Approximate solution of cryptosporidiosis model

Abstract

Cryptosporidium is associated with waterborne transmission mechanism through the faecal–oral path in many recreational water facilities. We investigate the probable approximate solution of integer and noninteger systems of nonlinear ordinary differential equations representing cryptosporidiosis dynamics. The approximate or estimate solution is derived through recent developed analytic method, the homotopy decomposition method (HDM). The algorithm is systemically explained and demonstrated with some numerical examples. The numerical results indicate that the approximate solution is of continuous function form in the light of noninteger-order derivative. The integer-order numerical solution of parameters values varied and investigated which show similar solution in each case. The method employed to obtain the solution to this problem is robust, easy, reliable and quick in terms of time.

Background

Human cryptosporidiosis is caused by cryptosporidium protozoan and constitutes a large number of gastrointestinal disease usually connected with recreational water use as the case in Australia [1, 2] as well as other parts of the world [see e.g. [3–6] and references therein]. Cryptosporidiosis is characterized with severe watery diarrhoea; however, asymptomatic infection may arise which becomes the source of infection [1]. Cryptosporidiosis is transmitted through interaction with contaminated water, food and surfaces. Allowing water to get through the mouth into the stomach in recreational swimming is the easiest way of contracting the disease. Crypto is extremely infectious and if not cured one can get the infection again and may infect others. Cryptosporidium is well identified with waterborne transmission mechanism via the faecal–oral path in many recreational water facilities. It is established that the rate of infection is very low in cryptosporidiosis. For instance, it is estimated that it has a low infective dose ranging between 10 and 30 in a healthy adult [7–10]. The disease is capable of resisting to just halogen disinfection which constitutes the recommended level for treating water recreational facilities [11]. If anyone is diagnosed of cryptosporidiosis then the person is likely to have a weak immune system which is a symptom of HIV.

In this paper, we present an SIR model proposed in [25]. The model is characterized by four components which are susceptible human, infected human, recovered human and environment where the bacteria live. $\Lambda$ is the recruitment rate into the susceptible class. Recovered individual may lose their immunity to the disease at the $\omega$. Recovery rate due to treatment is denoted by $\sigma$. The natural and mortality rates due to the disease are $\mu$ and $\psi,$ respectively. $E_c$ is the microbe population, and the contact rate of the microbe population is denoted by $\nu$. The concentration of microbe population in the environment is represented by K. The rate of cryptosporidiosis infected to the environment is denoted by $\pi$. $\mu _b$ is the mortality rate of the microbes. $\rho$ is the rate of contact with the environment

$$\begin{aligned} \begin{array}{ll} \left\{ \begin{array}{l} \frac{\rm{d}}{{dt}}S = \Lambda + \omega R - \mu S - \left( {\frac{{\nu I}}{{K + I}} + \rho E_c } \right) S \\ \frac{{\rm{d}}}{{{\rm{d}}t}}I = \left( {\frac{{\nu I}}{{K + I}} + \sigma E_c } \right) S - \left( {\mu + \psi + \sigma } \right) I \\ \frac{{\rm{d}}}{{{\rm{d}}t}}R = \sigma I - \left( {\mu + \psi } \right) R \\ \frac{{\rm{d}}}{{{\rm{d}}t}}E = \pi I - \mu _b E \\ \end{array} \right. \end{array} \end{aligned}$$

(1)

subject to the initial conditions

$$\begin{aligned} S(0) \ge 0,\quad I(0) \ge 0,R(0) \ge 0 \quad \text {and}\, \quad E(0) \ge 0 \end{aligned}.$$

Mathematical models, in general, are highly nonlinear, and obtaining the exact solution usually becomes a challenge. Most researchers resort to numerical solutions. In recent times, there have been several analytical approximation techniques to address these problems. The assistance of computer-aided techniques is growing at a fast rate and numerical simulations have become inevitable. Non-numerical problems are still very crucial because of their role in the socioeconomy of every nation (see e.g. [14–17] references therein). There are several, however, alternative analytical asymptotic techniques which include the nonperturbation modified, Lindstedt–Poincare technique [15], variation iteration technique [17], Adomian decomposition method [18] and homotopy perturbation method [17, 19].

Recently, authors in [13] used HDM to investigate HIV infection of CD4+ T cells and obtained approximate solutions and compared the results with other existing methods. In their study, they found that HDM is as better as other well-known methods as mentioned in the literature. Authors in [20] employed HDM to examine Tuberculosis using both integer and fractional derivative and obtained solutions that are of continuous functions of the noninteger-order derivative. Author in [20] used HDM to investigate cryptosporidiosis model of both integer and fractional order and obtained solutions that are continuous functions of the noninteger-order derivative.

The purpose of this paper is to present approximate analytical solutions for the standard form and fractional aspect of (1) in addition to (2) using the relative new analytical method called homotopy decomposition method (HDM).

The paper is organized as follows: In “Background” section , the basic ideas of homotopy decomposition method are presented. In “Fundamental information about homotopy decomposition method” section, the application of HDM for system for cryptosporidiosis population dynamics is presented. “Stability analysis” section deals with the application of the HDM for the system of fractional cryptosporidiosis model dynamics. In “Application of the HDM to the model with integer-order derivative” section, the conclusion is drawn.

Fundamental information about homotopy decomposition method

To show the basic notion of this method, we take into consideration a universal nonlinear nonhomogeneous partial differential equation characterized with the following form [12]:

$$\begin{aligned} \frac{{\partial ^{m} U(x,t)}}{{\partial t^{m} }} = L\left( {U(x,t)} \right) + N\left( {U(x,t)} \right) + f(x,t), \quad m = 1,2,3,\ldots , \end{aligned}$$

(2)

By focusing on the primary condition, we have

$$\begin{aligned} \frac{{\partial ^{i} (x,0)}}{{\partial t^{i} }} = y_{i} ,\quad \frac{{\partial ^{m} U(x,0)}}{{\partial t^{m - 1} }} = 0 \qquad i = 0,1,2,\ldots m - 2 \end{aligned}$$

(3)

where m denotes the order of the derivative, f represents an identified function, N denotes the common nonlinear differential operator, L represents a linear differential operator, and m is the order of the derivative. The initial process here is to ensure that the inverse operator $\partial ^{m} /\partial t^{m}$ is applied on both sides of (3) so that we obtain [12]

$$\begin{aligned} U(x,t) = \sum \limits _{k = 0}^{m - 1} {\frac{{t^k }}{{k!}}} \frac{{d^k u(x,0)}}{{dt^k }} + \int \limits _0^{t} {\int \limits _0^{t_1 } {\cdots \int \limits _0^{t_{m - 1} } {L\left( {U(x,\tau )} \right) } } } + N\left( {U(x,\tau )} \right) + f(x,\tau )\ldots \mathrm{{d}}t \end{aligned}$$

(4)

The multi-integral in Eq. (3) can be reorganized as

$$\begin{aligned}&\int \limits _0^{t} {\int \limits _0^{t_1 } {\cdots \int \limits _0^{t_{m - 1} } {L\left( {U(x,\tau )} \right) } } } + N\left( {U(x,\tau )} \right) + f(x,\tau )\ldots \mathrm{{d}}t \\ \nonumber&= \frac{1}{{(m - 1)!}}\int \limits _0^{t} {(t - } \tau )^{m - 1} L\left( {U(x,\tau )} \right) + N\left( {U(x,\tau )} \right) + f(x,\tau )\ldots \mathrm{{d}}\tau \end{aligned}$$

(5)

So, Eq. (3) can be reformulated as

$$\begin{aligned} U(x,t)&= \sum \limits _{k = 0}^{m - 1} {\frac{{t^k }}{{k!}}} y_i (x) + \frac{1}{{(m - 1)!}}\int \limits _0^t {(t - } \tau )^{m - 1} L\left( {U(x,\tau )} \right) \\ \nonumber&\quad + N\left( {U(x,\tau )} \right) + f(x,\tau )\ldots \mathrm{{d}}t \end{aligned}$$

(6)

Employing the homotopy scheme, the solution of the above-mentioned integral equation is expressed in series form as follows:

$$\begin{aligned} U(x,t)&= \sum \limits _{k = 0}^{m - 1} {p^m U_n } (x,t),\\ \nonumber U(x,t)&= \mathop {\lim }\limits _{p \rightarrow 1} U(x,t,p) \end{aligned}$$

(7)

and the nonlinear term can be decomposed by

$$\begin{aligned} \mathrm{{NU}} (r,t) = \sum \limits _{n = 1}^\infty {P^n \mathfrak {R}_n } (U) \end{aligned},$$

where $p\in \left( {0,1} \right]$ denotes an implanting parameter and $\mathfrak {R}_n (U)$ represents the polynomial that can be engendered by

$$\begin{aligned} \begin{array}{l} \mathfrak {R}_n (U)\left( {U_0 \ldots , U_n } \right) = \frac{1}{{n!}}\frac{{\partial ^n }}{{\partial p^n }}\left[ {N\left( {\sum \limits _{j = 0}^{n} {p^{j} U_{j} (x,t)} } \right) } \right] \\ n = 0,1,2,3 \\ \end{array} \end{aligned}$$

(8)

The homotopy decomposition method is composed of the decomposition method and Abel integral which is given as

$$\begin{aligned} \sum \limits _{n = 0}^\infty {p^{n} U_n } (x,t)&= \mathrm{T}(x,t) + p\frac{1}{{(m - 1)!}} \\ \nonumber&\quad \times \int \limits _0^{1} {(t - \tau )^{m - 1} } \left[ {f(x,t) + \left( {\sum \limits _{n = 0}^\infty {p^n U_n } (x,t)} \right) + \sum \limits _{n = 0}^\infty {p^{n} \mathfrak {R}_n } (U)} \right] \mathrm{{d}}\tau \end{aligned}$$

(9)

with

$$\mathrm{T}(x,t) = \sum \limits _{k = 0}^{m - 1} {\frac{{t^{k} }}{{k!}}y_i } (x).$$

Matching the term of identical powers of $p_{i}$ leads to solution of different orders. The estimate of the approximation is $\mathrm{T}(x,t)$ that is precisely the Taylor series presenting the exact solution of order m. It is worthy to note that the initial guess or estimate assures the uniqueness of the series decompositions [20, 21].

Stability analysis

The stability analysis of a model is very essential, which allows to establish the behaviour of the model. The disease-free equilibrium is obtained by setting system 1 to be equal to zero and solving

$$\begin{aligned} E_0 = \left( {S^* ,I^* ,R^* ,E^* } \right) = \left( {\frac{\Lambda }{\mu },0,0,0} \right) \end{aligned}.$$

The linear stability of $E_0$ is determined by applying the next-generation operator technique [26] on system 1, and the reproduction number is obtained as follows:

$$\begin{aligned} R_{0} = \frac{{\nu \Lambda \sqrt{\mu _{b} } + \sqrt{\nu ^{2} \Lambda ^{2} \mu _{b} + 4K^2 \pi \mu ^{2} \rho (\mu + \sigma + \psi ))} }}{{2K\mu \sqrt{\mu_{b} } (\mu + \sigma + \psi )}} \end{aligned}.$$

Theorem 1

The disease free equilibrium of the model 1, given by $R_0$ , is locally asymptotically stable if $R_0<1,$ and unstable if $R_0>1.$

Proof

$$\begin{aligned} \left\{ \begin{array}{l} S^* = \frac{{\Lambda + \omega R^* }}{{\mu + \beta ^* }}, \\ R^* = \frac{{\sigma I^* }}{{\omega + \mu }}, \\ E^* = \frac{{\pi I^* }}{{\mu _b }}. \\ \end{array} \right. \end{aligned}$$

(10)

$$\begin{aligned} q(I^* ) = I^* \left( {G_1 (I^* )^2 + G_2 (I^* ) + G_3 } \right) = 0 \end{aligned}$$

(11)

$$G_{1} = \rho \rho \left( {\mu (\mu + \sigma + \psi } \right) + \left( {\mu + \psi )\psi } \right) ,$$

$$\begin{aligned} G_2 = \frac{{K\mu _b \mu \left( {\mu + \sigma + \psi } \right) }}{{\nu \mu _b + K\rho \rho }}\left( {R_p - R_0 } \right) , \end{aligned}$$

$$\begin{aligned} G_3 = K\mu _b \mu (\mu + \omega )(\mu + \sigma + \psi )(1 - R_0 ) \end{aligned}$$

$$\begin{aligned} R_p = \frac{{(\nu \mu _b + K\rho \rho )(K\rho \rho [\mu ^2 + (\mu + \psi )(\sigma + \psi )] + \mu \mu _b (\nu + \mu )(\mu + \sigma + \psi + Q))}}{{K\mu \mu _b (\mu + \sigma + \psi )}} \end{aligned}$$

Theorem 2

For $K = 0$, the system 1 has no endemic equilibrium and for $K>0$ the model exhibits two conditions: a transcritical bifurcation if $R_p \ge 1$ and a backward bifurcation if $R_p<1.$

$\square$

For more details about the stability analysis of system 1, see [20] reference therein.

Application of the HDM to the model with integer-order derivative

This section discusses the usage of HDM to derive the set of mathematical equations based on cryptosporidiosis population dynamics model. Following the steps involved in HDM method, we arrive at the following integral equation;

$$\begin{aligned} \begin{array}{l} p^0 = S_0 (t) = S(0), \\ p^0 = I_0 (t) = I(0), \\ p^0 = R_0 (t) = R(0), \\ p^0 = E_0 (t) = E(0) \\ \end{array} \end{aligned}$$

(12)

$$\begin{array}{*{20}l} {p^{1} :S_{1} (t) = \int\limits_{0}^{t} {(\Lambda + \omega R_{0} (\tau ) - \mu S_{0} (\tau ) - \left( {\frac{{\nu I_{0} }}{{K + I_{0} }} + \rho E_{{c0}} } \right)} S_{0} (\tau ))d\tau ,{\text{ }}S_{0} (0) = 0,} \\ {p^{1} :I_{1} (t) = \int\limits_{0}^{t} {\left( {\frac{{\nu I_{0} }}{{K + I_{0} }} + \sigma E_{{c0}} } \right)S_{0} (\tau ) - \left( {\mu + \psi + \sigma } \right)} I_{0} (\tau )){\text{d}}\tau ,\quad {\text{I}}_{{\text{0}}} ({\text{0}}){\text{ = 0}},} \\ {p^{1} :R_{1} (t) = \int\limits_{0}^{t} {\sigma I_{0} (\tau ) - \left( {\mu + \psi } \right)} R_{0} (\tau )){\text{d}}\tau ,\quad {\text{R}}_{{\text{0}}} ({\text{0}}){\text{ = 0}},} \\ {p^{1} :E_{1} (t) = \int\limits_{0}^{t} {\pi I_{0} (\tau ) - \mu _{b} } E_{0} (\tau )){\text{d}}\tau ,\quad {\text{E}}_{{\text{0}}} ({\text{0}}){\text{ = 0}}.} \end{array}$$

(13)

$$\begin{aligned} p^n :S_n (t)&= \int \limits _0^t ( {\Lambda + \omega R_{(n - 1)} (\tau )} - \mu S_{(n - 1)} (\tau ) \\ \nonumber \qquad&- \sum \limits _{j = 0}^{n - 1} {\left( \frac{{\nu I_j }}{{K + I_j }} + \rho E_{cj} \right) } S_{n - j - 1} {(\tau )} d\tau, \quad S_{n - 1} (0) = 0, \\ \nonumber p^n :I_n (t)&= \int \limits _0^t \left( {\sum \limits _{j = 0}^{n - 1}} {\left( {\frac{\nu I_j }{K + I_j }} + \sigma E_{cj} \right) } S_{n - j - 1} (\tau ) - ( {\mu + \psi + \sigma }) I_{n - 1} (\tau )\right) d\tau, \quad I_{n - 1} (0) = 0,\\ \nonumber p^n :R_n (t)&= \int \limits _0^t ( {\sigma I_{n - 1} (\tau )} - ( {\mu + \psi }) R_{n - 1} {(\tau )} d\tau,\quad R_{n - 1} (0) = 0,\\ \nonumber p^n :E_n (t)&= \int \limits _0^t ( {\pi I_{n - 1} (\tau )} - \mu _b E_{n - 1} {(\tau )}))d\tau, \quad E_{n - 1} (0) = 0. \end{aligned}$$

(14)

Using integration techniques, we arrived at the following components:

$$\begin{aligned} S_0 (t) = S(0);I_0 (t) = I(0);R_0 (t) = R(0);E_0 (t) = E(0) \end{aligned}$$

$$\begin{aligned} \begin{array}{l} S_1 (t) = \left( {\Lambda + \omega R_0 - \mu S_0 - \left( {\frac{{\nu I_0 }}{{K + I_0 }} + \rho E_{c0} } \right) S_0 } \right) t,\\ I_1 (t) = \left( {\left( {\frac{{\nu I_0 }}{{K + I_0 }} + \sigma E_{c0} } \right) S_0 - \left( {\mu + \psi + \sigma } \right) I_0 } \right) t,\\ R_1 (t) = \left( {\sigma I_0 - \left( {\mu + \psi } \right) R_0 } \right) t,\\ E_1 (t) = \left( {\pi I_0 - \mu _b E_0 } \right) t. \end{array} \end{aligned}$$

(15)

For the purpose of simplification, we let

$$\begin{aligned} \begin{array}{l} a_0 = \left( {\Lambda + \omega R_0 - \mu S_0 - \left( {\frac{{\nu I_0 }}{{K + I_0 }} + \rho E_{c0} } \right) S_0 } \right) ,\\ b_0 = \left( {\left( {\frac{{\nu I_0 }}{{K + I_0 }} + \sigma E_{c0} } \right) S_0 - \left( {\mu + \psi + \sigma } \right) I_0 } \right) ,\\ c_0 = \left( {\sigma I_0 - \left( {\mu + \psi } \right) R_0 } \right) ,\\ d_0 = \left( {\pi I_0 - \mu _b E_0 } \right) . \end{array} \end{aligned}$$

(16)

By applying integrating $S_1 (t),I_1 (t),R_1 (t)$ and $E_1 (t),$ we have

$$\begin{aligned} S_2 (t)&= \frac{{t^2 }}{2}\left( { + \omega c - \mu a - \frac{{\nu I_0 }}{{K + I_0 }}a - \frac{{\nu b}}{{K + b}}S_0 - a\rho E_{c0} - \rho dS_0 } \right) \\ \nonumber&= S_2 (t) = \frac{{t^2 }}{2}a_1 \\ \nonumber I_2 (t)&= \frac{{t^2 }}{2}\left( {\frac{{\nu I_0 }}{{K + I_0 }}a + \frac{{\nu b}}{{K + b}}S_0 + \rho E_{c0} + \rho dS_0 - \left( {\mu + \psi + \sigma } \right) b} \right) \\ \nonumber&= I_2 (t) = \frac{{t^2 }}{2}b_1 \\ \nonumber R_2 (t)&= \frac{{t^2 }}{2}\left( {\sigma a - \left( {\mu + \psi } \right) c} \right) \\ \nonumber&= R_2 (t) = \frac{{t^2 }}{2}c_1 \\ \nonumber E_2 (t)&= \frac{{t^2}}{2}\left( {\pi a - \mu _b d} \right) \\ \nonumber&= E_2 (t) = \frac{{t^2 }}{2}d_1 \end{aligned}$$

In general, the following recursive formulas are obtained:

$$\begin{aligned} \begin{array}{l} S_n (t) = \frac{{t^n }}{{n!}}a_n ,\\ I_n (t) = \frac{{t^n }}{{n!}}b_n ,\\ R_n (t) = \frac{{t^n }}{{n!}}c_n ,\\ E_n (t) = \frac{{t^n }}{{n!}}d_n ,\\ \end{array} \end{aligned}$$

(17)

where $a_n ,b_n ,c_n$ and $d_n$ rely on the fixed set of empirical parameters. It thus, in principle, implies that the approximate solution of the system (1) is obtained as

$$\begin{aligned} \begin{array}{l} S_n (t) = \sum \limits _{n = 0}^N {\frac{{t^n }}{{n!}}a_n ,}\\ I_n (t) = \sum \limits _{n = 0}^N {\frac{{t^n }}{{n!}}b_n ,}\\ R_n (t) = \sum \limits _{n = 0}^N {\frac{{t^n }}{{n!}}c_n ,}\\ E_n (t) = \sum \limits _{n = 0}^N {\frac{{t^n }}{{n!}}d_n } \end{array} \end{aligned}$$

(18)

The total human population is assumed to be 140; the initial susceptible human population is 120; the initial infected human population is 5 and initial human recovered population 15. The initial cryptosporidiosis population is 12. The human recruitment rate is $\Lambda =0.0004$; individual may lose their immunity to the disease at $\omega =0.001$; recovery rate due to treatment is $\sigma =0.07$; natural mortality is $\mu =0.00055$; mortality rate due to the disease is $\psi =0.006$; contact rate of the microbe population is $\nu =0.5$; the microbe population is $K=1000$; cryptosporidiosis infection contribution to the environment is $\pi =0.045$; mortality rate of the microbe is $\mu _b=0.033$ and the rate of humans contact with the environment is $\rho =0.045.$

We make an assumption that there is a constant population. Figure 1 depicts the approximate solution of the system (1). In Fig. 1, the susceptible population with time decreases as more humans get infected and others die due to natural death. This is envisaged in biologically feasible situation. The infected humans in Fig. 1 increase in a short period and then decrease. This could be the fact that people get to know about the disease and try to avoid making contact with the microbe. The recovered humans appeared to reduce and could be attributable to difficulty in diagnosing a patient with cryptosporidiosis even in the advanced world. The microbe population decreases with time and this could be the fact that health authorities may design a problem to fight against the microbe. We varied some of the parameters in Fig. 2 in order to observe the dynamics in the integer-order situation. In Fig. 2, susceptible, recovered and microbe population are quickly moving towards the origin. However, the infective population in Fig. 2 is relatively slow moving towards the origin. This is expected as susceptible humans get infected and then slowly the infected get recovered. It can be seen from Fig. 2 that the microbe population also reduces because people get the awareness and prevent themselves.

Fig. 1

Approximate solution for $\alpha = 1,\beta =1, \gamma =1,\xi =1$

Fig. 2

Approximate solution for $\alpha = 1,\beta =1, \gamma =1,\xi =1$

Application of the HDM to the model with noninteger-order derivative

In recent times, fractional calculus has been identified as a powerful to tool to model many physical and engineering processes, which are best described in terms of fractional differential equation [12]. It is remarkable to note that the usual standard mathematical models in the form of integer-order derivatives fall short of vividly describing many physical problems. Fractional calculus for the past few years has become indispensable in many field of endeavour which include mathematics, biology, chemistry, food science, mechanics, electricity, electronics, image processing, control theory and many more. Some of the vital topics include fractional filters, computational fractional derivative equations, nonlocal phenomena; porous media, biomathematics and fractional phase-locked loops (see [19, 22–24]).

Properties and definitions

Definition 1 A real function $f(x),x > 0,$ is said to be in the space $C_\mu ,\mu \in \mathbb {R}$ , if there exists a real number $p > \mu$ , such that $f(x) = x^p h(x)$ , where $h(x) \in C\left[ {0,\infty } \right) ,$ and it is said to be in space $C_\mu ^m$ if $f^{(m)} \in C_\mu ,m \in \mathbb {N}$

Definition 2

The Riemann–Liouville fractional integral operator of order $\alpha \geqslant 0$ , of a function $f \in C_\mu ,\mu \geqslant - 1,\alpha ,\beta \geqslant 0,$ is defined as follows:

$$\begin{aligned} J^\alpha J^\beta f(x) = J^{\alpha + \beta } f(x) \end{aligned}$$

$$\begin{aligned} J^\alpha J^\beta f(x) = J^\beta J^\alpha f(x)J^\alpha x^\gamma = \frac{{\Gamma (\gamma + 1)}}{{\Gamma (\alpha + \gamma + 1)}}x^{\alpha + \gamma } \end{aligned}.$$

Lemma 1

If $m - 1 < \alpha \leqslant m,m \in \mathbb {N}$ and $f \in C_\mu ^m ,\mu \geqslant 1,$ then

$$\begin{aligned} D^\alpha J^\alpha f(x) = f(x), \end{aligned}$$

$$\begin{aligned} J^\alpha D^\alpha f(x) = f(x) - \sum \limits _{k = 0}^{m - 1} {f^{(k)} (0^ + )\frac{{x^k }}{{k!}}} ;x > 0 \end{aligned}$$

Definition 3

(partial derivatives of fractional order). Assume now that f(x) is a function of n variables $x_i i = 1,\ldots ,n$ also of class C on $D \in \mathbb {R}_n$. We define partial derivative of order $\alpha$ for f respect to $x_i$ the function

$$\begin{aligned} a\partial _{\underline{x} }^\alpha f = \frac{1}{{\Gamma (m - \alpha )}}\int \limits _a^{x_j } {(x_i } - t)^{m - \alpha - 1} a\partial _{x_i }^m f(x_j )\left| {_{x_j = t} } \right. \mathrm{{d}}t \end{aligned}$$

where $\partial _{x_i }^m$ is the usual partial derivative of integer-order m.

Approximate solution of fractional version

The system 1 that is transformed into fractional derivative is expressed as

$$\left\{ \begin{gathered} \frac{{{\text{d}}^{\theta } }}{{{\text{d}}t^{\theta } }}S = \Lambda + \omega R(t) - \mu S(t) - \left( {\frac{{\nu I}}{{K + I}} + \rho E_{c} } \right)S(t),\quad {\text{0}} < \theta \le {\text{1}} \hfill \\ \frac{{{\text{d}}^{\vartheta } }}{{{\text{d}}t^{\vartheta } }}I = \left( {\frac{{\nu I}}{{K + I}} + \sigma E_{c} } \right)S(t) - \left( {\mu + \psi + \sigma } \right)I(t),\quad {\text{0}} < \vartheta \le {\text{1}} \hfill \\ \frac{{{\text{d}}^{\upsilon } }}{{{\text{d}}t^{\upsilon } }}R = \sigma I(t) - \left( {\mu + \psi } \right)R(t),\quad {\text{0}} < \upsilon \le {\text{1}} \hfill \\ \frac{{{\text{d}}^{\varpi } }}{{{\text{d}}t^{\varpi } }}E = \pi I - \mu _{b} E,\quad {\text{0}} < \varpi \le {\text{1}} \hfill \\ \end{gathered} \right.$$

(19)

Following the discussion presented earlier, we arrive at the following equations:

$$\begin{aligned} \begin{array}{l} p^0 = S_0 (t) = S(0), \\ p^0 = I_0 (t) = I(0), \\ p^0 = R_0 (t) = R(0), \\ p^0 = E_0 (t) = E(0) \\ \end{array} \end{aligned}$$

(20)

$$\begin{aligned} p^{1} :S_{1} (t) &= \frac{1}{{\Gamma (\theta )}}\int\limits_{0}^{t} {(t - \tau )^{{\theta - 1}} \times } \left( {\Lambda + \omega R_{0} (\tau ) - \mu S_{0} (\tau )} \right. \\ &\qquad \left. - \left( {\frac{{\nu I_{0} }}{{K + I_{0} }} + \rho E_{{c0}} } \right)S_{0} (\tau ), \right)d\tau \quad S_{1} (0) = 0, \end{aligned}$$

$$\begin{aligned} p^1 :I_1 (t) &= \frac{1}{\Gamma (\vartheta )}\int \limits _0^t ({t - \tau })^{\vartheta - 1} \\ \\ &\quad \times \left( \left( {\frac{{\nu I_0 }}{{K + I_0 }} + \sigma E_{c0}}\right) S_0 (\tau ) - \left( {\mu + \psi + \sigma } \right) I_0 (\tau )\right)d\tau, \quad I_1 (0) = 0, \end{aligned}$$

$$\begin{aligned} p^1 :R_1 (t) = \frac{1}{{\Gamma (\upsilon )}}\int \limits _0^t {\left( {t - \tau } \right) ^{\upsilon - 1} \times \left( {\sigma I_0 (\tau )} \right. - \left( {\mu + \psi } \right) } R_0 (\tau ))d\tau, \quad R_1 (0) = 0, \end{aligned}$$

$$\begin{aligned} p^1 :E_1 (t) = \frac{1}{{\Gamma (\varpi )}}\int \limits _0^t {\left( {t - \tau } \right) ^\varpi \times \left( {\pi I_0 (\tau )} \right. - \mu _b } E_0 (\tau ))d\tau,\quad E_1 (0) = 0. \end{aligned}$$

$$\begin{aligned} p^n :S_n (t) &= \frac {1} {\Gamma (\theta )} \int \limits _0^t ({t - \tau }) ^{\theta - 1} \times \left({\Lambda + \omega R_{(n - 1)} (\tau)} - \mu S_{(n - 1)} (\tau) \right. \\ & \quad \left.- \sum \limits _{j = 0}^{n - 1} {\left( {\frac{{\nu I_j }}{{K + I_j }} + \rho E_{cj} } \right) } S_{n - j - 1} {(\tau )} \right) d\tau, \quad S_{n - 1} (0) = 0, \end{aligned}$$

$$\begin{aligned} {p^n}:{I_n}(t)\, & = \frac{1}{{\Gamma (\vartheta )}}\int\limits_0^t {{{\left( {t - \tau } \right)}^{\vartheta - 1}} \times } \left( {\sum\limits_{j = 0}^{n - 1} {\left( {\frac{{\nu {I_j}}}{{K + {I_j}}} + \sigma {E_{cj}}} \right){S_{n - j - 1}}(\tau )\,} } \right)\\ & - \left( {\mu + \psi + \sigma } \right){I_{n - 1}}\left. {(\tau )} \right)d\tau ,{I_{n - 1}}(0) = 0 \end{aligned}$$

$$\begin{aligned} p^n :E_n (t) = \frac{1}{{\Gamma (\upsilon )}}\int \limits _0^t {\left( {t - \tau } \right) ^{\upsilon - 1} \times \left( {\pi I_{n - 1} (\tau )} \right. - \mu _b } E_{n - 1} \left. {(\tau )} \right) )d\tau, \quad E_{n - 1} (0) = 0, \end{aligned}$$

$$\begin{aligned} p^n :R_n (t) = \frac{1}{{\Gamma (\varpi )}}\int \limits _0^t {\left( {t - \tau } \right) ^{\varpi - 1} \times \left( {\sigma I_{n - 1} (\tau )} \right. - \left( {\mu + \psi } \right) } R_{n - 1} \left. {(\tau )} \right) d\tau, \quad R_{n - 1} (0) = 0. \end{aligned}$$

The figures depict the solutions obtained for different values of the fractional order derivatives. The approximate solutions of the system (1) are shown in Figs. 2, 3, 4 and 5, 6, 7 respectively. It is obvious from Figs. 2, 3, 4 and 5 that the fractional derivatives approximate solutions far better than that of Fig. 1, which is the integer-order derivative approximate solution of the system (1). In Fig. 1, the solutions are quickly converging towards the time axes but in Figs. 2, 3, 4 and 5 one can observe a continuous solution as they move away from being asymptotic to the time axes. The numerical solutions indicate that the approximate solutions are of continuous functions in character of the noninteger-order derivatives. It is worthy of interest to note that the usual mathematical models of integer-order derivatives such nonlinear models usually fall short in many instance in terms of vivid description of the situation. It is prudent, therefore, to entreat people to make use of fractional models in order to vividly represent problems as it is in their natural settings.

Fig. 3

Approximate solution for $\alpha = 0.5,\beta =0.65, \gamma =0.25,\xi =0.75$

Fig. 4

Approximate solution for $\alpha = 0.5,\beta =0.95, \gamma =0.95,\xi =0.35$

Fig. 5

Approximate solution for $\alpha = 0.02,\beta =0.5, \gamma =0.5,\xi =0.95$

Fig. 6

Approximate solution for $\alpha = 0.5,\beta =0.5, \gamma =0.5,\xi =0.5$

Fig. 7

Approximate solution for $\alpha = 0.2,\beta =0.04, \gamma =0.35,\xi =0.6$

Conclusion

The cryptosporidiosis model presented in this paper was investigated in the instances of both integer- and noninteger-order derivatives perspective. The model was solved using a recently developed and iterative technique called homotopy decomposition method. The detailed fundamental characteristics of HDM are presented. The noninteger approximate solutions turn to be increasing continuous functions of the fractional derivative. The algorithm for cryptosporidiosis models is very friendly, effective, simple, reliable and quick. The numerical results for the both instances exhibit the real biological dynamics of the problem solved.