A Partial Lagrangian Approach to Mathematical Models of Epidemiology

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 secondorder 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 HIVmodels. 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.


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][8][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 2 Mathematical Problems in Engineering 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][17][18][19].A separate strand of the literature has looked at the dynamics of these models through the stability analysis of the dynamical systems [20][21][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][24][25][26][27][28][29][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 firstorder 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.

Preliminaries
We now provide an overview of the partial Lagrangian approach as proposed in [32,33].
The total derivative operator with respect to  in this context is defined as so that   (  ) = q  , . . .,   ( (−1) ) =  () .The following are the basic operators defined in the space A which is the vector space of differential functions.
The Lie-Bäcklund operator  is given by where in which   0 ≡   .The Euler-Lagrange operator is defined by The characteristic form of the Lie-Bäcklund operator (3) is where   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  = Ṡ 2 /2.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   = 0, 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 2 (partial Noether operator [33]).The operator  as in (3), which satisfies where  is the gauge term, is referred to as a partial Noether operator corresponding to the partial Lagrangian .In the special case when /  = 0 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]  =  () −  (11) for suitable gauge function  or in expanded form by where   is the characteristic of the conservation law related to this first integral .

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.

The First Integrals and Exact
with the initial conditions (0) =  1 ≥ 0, (0) =  2 ≥ 0, (0) =  3 ≥ 0, 0 ≤  ≤ 1, and 0 ≤  ≤ 1 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 have The 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 as From ( 13), we have and this converts (14) to the following second-order ODE in : Mathematical Problems in Engineering 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  = Ṡ 2 /2 and the corresponding partial Euler-Lagrange equation is where The partial Noether operator  = / + / corresponding to  satisfies (10); that is, where  = (, ),  = (, ), and  = (, ).Now we separate (22) with respect to powers of Ṡ 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 by and thus only one first integral is functionally independent.

The Exact Solutions of SIR
Model.Now we utilize  1 and derive the exact solutions to (19).Setting  1 =  1 , we have which is integrable by quadratures where  1 and  2 are arbitrary constants.Now, the solution for the infected  can be derived from ( 18) and finally  from ( 17) is given by An interesting solution exists for the special case when  = 0 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.

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  1 = 20,  2 = 15, and  3 = 10, respectively, and  = 0.01 as adopted in [17].The values of arbitrary constants are  1 = −0.35 and  2 = 0.0214.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 [0, 10] as follows: System ( 13)-( 15) follows implicit scheme as follows: with the initial conditions In order to implement the aforementioned scheme, we have developed a code in MATLAB and used  = 200 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.

The First Integrals and Exact Solutions of SIR Model with
Demography.In many infectious diseases new susceptible agents enter into the population and therefore deaths must be included in the model (this means we consider the model with demography).Assume that the death rate at any epidemiological compartment is  ≥ 0 and a birth rate equals the death rate, thus ensuring that the total population size is constant .The SIR model, including births and deaths, is represented by the following dynamical system of ODEs: with the initial conditions (0) =  1 ≥ 0, (0) =  2 ≥ 0, (0) =  3 ≥ 0, 0 ≤  ≤ 1, and 0 ≤  ≤ 1 and where dot denotes differentiation with respect to the time and  and  denote the infection and the recovery rates, respectively.As adopted in the standard literature, if we add (35)-(37) and integrate the resultant equation, this gives where  0 is the arbitrary constant of integration and it must be chosen as zero to ensure that the total population size is constant, namely, .We note that  occurs only in the last equation and hence the  equation can be omitted by From (36), we have

First Integrals via Partial Lagrangian Approach.
We apply the partial Lagrangian approach to construct first integrals of the second-order ODE (41).We utilize these first integrals to find exact solutions for  and then  can be computed from (40) and finally we have () = −()−().
The partial Euler-Lagrange equation corresponding to the partial Lagrangian  = İ 2 /2 of (41) is given by where If we assume that  = (,),  = (, ), and  = (, ), then the partial Noether determining equation (10) corresponding to  after separation with respect to powers of İ results in the following overdetermined system for , , and : The solution of the first three equations of system (44) yields The last equation of system (44), with knowledge of , , and  from system (45), gives  = 0 and where  1 ,  2 , and  3 are arbitrary constants and without loss of generality we can choose  3 = 0.The unknown , , and  from system (45) with the help of (47) result in Finally we determine the following two first integrals with  = 0: For the case  ̸ = 0, no first integral exists.Notice that the integrals are connected by and therefore only one first integral is functionally independent.

The Exact Solutions and Transition Dynamics of SIR
Model with Demographic Growth.Now we utilize  1 and derive the exact solutions to (41) for the  = 0 case.Setting  1 =  1 , we have and this can be rewritten as Equation ( 51) is the familiar Bernoulli differential equation.Equation (40) with the aid of (51) takes the following form: Finally,  from (39) gives An interesting particular solution can be derived if we set  1 = 0. Then the implicit solution of Bernoulli's equation ( 51) is given by In this case  = 0 and  takes the following form: The time derivative of (52) with  1 = 0 yields It is clear from (56) that the growth rates of susceptible and infected agents in the population are negatively related.This makes true sense because if the number of susceptible agents in the population increases, then the number of infected agents in population will decrease and vice versa.This is an interesting result which has not been pointed out before in the literature.
The exact solutions obtained can be used as benchmarks for any numerical solution to this SIR model with demographic growth.

The Comparison of Exact and Numerical Solutions of SIR
Model with Demography.Now we compare exact solutions (54) and (55) with the numerical integration of system (35)-(37).We take initial values as  1 = 15,  2 = 10, and  3 = 0, respectively, and  = 0.01 and  = 0.20 as adopted in [17].The values of arbitrary constants are  1 = 0 and  2 = −0.1.Similar procedures as those we used for SIR model without demography have been employed for system equations ( 35)-(37) in order to obtain the numerical simulations and compared with the exact solution curves in Figures 3 and 4. The exact and numerical solutions overlap.

An HIV-Transmission Model with and without Demography
In this section, we derive the first integrals and exact solutions of the important model proposed by Anderson [11], which describes the HIV transmission in male homosexual/bisexual cohorts.The partial Lagrangian approach is employed to obtain the first integrals for the HIV model with and without demography.We utilize these first integrals to find the exact solutions of HIV model with and without demographic growth.model is represented by the following system of nonlinear ODEs:

An HIV-Transmission
where  is AIDS related death, 1/] is the average incubation period for infected individual to develop AIDS, and  is the average probability that an infected individual will infect a susceptible partner.We can express  3 () from (57) in terms of  1 () and  2 () and thus we have Adding ( 57) and (58) results in Equation ( 59) with the help of ( 60) and (61) yields the following second-order ODE in  1 (): First-order system (57)-( 59) is equivalent to coupled system (61) and (62) but no first integrals exist for this system unless  = .Equation (62) for  =  case reduces to the following second-order ODE in terms of  1 (): 4.1.1.First Integrals.We derive the first integrals of (63) by the partial Lagrangian approach.Then we find exact solutions for  1 with the help of these first integrals.Equation (61) results in  2 and finally (60) gives  3 .Equation (63) admits the partial Lagrangian  = u 2 1 /2 and the corresponding partial Euler-Lagrange equation is where If we assume that  = (, 1 ),  = (, 1 ), and  = (,  1 ), then the partial Noether operators determining equation after separation with respect to powers of u 1 results in the following system: The solution of system (66) provides the following partial Noether operators and gauge terms: The first integral with , , and  from (67) can be written as follows: ) . (68) We easily notice that and we can take  1 and  3 as functionally independent first integrals for second-order ODE (63).There is an additional dependence of  5 on  1 and  3 .

The Exact Solutions and Transition Dynamics of HIV
Model.Now we utilize the derived first integrals to construct solutions of second-order ODE (63) which eventually yields the solutions to the dynamical system of first-order ODEs (57)-(59).
Setting  1 =  1 , we have and this gives Equation (61), after substituting  1 from (71), gives the following solution: and finally  3 can be computed from (60).If we choose  2 = 0 then the solutions for  2 and  3 take the simpler forms and we obtain the following solutions: (73) We derive the same solutions from the other first integrals as are obtained from the first integral  1 .
It is worthy to mention here that if we use one first integral to construct a solution, then we obtain a Bernoulli differential equation.The solution of Bernoulli's differential equation yields the solution of our second-order ODE.We can obtain the same solution by using the two functionally independent first integrals  1 and  3 .In this event no integration is required in order to obtain the solution.

An HIV-Transmission Model with Demography.
The model is represented by the following nonlinear ODEs [26,30]: where  1 () is that portion of the population which is HIV negative,  2 () is that portion of the population which is HIV positive, and  3 () is the portion of the population which has AIDS.The parameter  is the death rate from other causes,  is the death rate from AIDS, ] is the rate at which HIV positives develop AIDS,  is the rate of infection, and  is the rate of change of the partner.We can express  3 () from (74) in terms of  1 () and  2 () and thus we have Adding ( 74) and (75) results in Equation ( 76) with the help of ( 77) and (78) gives the following second-order ODE in  1 () provided  =  +  (see [26]): We derive the first integrals of (79) by the partial Lagrangian approach.Then we find exact solutions for  1 with the aid of these first integrals.Equation (78) results in  2 and, finally, (77) gives  3 .
If we assume that (,  1 ), (,  1 ), and (,  1 ), then the partial Noether operators determining equation after separation with respect to powers of u 1 results in the following system: (81)

Figure 3 :
Figure 3: Exact and numerical solutions for () for SIR model with demography.

Figure 4 :
Figure 4: Exact and numerical solutions for () for SIR model with demography.