Mathematical Problems in Engineering

Volume 2015, Article ID 602915, 11 pages

http://dx.doi.org/10.1155/2015/602915

## A Partial Lagrangian Approach to Mathematical Models of Epidemiology

^{1}Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan^{2}Department of Mathematics, School of Science and Engineering, LUMS, Lahore Cantt 54792, Pakistan^{3}DST-NRF Centre of Excellence in Mathematical and Statistical Sciences, School of Computer Science and Applied Mathematics,
University of the Witwatersrand (Wits), Johannesburg 2050, South Africa

Received 31 March 2015; Revised 11 June 2015; Accepted 11 June 2015

Academic Editor: Ruben Specogna

Copyright © 2015 R. Naz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper analyzes the first integrals and exact solutions of mathematical models of epidemiology via the partial Lagrangian approach by replacing the three first-order nonlinear ordinary differential equations by an equivalent system containing one second-order equation and a first-order equation. The partial Lagrangian approach is then utilized for the second-order ODE to construct the first integrals of the underlying system. We investigate the SIR and HIV models. We obtain two first integrals for the SIR model with and without demographic growth. For the HIV model without demography, five first integrals are established and two first integrals are deduced for the HIV model with demography. Then we utilize the derived first integrals to construct exact solutions to the models under investigation. The dynamic properties of these models are studied too. Numerical solutions are derived for SIR models by finite difference method and are compared with exact solutions.

#### 1. Introduction

Epidemiology has become an exciting area for the modern application of mathematics. Mathematical models play a vital role in analyzing the spread and control of different diseases. Graunt [1] was the first scientist who tried to quantify causes for death in 1662 and this analysis led to the foundation of modern epidemiological theory. The first epidemiological mathematical model describing infectious disease was proposed by the Swiss mathematician Bernoulli in 1760. Bernoulli studied the impact of immunization with cowpox upon the expectation of life of the immunized population and argued the importance of variolation [2, 3].

In 1927, Kermack and McKendrick [4] developed infectious disease models to study the Great Plague of London for the period of 1665-1666. These models served as the foundation of theoretical models in epidemiology [5]. The population is divided into three categories (compartments) as susceptible, infected, and recovered [4] and the model is called the SIR model. Later on, Brauer [6] formulated the SIR model which included births and deaths. The systems of nonlinear differential equations arising from the SIR models represent the Lotka-Volterra type equations [7–9]. In 1981, the human immunodeficiency virus (HIV) appeared which causes the sexually transmitted disease, namely, the acquired immunodeficiency syndrome (AIDS). The results reported in the analysis of HIV transmission in San Francisco [10] were replicated through a mathematical model developed by Anderson [11]. Song et al. [12] looked at the transmission dynamics of the tuberculosis model while Song et al. [13] discussed the global dynamics of the tuberculosis model with a density dependent demographic growth. A tremendous variety of models have been formulated for infectious diseases [14]. The study of mathematical models of epidemiology is essential in order to uncover the essential aspects of infectious diseases spread and helps public health decision makers to compare, plan, evaluate, and implement different control programs [14, 15].

Most of the mathematical models in epidemiology give rise to a system of linear and nonlinear first-order ordinary differential equations (ODEs) and several qualitative and quantitative techniques have been applied to discuss the dynamic properties of these models. One branch of the literature has applied numerical methods to study the dynamics of these models which include [16–19]. A separate strand of the literature has looked at the dynamics of these models through the stability analysis of the dynamical systems [20–22]. We apply Lagrangian methods to study the additional features of first integrals and reductions relating to the dynamics of these models. A small number of papers have utilized Lie symmetry methods for epidemiology models [23–30]. The optimal control problems arising in epidemics can be solved by the newly developed partial Hamiltonian approach [31].

Any system of first-order ODEs possesses an infinite number of Lie point symmetries and in order to reduce the number of symmetries from infinite to finite, the Lie group theorists replace this system by a system containing a second-order ODE [30]. Nucci and Leach [23] derived an explicit solution of the SIS model by the Painlevé analysis and the Lie theory. The second-order ODE for the SIS model possesses only one trivial Lie point symmetry and thus hidden Lie symmetries were computed by increasing the order of the second-order ODE by the Riccati type transformation [23]. Nucci [24] has shown that the same results for the SIS model can be derived by using the Jacobi last multiplier approach. The Jacobi last multiplier yields a transformation which converts the system of first-order ODEs for the SIS model to a coupled system of one first-order and one second-order ODE which admits more Lie symmetries. The Lie group analysis of a mathematical model which describes HIV transmission was performed by Torrisi and Nucci [25], Nucci [26], and Edwards and Nucci [27]. Leach and Andriopoulos [28] utilized the Lie group method to predict the cause of infectious diseases and compare the effects of different control strategies for SARS (Severe Acute Respiratory Syndrome) epidemic of 2002-2003. Lie integrable cases of the simplified multistrain/two-stream model for tuberculosis and dengue fever were discussed by Nucci and Leach [29]. Nucci and Leach [30] performed the singularity and symmetry analysis of the SIR model with and without demographic growth, the model describing the transmission of HIV, and the model describing the transmission of tuberculosis with and without demographic growth. To the best of our knowledge the direct search for first integrals for mathematical models in epidemiology has not yet been considered in the literature and is the subject of this work. We apply the partial Lagrangian approach which was developed in [32, 33] to construct first integrals. This approach is similar to the Noether method and works for even order ODEs. However, one requires a partial Lagrangian instead of the standard Lagrangian. The partial Noether condition which is similar to the Noether condition is then invoked for each of the partial Noether operators in order to explicitly (by means of a formula) determine the corresponding first integrals. The approach is useful since it uses a partial Lagrangian which is easier to work out than the standard Lagrangian which may be not possible or difficult to obtain (see the examples in [33]). We show its effectiveness here. Then, we utilize the derived first integrals to obtain exact solutions of the epidemiological models under consideration.

The outline of this paper is as follows. In Section 2, we give an overview of the partial Lagrangian approach. Section 3 deals with the first integrals of the SIR model without and with demography which are derived via the partial Lagrangian approach. We utilize these first integrals to find the reductions and exact solutions of the model. In Section 4, we study several features associated with the dynamics of the HIV model. Finally, concluding remarks are presented in Section 5.

#### 2. Preliminaries

We now provide an overview of the partial Lagrangian approach as proposed in [32, 33].

Consider a th-order ODE system,where is the independent variable, , are the dependent variables, and , is the th total derivative of with respect to the independent variable .

The total derivative operator with respect to in this context is defined asso that .

The following are the basic operators defined in the space which is the vector space of differential functions.

The Lie-Bäcklund operator is given bywherein which .

The Euler-Lagrange operator is defined by

The characteristic form of the Lie-Bäcklund operator (3) iswhere is the* Lie characteristic function*

The partial Noether method for the construction of first integrals (see [32, 33]) is applied to even order ODEs that have no Lagrangian or ODEs which satisfy a partial Euler-Lagrangian system. This method also applies to even order ODEs for which a Lagrangian is sometimes difficult to determine. For example, the second-order ODE (19) discussed at length below is of even order two and has a standard Lagrangian which is not obvious. However a partial Lagrangian for it is rather simple and one can take it to be . For odd order ODEs, such as a third-order ODE, one can only use the Noether or partial Noether method if one converts this ODE to fourth order by introducing a potential variable, for example, , for , or a system containing a second- and first-order ODE. For the system of three first-order ODEs studied herein we convert this system to one which has a second-order ODE (see below).

*Definition 1 (partial Lagrangian). *Suppose that the th-order differential system (1) can be expressed asA function, , if it exists, is known as a partial Lagrangian for system (1) ifprovided for some . Here is an invertible matrix. If one has that , then one has the standard Lagrangian and Noether’s theorem is applicable.

*Definition 2 (partial Noether operator [33]). *The operator as in (3), which satisfieswhere is the gauge term, is referred to as a partial Noether operator corresponding to the partial Lagrangian . In the special case when one has the Noether operator which gives the divergence invariance of the functional associated with the standard Lagrangian .

Theorem 3 (partial Noether first integral [33]). *The first integral of system (1) associated with the partial Noether operator corresponding to the partial Lagrangian is determined from [33]**for suitable gauge function or in expanded form by**where is the characteristic of the conservation law related to this first integral .*

#### 3. The First Integrals and Closed Form Solutions of SIR Model without and with Demography

In this section, we study the SIR model first analyzed by Kermack and McKendrick [4] and this model may be applied to diseases which cause permanent immunity after recovering from the disease. The model is applicable to diseases like diphtheria, typhoid fever, measles, mumps, smallpox, and chickenpox. We analyze the two variants of the model: the SIR model without demography and the SIR model with demography. The partial Lagrangian approach is utilized to obtain the first integrals for the SIR model with and without demographic growth. We utilize these first integrals to find the exact solutions of the SIR model with and without demographic growth.

##### 3.1. The First Integrals and Exact Solutions of SIR Model without Demography

In this model a fixed population (no demography) is considered and is divided into three compartments, namely, susceptible , infected , and recovered such that . The SIR model without demography describing the effects of the Black Death in the seventeenth century is represented by the following dynamical system of ODEs:with the initial conditions , , , , and and where dot denotes differentiation with respect to the time and and denote the infection and the recovery rates, respectively. If we add (13)–(15), we haveThe model is consistent with and thus the total population is an arbitrary positive constant of integration. Notice that occurs only in the last equation and hence the equation can indeed be omitted asFrom (13), we haveand this converts (14) to the following second-order ODE in :

###### 3.1.1. First Integrals via Partial Lagrangian Approach

We apply the partial Lagrangian approach to construct first integrals of the second-order ODE (19) and we utilize these first integrals to deduce exact solutions for . Once is known to us then can be computed from (18) and finally .

Equation (19) admits the partial Lagrangian and the corresponding partial Euler-Lagrange equation iswhere

The partial Noether operator corresponding to satisfies (10); that is,where , , and . Now we separate (22) with respect to powers of and we obtain the following overdetermined system for , , and :

The solution of system (23) yields the following partial Noether operators and gauge terms:Formula (12), with , , and from (24), results in the following two first integrals:Notice that the integrals are connected byand thus only one first integral is functionally independent.

###### 3.1.2. The Exact Solutions of SIR Model

Now we utilize and derive the exact solutions to (19). Setting , we havewhich is integrable by quadratureswhere and are arbitrary constants. Now, the solution for the infected can be derived from (18) and finally from (17) is given byAn interesting solution exists for the special case when and thus (28) gives rise to the following implicit solution:Exact solution (31) is new and not reported in the literature before. The Painlevé analysis and the Lie symmetry analysis of the SIR model were performed by Leach and Andriopoulos [28]. It is noted that the exact solutions of the SIR model with no demography can be utilized as benchmarks for numerical solutions that are deduced for this model.

###### 3.1.3. The Comparison of Exact and Numerical Solutions of SIR Model

Now we compare exact solutions (31) with the numerical integration of system (13)–(15). We take initial values as , , and , respectively, and as adopted in [17]. The values of arbitrary constants are and . We have employed finite difference method in order to solve system (13)–(15) numerically. In particular, we have used backward finite differences to approximate the first-order derivatives which gives us an implicit scheme for the solution of equation system. For a fixed time step , we have partitioned the time domain as follows:

System (13)–(15) follows implicit scheme as follows:with the initial conditionsIn order to implement the aforementioned scheme, we have developed a code in* MATLAB* and used time nodes to execute the simulations. The exact and numerical solutions of and are graphically represented by Figures 1 and 2. It can be observed that the numerical results are in complete agreement with the exact solution curves.