A fractional order model for obesity epidemic in a nonconstant population
 Elif Demirci^{1}Email author
https://doi.org/10.1186/s1366201711350
© The Author(s) 2017
Received: 16 September 2016
Accepted: 8 March 2017
Published: 17 March 2017
Abstract
In this paper, we propose a fractional order epidemic model for obesity contagion. The population size is assumed to be nonconstant, which is more realistic. The model considers vertical transmission of obesity and also obesityrelated death rate. We give local stability analysis of the model. Finally, some numerical examples are presented.
Keywords
MSC
1 Introduction
Obesity is one of the major risk factors for many chronic, fatal diseases including cancer, diabetes mellitus and cardiovascular disorders. According to the World Health Organization, worldwide obesity has doubled since 1980, and in 2008, 11% of adults aged 20 and over were obese [1].
Although there are some other reasons (e.g. genetic reasons, endocrine disorders), the main reason for obesity is excessive food intake and lack of physical activity. These reasons are closely related to the lifestyles of the individuals within a population. Therefore, obesity can be considered as socially contagious. In [2] a detailed analysis of the obesity epidemic in the U.S. is given. Santonja et al. [3] and Ejima et al. [4] also considered obesity as an epidemic disease and gave mathematical models to explain the spread of obesity. In both of these models integer order differential equations are used and the total population size is assumed to be constant.
The epidemic models where the total population is assumed to be constant are classical models given for shortterm epidemics. When the epidemic disease arises and vanishes in a short time like influenza, this kind of models give realistic results. But in the case of longterm effective diseases like hepatitis, rabies, rubella and so on, limiting the population to be constant would be a very strong assumption that affects the realism of the model. In this paper, we propose a new mathematical model in which we assume that the population is nonconstant. What is more, with a particular choice of the natural death rate function in the model, it gives a classical logistic growth for the population.
We also consider the memory dependence on the obesity contagion. The memory effect in the spread of obesity is discussed in detail in [5]. In recent years, it has frequently been observed in modeling memorydependent processes of physical and life sciences that models based on fractional order derivatives provide better agreement between solutions and real data [6–8]. Therefore, it is reasonable to use fractional order models to understand the spread of obesity in a population. Also note that the fractional order model we give is a generalization of an integer order model, and if the order of the fractional model is one, it reduces to its integer order counterpart.
In this paper, we consider continuous fractional differential equation systems to understand the spread of obesity in a population. Discrete fractional systems are also being used to model some real life problems, and their stability results are given in recent years [9, 10]. Stability of the proposed model in this paper is examined using the method given by Matignon [11]. Some other stability results can be found in [12–15].
This paper is organized as follows. Section 2 is devoted to the model construction. In this section we propose a new fractional order epidemic model including diseasedependent death rate within a nonconstant population size. We also consider the tendency to obesity at birth as a result of bad nutritional habits during pregnancy, by means of vertical transmission. In Section 3, a detailed local stability analysis for the model is given. Finally, in Section 4, we give some numerical examples to illustrate our results.
2 Mathematical model
We first give some basic definitions of fractional calculus.
Definition 1
Fractional derivative has several different definitions [16]. In this paper we use the Caputo definition due to its advantages in applied problems. The Caputo definition of fractional derivative allows us to use initial conditions of the classical form, avoiding solvability problems.
Definition 2
 p::

probability of having an overweight baby;
 b::

natural birth rate;
 \(d(\cdot)\)::

natural death rate function (dependent on the total population);
 α::

transmission rate of the disease by social contact;
 β::

rate at which an overweight individual moves to the obese class;
 \(k, r\)::

treatment rates for overweight and obese individual, respectively;
 θ::

obesityrelated (from the diseases that are caused by obesity) death rate.
We should note that for the diseasefree case (i.e. \(W_{3}=0\)), if d is a linear function, then the total population has logistic growth.
Theorem 3
Proof
It is easy to see the existence and uniqueness of the solution of the initial value problem (1)(3) in \(( 0,\infty ) \). We will show that the domain \(R_{+}^{3}\) remains positively invariant.
3 Equilibrium points and stability
Theorem 4
The DFE of system (4) (if exists) is asymptotically stable if \(R_{0}<1\).
Proof
4 Numerical results
In this section we consider four sets of parameters to discuss different cases. We use the solution technique given in [21] to evaluate the numerical solutions of the system for μ values \(1, 0.8\) and 0.6. Two main theorems about this technique are given below.
Theorem 6
[21]
Theorem 7
[21]
Parameter values for Case 14
This linear death rate function lets the total population N have logistic growth. For the exact solution of a fractional logistic equation, [26] used the Carleman embedding technique, but there is a controversy between the results of West and Area et al. [27]. However, in this paper we use a totally different technique [21] to find the solution of the system including the logistic equation.
Case 4: We now consider the case where vertical transmission of the disease does not exist, i.e. \(p=0\). The value of p is also related to the disease control and prevention strategies. For this case, the DFE exists. For \(F_{0}^{\ast}= ( 243.926,0,243.926 ) \), the basic reproductive number \(R_{0}=0.84507\), which states that \(F_{0}^{\ast}\) is asymptotically stable for all \(\mu\in(0,1]\).
5 Conclusions
In this paper, a fractional order mathematical model of obesity epidemic, including vertical transmission within a nonconstant population size, is proposed. The order of the proposed system is a free parameter which can be used to have a better fit between the real data and a theoretical formulation of the solutions. We should also note that fractional order models give more realistic predictions in modeling procedures with short memory effect [16, 19]. Since epidemic dynamics of obesity can be considered as a memorydependent process, fractional order systems may be good tools for modeling the contagion of obesity. Obesity is one of the major health problems all over the world. Because of the economic impact of obesityrelated diseases, the dynamics of obesity epidemic is important for countries. In the United States, nationwide excess medical costs for obesity is as much as $147 billion annually for adults and $14.3 billion annually for children [28]. Due to these economic reasons, disease control is very important for countries. The parameters k and r in our model are control parameters for disease control. In the final part of the paper, we simulated the system for different parameter values given in Table 1. We should point out that by adjusting the control parameters r and k, disease can be kept under control.
Declarations
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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