# Approximate solution of cryptosporidiosis model

- Ebenezer Bonyah
^{1, 3}Email author, - Abdon Atangana
^{2}, - Kazeem Okosun
^{3}and - Muhammad Altaf Khan
^{4}

**3**:2

https://doi.org/10.1186/s40540-016-0018-2

© The Author(s) 2016

**Received: **20 July 2016

**Accepted: **28 October 2016

**Published: **11 November 2016

## 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.

## Keywords

## 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.

*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

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

*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]

*m*. It is worthy to note that the initial guess or estimate assures the uniqueness of the series decompositions [20, 21].

## Stability analysis

###
**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*

###
**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

## 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**

###
**Lemma 1**

*If*\(m - 1 < \alpha \leqslant m,m \in \mathbb {N}\)

*and*\(f \in C_\mu ^m ,\mu \geqslant 1,\)

*then*

###
**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

*m*.

### Approximate solution of fractional version

## 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.

## Declarations

### Authors' contributions

The authors have contributed equally for the production of this manuscript. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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