclude that there exists a unique \(V_{1}\in (0,\dfrac{d}{c})\) such that \(g(V_{1})=0.\) If \(\mathscr {R}_{0}>1,\) the system (5) has infected steady state \(E_{1}=(S_{1},L_1,I_{1},V_{1},B_{1}),\) where

$$\begin{aligned} S_{1}&=\dfrac{p(h+\nu )}{\alpha (\nu +\rho h)} \left[ \dfrac{r}{m}+\dfrac{q\eta }{m(d-cV_{1})}\right] \\ L_{1}&=\dfrac{p(1-\rho )}{\nu +\rho h}\left[ \dfrac{rV_{1}}{m}+\dfrac{q\eta V_{1}}{m(d-cV_{1})}\right] \\ I_{1}&=\dfrac{rV_{1}}{m}+\dfrac{q\eta V_{1}}{m(d-c V_{1})}, \\ B_{1}&=\dfrac{\eta }{d-c V_{1}}. \end{aligned}$$

\(\square \)

2.3 Global properties

Let the positive definite function \(G(z)=z-1-\ln z\).

Theorem 1

If \(\mathscr {R}_{0}\le 1,\) then the disease-free equilibrium \(E_{0}\) is globally asymptotically stable (G. A. S.).

Proof

As in [8], let the Lyapunov function given by

$$\begin{aligned} F_{0}(S, L, I, V, B)=S_{0}G\bigg (\dfrac{S}{S_{0}}\bigg )+\dfrac{\nu }{\rho h+\nu } L +\dfrac{h+\nu }{\rho h+\nu }I+\dfrac{\alpha S_{0}}{r+qB_{0}}V+\dfrac{q\alpha S_{0}}{c (r+qB_{0})}B_{0}G\bigg (\dfrac{B}{B_{0}}\bigg ). \end{aligned}$$

Calculating \(\dot{F}_{0} \) along system (5) we obtain

$$\begin{aligned} \dot{F}_0& = \dfrac{(S-S_0)}{S}\bigg (\mu -a S- \alpha S V\bigg )+\dfrac{\nu }{\rho h+\nu } \bigg ((1-\rho )\alpha S V -(h+\nu )L\bigg )\\{} & {} + \dfrac{h+\nu }{\rho h+\nu } \bigg (\rho \alpha S V +\nu L-p I\bigg ) + \dfrac{\alpha S_{0}}{r+qB_{0}}\bigg (m I-rV-q B V\bigg )\\{} & {} + \dfrac{q\alpha S_{0}}{c (r+qB_{0})} \dfrac{(B-B_0)}{B} \bigg (\eta +c B V -d B\bigg )\\& = -a\dfrac{(S-S_0)^2}{S} - \alpha S V + \alpha S_0V +\dfrac{\nu (1-\rho )}{\rho h+\nu }\alpha S V -\dfrac{\nu (h+\nu )}{\rho h+\nu } L \\{} & {} + \dfrac{\rho (h+\nu )}{\rho h+\nu } \alpha S V+ \dfrac{\nu (h+\nu )}{\rho h+\nu } L - \dfrac{p(h+\nu )}{\rho h+\nu } I + \dfrac{m \alpha S_{0}}{r+qB_{0}}I - \dfrac{r+q B}{r+qB_{0}} \alpha S_{0} V \\{} & {} - \dfrac{qd\alpha S_{0}}{c (r+qB_{0})} \dfrac{(B-B_0)^2}{B} + \dfrac{qcB}{c (r+qB_{0})} \dfrac{(B-B_0)}{B} \alpha S_{0} V\\& = -a\dfrac{(S-S_0)^2}{S} - \dfrac{p(h+\nu )}{\rho h+\nu } I + \dfrac{m \alpha S_{0}}{r+qB_{0}}I - \dfrac{qd\alpha S_{0}}{c (r+qB_{0})} \dfrac{(B-B_0)^2}{B}\\{} & {} + \alpha S_0V - \dfrac{r+q B}{r+qB_{0}} \alpha S_{0} V + \dfrac{q}{(r+qB_{0})} (B-B_0) \alpha S_{0} V\\& = -a\dfrac{(S-S_0)^2}{S} +\dfrac{p(h+\nu )}{\rho h+\nu } I\left( \dfrac{m \alpha S_{0}}{r+qB_{0}}\dfrac{\rho h+\nu }{p(h+\nu )} -1\right) - \dfrac{qd\alpha S_{0}}{c (r+qB_{0})} \dfrac{(B-B_0)^2}{B} \\& = -a\dfrac{(S-S_0)^2}{S} +\dfrac{p(h+\nu )}{\rho h+\nu } I({\mathscr {R}}_0 - 1)-\dfrac{qd\alpha S_0}{r (r+qB_0)}\dfrac{(B-B_0)^2}{B}. \end{aligned}$$

If \(\mathscr {R}_{0} \le 1\) we get \(\dot{F}_{0}\le 0\) for all \(S, L, I, V, B>0\). Let \(D_{0}=\{(S,~L,~ I,~ V,~ B): \dot{F}_{0}=0\}\) , \(D_{0}=\{E_{0}\}\). \(E_{0}\) is then G. A. S. \(\square \)

Theorem 2

If \(\mathscr {R}_{0} > 1,\) then the endemic equilibrium \(E_{1}\) is G. A. S.

Proof

Let the Lyapunov function given by

$$\begin{aligned} \begin{array}{llll} F_{1}(S, L, I, V, B) &{}=&{} S_{1}G\left( \dfrac{S}{S_1}\right) +\dfrac{\nu }{\rho h+\nu }L_{1}G\left( \dfrac{L}{L_1}\right) +\dfrac{h+\nu }{\rho h+\nu } I_1G\left( \dfrac{I}{I_1}\right) \\ &{}&{}+\dfrac{\alpha S_1 V_1}{m I_1} V_1 G\left( \dfrac{V}{V_1}\right) +\dfrac{q \alpha S_1 V_1}{c m I_1} B_1G\left( \dfrac{B}{B_1}\right) . \end{array} \end{aligned}$$

The derivative of \(F_1\) with respect to the time t along the system (5) is given by

$$\begin{aligned} \begin{array}{llll} \dot{F}_{1} &{}=&{}\left( 1-\dfrac{S_1}{S}\right) \left( \mu -a S- \alpha S V \right) + \dfrac{\nu }{\rho h+\nu } \left( 1-\dfrac{L_1}{L}\right) \left( (1-\rho )\alpha S V -(h+\nu )L\right) \\ &{}&{} + \dfrac{h+\nu }{\rho h+\nu } \left( 1-\dfrac{I_1}{I}\right) \left( \rho \alpha S V +\nu L-p I\right) + \dfrac{\alpha S_1 V_1}{m I_1} \left( 1-\dfrac{V_1}{V}\right) \left( m I-rV-q B V\right) \\ &{}&{} + \dfrac{q \alpha S_1 V_1}{c m I_1} \left( 1-\dfrac{B_1}{B}\right) \left( \eta +c B V -d B\right) . \end{array} \end{aligned}$$

Using the fact that

$$\begin{aligned} \mu& = a S_1 + \alpha S_1 V_1 , \\ (1-\rho )\alpha S_1 V_1& = (h+\nu )L_1, \\ \rho \alpha S_1 V_1 +\nu L_1& = p I_1, \\ m I_1& = rV_1 + q B_1 V_1, \\ \eta +c B_1 V_1& = d B_1 , \\ \nu L_1& = \dfrac{\nu (1-\rho )}{(h+\nu )} \alpha S_1 V_1 ,\\ p I_1& = \dfrac{\rho h+\nu }{(h+\nu )} \alpha S_1 V_1, \end{aligned}$$

the derivative of \(F_1\) is simplified to

$$\begin{aligned} \dot{F}_{1}& = \left( 1-\dfrac{S_1}{S}\right) \left( a S_1 + \alpha S_1 V_1 -a S- \alpha S V \right) + \dfrac{\nu }{\rho h+\nu } \left( 1-\dfrac{L_1}{L}\right) \left( (1-\rho )\alpha S V -(h+\nu )L\right) \\{} & {} + \dfrac{h+\nu }{\rho h+\nu } \left( 1-\dfrac{I_1}{I}\right) \left( \rho \alpha S V +\nu L-p I\right) + \dfrac{\alpha S_1 V_1}{m I_1} \left( 1-\dfrac{V_1}{V}\right) \left( m I-rV-q B V\right) \\{} & {} + \dfrac{q \alpha S_1 V_1}{c m I_1} \left( 1-\dfrac{B_1}{B}\right) \left( d B_1 - c B_1 V_1 + c B V -d B\right) \\& = -a \dfrac{(S-S_1)^2}{S} + \alpha S_1 V_1 - \alpha S V - \dfrac{S_1}{S} \alpha S_1 V_1 + \alpha S_1 V \\{} & {} + \dfrac{\nu (1-\rho )}{\rho h+\nu } \alpha S V -\dfrac{\nu (h+\nu )}{\rho h+\nu }L - \dfrac{\nu (1-\rho )}{\rho h+\nu }\dfrac{L_1}{L} \alpha S V +\dfrac{\nu (h+\nu )}{\rho h+\nu }L_1\\{} & {} + \dfrac{\rho (h+\nu )}{\rho h+\nu } \alpha S V +\dfrac{\nu (h+\nu )}{\rho h+\nu } L- \dfrac{p(h+\nu )}{\rho h+\nu } I \\{} & {} - \dfrac{\rho (h+\nu )}{\rho h+\nu } \dfrac{I_1}{I} \alpha S V -\dfrac{\nu (h+\nu )}{\rho h+\nu } \dfrac{I_1}{I} L + \dfrac{p(h+\nu )}{\rho h+\nu } I_1 \\{} & {} + \dfrac{I}{I_1} \alpha S_1 V_1 -\dfrac{rV}{m I_1} \alpha S_1 V_1 - \dfrac{q B V}{m I_1} \alpha S_1 V_1 - \dfrac{I}{I_1} \dfrac{V_1}{V} \alpha S_1 V_1 +\dfrac{rV_1}{m I_1} \alpha S_1 V_1 + \dfrac{q B V_1}{m I_1} \alpha S_1 V_1 \\{} & {} - \dfrac{d q \alpha S_1 V_1}{c m I_1} \dfrac{(B-B_1)^2}{B} + \dfrac{q B V}{m I_1} \alpha S_1 V_1 - \dfrac{q B_1 V_1}{m I_1} \alpha S_1 V_1 - \dfrac{q B_1 V}{m I_1} \alpha S_1 V_1 + \dfrac{q B_1^2 V_1}{m I_1 B} \alpha S_1 V_1 \\& = -a \dfrac{(S-S_1)^2}{S} + \alpha S_1 V_1 - \alpha S V - \dfrac{S_1}{S} \alpha S_1 V_1 + \alpha S_1 V \\{} & {} + \dfrac{\nu (1-\rho )}{\rho h+\nu } \alpha S V - \dfrac{\nu (1-\rho )}{\rho h+\nu }\dfrac{L_1}{L} \alpha S V +\dfrac{\nu (h+\nu )}{\rho h+\nu }L_1 \\{} & {} + \dfrac{\rho (h+\nu )}{\rho h+\nu } \alpha S V - \dfrac{p(h+\nu )}{\rho h+\nu } I - \dfrac{\rho (h+\nu )}{\rho h+\nu } \dfrac{I_1}{I} \alpha S V -\dfrac{\nu (h+\nu )}{\rho h+\nu } \dfrac{I_1}{I} L +\dfrac{p(h+\nu )}{\rho h+\nu } I_1 \\{} & {} + \dfrac{I}{I_1} \alpha S_1 V_1 -\dfrac{rV}{m I_1} \alpha S_1 V_1 - \dfrac{I}{I_1} \dfrac{V_1}{V} \alpha S_1 V_1 +\dfrac{rV_1}{m I_1} \alpha S_1 V_1 + \dfrac{q B V_1}{m I_1} \alpha S_1 V_1 \\{} & {} - \dfrac{d q \alpha S_1 V_1}{c m I_1} \dfrac{(B-B_1)^2}{B} - \dfrac{q B_1 V_1}{m I_1} \alpha S_1 V_1 - \dfrac{q B_1 V}{m I_1} \alpha S_1 V_1+ \dfrac{q B_1^2 V_1}{m I_1 B} \alpha S_1 V_1 \\& = -a \dfrac{(S-S_1)^2}{S}-\dfrac{q\eta }{m c B_1 I_1} \alpha S_1 V_1 \dfrac{(B-B_1)^2}{B} \\{} & {} + \dfrac{(1-\rho )\nu }{\rho h+\nu }\alpha S_1 V_1 \left( 4-\dfrac{S_1}{S}-\dfrac{S V L_1}{S_1 V_1 L} -\dfrac{LI_1}{L_1I}-\dfrac{V_1I}{VI_1} \right) \\{} & {} + \dfrac{\rho (h+\nu )}{\rho h+\nu }\alpha S_1 V_1 \left( 3-\dfrac{S_1}{S}-\dfrac{S V I_1}{S_1 V_1 I}-\dfrac{V_1 I}{V I_1}\right) . \end{aligned}$$

Therefore, using the following rule

$$\begin{aligned} \dfrac{1}{n}\sum _{i=1}^{n}a_{i}\ge \root n \of {\prod _{i=1}^{n}a_{i}}, \end{aligned}$$
(8)

we obtain \(\dot{F}_{1} \le 0 \) for all \(S, L, I, V, B>0\) and \(\dot{F}_{1} = 0 \) at \(E_{1}\). \(E_{1}\) is G. A. S. \(\square \)

3 Solution of DTM on CHIKV model

Let the CHIKV dynamics model given in Sect. 2

$$\begin{aligned} \left\{ \begin{array}{llll} \dot{S}(t) &{} =\mu -a S(t)- \alpha S(t) V(t) , \\ \dot{L}(t) &{} =(1-\rho )\alpha S(t) V(t) -(h+\nu )L(t), \\ \dot{I}(t) &{} =\rho \alpha S(t) V(t) +\nu L(t)-p I(t), \\ \dot{V}(t) &{} =m I(t)-rV(t)-q B(t) V(t), \\ \dot{B}(t) &{} =\eta +c B(t) V(t) -d B(t). \end{array} \right. \end{aligned}$$
(9)

Follow [21, 22] we consider the time domain \( [0,\,T]\) with \(T>0\), we consider \(t_k=k\dfrac{T}{M}\) and \(0\le k\le M\) the time step. Applying the DTM which offers the solution \(\big (S,L,I,V,B\big )\) by the series

$$\begin{aligned} X(t)=\displaystyle \sum _{i=0}^{\infty }X_k(i)(t-t_k)^i, \end{aligned}$$
(10)

with X is one of the variables S, V, I, L or B.

We put \(X_0=X(0)\) the initial condition, for \(i\ge 0\) and for each step \(k\le M\) the initial condition is given by \(X_k(0)=X(t_{k-1})\), with X is last calculed solution at step \(k-1\) in the interval \([t_{k-2},\,t_{k-1}]\).

Then, the DTM is follow

$$\begin{aligned} S_k(i+1)& = \frac{1}{1+i} \bigg [\mu \delta (i)-aS_k(i)-\alpha \sum _{l=0}^iS_k(l) V_k(i-l)\bigg ],\\ L_k(i+1)& = \frac{1}{1+i} \bigg [ (1-\rho ) \alpha \sum _{l=0}^iS_k(l) V_k(i-l) - (h + \nu )L_k(i)\bigg ], \\ I_k(i+1)& = \frac{1}{1+i} \bigg [\rho \alpha \sum _{l=0}^iS_k(l) V_k(i-l)+ \nu L_k(i) - pI_k(i)\bigg ],\\ V_k(i+1)& = \frac{1}{1+i} \bigg [m I_k(i) - r V_k(i) - q\sum _{l=0}^iB_k(l) V_k(i-l)\bigg ], \\ B_k(i+1)& = \frac{1}{1+i} \bigg [\eta \delta (i)+ c\sum _{l=0}^iB_k(l) V_k(i-l) - d B_k(i)\bigg ]. \end{aligned}$$

Here, we have note \(X_k(i)\) is i-th term the differential transforms of the function X used in (9). We obtain the first iteration

$$\begin{aligned} S(1)& = \mu \delta (0)-aS(0)-\alpha S(0) V(0) \\ L(1)& = (1-\rho ) \alpha S(0) V(0) - (h + \nu )L(0) \\ I(1)& = \rho \alpha S(0) V(0)+ \nu L(0) - pI(0) \\ V(1)& = m I(0) - r V(0) - qB(0) V(0) \\ B(1)& = \eta \delta (0)+ cB(0) V(0) - d B(0) \end{aligned}$$

4 Numerical examples

This section is devoted to the evaluation of DTM capabilities with the multistage technique. Then, we simulate the transmission of the CHIKV epidemic. In fact, taking \(\mu =5,~ a=0.1,~ \alpha =8,~ \rho =0.5,~ h=0.1,~ \nu =1,~ p=0.1,~ m=0.2,~ r=0.1,~ q=1,~ \eta =2,~ c=0.1~\text {and}~ d=0.1\), and the initial conditions \(S(0)=2,~ L(0)=0.2,~ I(0)=0.4,~ V(0)=0.4~\text {and}~ B(0)=1.\) These parameters were chosen so that \(\mathscr {R}_{0}\approx 37.9918 > 1\) put themselves in the case where there will necessarily be an epidemic state; as indicated in Theorem 2 of the Sect. 2.

First, an important aspect of this work also deals with the value of errors incurred in solving the detected CHIKV problem. Table 2 summarizes the values of the stability points obtained by DTM, RK4, and the reference values. This shows that the DTM can play an important role in terms of accuracy. Moreover, these results are in agreement with the study presented in the Sect. 2 (proof of Lemma 2). Indeed, we concluded that if \(\mathscr {R}_{0} > 1\), the system (5) has an infected stable state \(E_1 = (S_1, L_1, I_1, V_1, B_1)\) with \(V_1=0.2655 \in [0, d/c]=[0,1]\) (value calculated by RK4, and we will assume that it is exact for the comparison).

Table 2 Summarized of absolute errors of stability point
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The curves in Figs. 1 and 2 compare the solutions obtained by the Runge Kutta 4 method (RK4) (solid line with dot markers) with the DTM solutions (solid line). We note that the curves of each solution overlap well two by two. This proves the effectiveness of the DTM with the support of the multi-step technique used.

Fig. 1
figure 1

The solution (S, L, I, V, B) of the problem compared with the Runge–Kutta method of order 4 in a domain of study large enough to show the convergence of the method to the equilibrium point

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Fig. 2
figure 2

The solutions S, I, V,  and L, B of the problem have been grouped together in the same figure according to their close value and are compared with the RK4 method

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Figure 3 agrees with the mathematical analysis we did at the beginning and shows when \(\mathscr {R}_{0}\) is close to zero \((\mathscr {R}_{0}\approx 0.091)\) then the solution quickly converges to \(E_0=(1,\,0.08,\, 0,\,0,\,19.08)\), it means that the solutions of the system eventually lead to the virus-free steady state \(E_0\). In this case, the CHIKV will be eliminated from the body. This result is consistent with the result of Theorem 1 that \(E_0\) is globally asymptotically stable.

Fig. 3
figure 3

The solution converges to \((S_1,0,0,0,B_1)\) for \(\mathscr {R}_{0}\) close to zero, it is a virus-free steady state

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Figure 4 shows the convergence of all the solutions of our system to the steady state \(E_1 = (S_1, L_1, I_1, V_1, B_1)\) in the case of \(\mathscr {R}_0>1\), and this is in agreement with the theoretical result that we have done in the Sect. 2 (Theorem 2) which shows the convergence of the five unknowns towards a positive solution and globally asymptotically stable \(E_1\) as soon as \(\mathscr {R}_0 > 1.\). It is an epidemic state. In Fig. 5 we have proposed to represent the concentration curve of uninfected monocytes S which grows remarkably at the beginning up to a maximum at \(t=1.7\) and rapidly changes its direction of growth to stabilize towards a limit value \(S_\textrm{lim}=1.823\). Almost the same for the concentration of latently infected monocytes L that converges to \(L_\textrm{lim}=2.187\). On the other hand for the Concentrations of CHIKV particles, V begins its pace to reach a minimum at \(t=1.4\) and it goes towards the limit value \(V_\textrm{lim}=0.330\). Figure 6 shows that if the rate \(\mu \) is increased (the rate of creation of uninfected susceptible host cells) by taking the values 0.5, 1.5, 2.5 afterward 3.5, the curve of the concentrations of uninfected monocytes take on the appearance of convergence more and more quickly. This can be obvious since the increase in the parameter \( \mu \) causes the increase of \(\mathscr {R}_0\) which passes from the value 3.8 (for \(\mu =0.5\)) and arrives at the value 26.6 (for \(\mu =3.5\)).

Fig. 4
figure 4

The solution converges for \(\mathscr {R}_{0}>1\); it is an epidemic state

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Fig. 5
figure 5

Behaviours of the variables S, L and V

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Figures 6 and 7 show that the increase in the parameters \(\eta \) and \(\mu \) which represent the constant antibody generation and proliferation rate leads to an increase in concentrations of B cells and uninfected monocytes.

Fig. 6
figure 6

Influence of the variation of parameter \(\mu \) on the concentrations of uninfected monocytes

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Fig. 7
figure 7

Influence of the variation of parameter \(\eta \) on the concentrations of B cells

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5 Conclusions

In this article, we have proposed a semi-analytical solution to a problem describing the spread of CHIKV within the human population. We proposed a differential transformation method which was reinforced by a multi-step strategy. By comparing the obtained results with the RK4 solution, we can conclude the practical and simple way of DTM. It can be interesting to use this method with more complicated dynamic systems with or without delay. The numerical examples produced to show that one can make a practical and effective mathematical and biological analysis of the problem of this virus.