Weak Equivalence Transformations for a Class of Models in Biomathematics

Abstract

A class of reaction-diffusion systems unifying several Aedes aegypti population dynamics models is considered. Equivalence transformations are found. Extensions of the principal Lie algebra are derived for some particular cases.

1. Introduction

In this paper we focus our attention on the following class of nonlinear advection-reaction-diffusion systems: 𝑢_𝑡=𝑓(𝑢)𝑢_𝑥𝑥+𝑔𝑢,𝑣,𝑢_𝑥,𝑔_𝑢𝑥𝑣≠0,_𝑡=ℎ(𝑢,𝑣). (1) These systems can describe the evolution of the densities 𝑢 and 𝑣 of two interacting populations where the balance equation for 𝑢 takes into account not only the reaction-diffusion effects but also some advection effects while the balance equation for density 𝑣 takes into account only the so-called reaction terms. The advection effects are due to the presence in the function 𝑔 of the gradient 𝑢_𝑥 and appear when the individuals of population 𝑢 feel external stimuli as, for instance, wind effects or water currents. Class (1) can be considered a generalization of the equations with the typical properties of the already known Aedes aegypti mathematical models [1–3]. We recall them shortly in the following.

The Aedes aegypti mosquitos are the main vector of dengue, a viral disease that causes the so-called dengue hemorrhagic fever characterized by coagulation problems often leading the infected individual to death. Since the subtropical zone climate and environmental conditions are favorable to the development of Aedes aegypti, dengue is a serious public health problem in many countries around the world. However, due to the global warming, the interest in such considered mosquitoes is not restricted to those places affected by the disease, but it is also of interest for those countries whose weather, in the next decades, can become similar to the current environmental found in the subtropical zone. Therefore, the interest in modeling such a vector is not only a theoretical deal but also a way for finding methods and alternatives to overcome and control the problems arising from the dispersal dynamics of the mosquitos and, consequently, the propagation of the disease.

The following system 𝑢_𝑡=𝑢^𝑝𝑢_𝑥𝑥−2𝜈𝑢^𝑞𝑢_𝑥+𝛾𝑘𝑣(1−𝑢)−𝑚₁𝑣𝑢,_𝑡=𝑘(1−𝑣)𝑢−𝑚₂+𝛾𝑣, (2) where 𝑝,𝑞∈ℝ, belongs to class (1). It was introduced in [2] as a generalization of a model studied in [1].

We recall that in (2), as well as in [1], 𝑢 and 𝑣 are, respectively, nondimensional densities of winged and aquatic populations of mosquitoes and 𝑘, 𝛾, 𝑚₁, and 𝑚₂ are nondimensional, in general, positive parameters, 𝜈∈ℝ. Specifically 𝑘 is the ratio between two constants 𝑘₁ and 𝑘₂, which are, respectively, the carrying capacity related to the amount of findable nutrients and the carrying capacity effect dependent on the occupation of the available breeder and 𝛾 denotes the specific rate of maturation of the aquatic form into winged female mosquitoes, while 𝑚₁ and 𝑚₂ are, respectively, the mortality of winged and the mortality of aquatic populations. Finally 𝜈 denotes a constant of velocity for flux due to wind currents that, in general, generate an advection motion of large masses of the winged population and consequently can facilitate a quick advance of the infestation. For further details, see [4] and references therein.

Another system belonging to (1) is 𝑢_𝑡=𝑢^𝑝𝑢_𝑥𝑥−2𝜈𝑢^𝑞𝑢_𝑥+𝛾𝑘𝛾𝑣+𝑘−𝑚₁𝑣𝑢,_𝑡=𝑘𝑢+𝑘−𝑚₂−𝛾𝑣, (3) which was also introduced in [2] starting from (2) and, due to the weak interaction between the aquatic and winged populations, by modifying the source terms.

The first equation of (3) gives the time rate of change of the mosquitoes density as a sum of the growth terms (𝛾/𝑘)𝑣, of the per capita death rate ((𝛾/𝑘)−𝜇₁)𝑢 and the diffusive-advective flux due to the movement of mosquito population.

The second equation gives the corresponding time rate of change of the density of aquatic population as a sum of the growth term 𝑘𝑢 of aquatic population, due to the new egg depositions of female mosquitoes, with per capita death rate (𝑘−𝜇₂)𝑣 of aquatic population. The term −𝛾𝑣 represents the loss due to the change of the aquatic into winged form; see [4].

The family of system (1) contains arbitrary functions or numerical parameters, which specifies the individual characteristics of phenomena belonging to large subclasses. In this sense, the knowledge of equivalence transformations can provide us with certain relations between the solutions of different phenomena of the same class and allows us to get symmetries in a quite direct way.

Following [5], an equivalence transformation is a nondegenerated change of independent and dependent variables 𝑡, 𝑥, 𝑢, and 𝑣 into 𝑡̂, 𝑥, 𝑢, and 𝑣: ̂𝑣,̂𝑣,̂𝑣,̂𝑣,𝑥=𝑥𝑥,𝑡,𝑢,𝑡=𝑡𝑥,𝑡,𝑢,𝑢=𝑢𝑥,𝑡,𝑢,𝑣=𝑣𝑥,𝑡,𝑢, (4) which maps a system of class (1) in another one of the same class, that is, in an equation preserving differential structure but, in general, with 𝑓(𝑢)≠𝑓(𝑢),𝑔𝑢,𝑣,𝑢𝑥≠𝑔𝑢,𝑣,𝑢_𝑥,ℎ(𝑢,𝑣)≠ℎ(𝑢,𝑣). (5)

Of course in the case 𝑓(𝑢)=𝑓(𝑢),𝑔𝑢,𝑣,𝑢𝑥=𝑔𝑢,𝑣,𝑢_𝑥,ℎ(𝑢,𝑣)=ℎ(𝑢,𝑣) (6) an equivalence transformation becomes a symmetry.

In this paper we look for certain equivalence transformations for the class of systems (1) in order to find symmetries for special systems belonging to (1) and to get information about constitutive parameters 𝑓, 𝑔, and ℎ appearing there. Moreover we wish to stress that, as it is known, an equivalence transformation maps solutions of an equation in solutions of the transformed equation [6]. Then in order to find solutions for a certain equation one can look for the equivalence transformations that bring the equation in simpler other ones whose solutions are well studied; see, for example, [7, 8] and references inside.

The plan of the paper is as follows. In the next section we provide some elements about equivalence transformations. In Section 3 we apply these concepts in order to obtain a set of weak equivalence generators. In Section 4, after having introduced a projection theorem, we show how to apply it to find symmetries of (1). In Section 5, after having introduced a special structure of the advection-reaction function 𝑔 that generalizes that one used in (3), we find extensions with respect to the principal Lie algebra. Conclusions and final remarks are given in Section 6.

2. Elements on Equivalence Transformations

In the past differential equation literature it is possible to find several examples of equivalence transformations. The direct search for the most general equivalence transformations through the finite form of the transformation is connected with considerable computational difficulties and quite often leads to partial solutions of the problem (e.g., [9, 10]).

A systematic treatment to look for continuous equivalence transformations by using the Lie infinitesimal criterion was suggested by Ovsiannikov [11].

In general, the equivalence transformations for class (1) can be considered as transformations acting on point of the basic augmented space𝐴≡𝑡,𝑥,𝑢,𝑣,𝑢_𝑡,𝑢_𝑥,𝑣_𝑡,𝑣_𝑥,𝑓,𝑔,ℎ. (7)

The previous elements allow us to consider, in the following, the one-parameter equivalence transformations as a group of transformations, acting on the basic augmented space  𝐴, of the type ̂,̂,̂,̂,̂𝑢̂𝑡,𝑢𝑥,𝑣𝑡,̂𝑣𝑥,,̂𝑢̂𝑡,𝑢𝑥,𝑣𝑡,̂𝑣𝑥,,̂𝑢̂𝑡,𝑢𝑥,𝑣𝑡,̂𝑣𝑥,,𝑥=𝑥𝑥,𝑡,𝑢,𝑣,𝜀𝑡=𝑡𝑥,𝑡,𝑢,𝑣,𝜀𝑢=𝑢𝑥,𝑡,𝑢,𝑣,𝜀𝑣=𝑣𝑥,𝑡,𝑢,𝑣,𝜀𝑓=𝑓𝑥,𝑡,𝑢,𝑣,𝑓,𝑔,ℎ,𝜀𝑔=𝑔𝑥,𝑡,𝑢,𝑣,𝑓,𝑔,ℎ,𝜀ℎ=ℎ𝑥,𝑡,𝑢,𝑣,𝑓,𝑔,ℎ,𝜀 (8) which is locally a 𝐶^∞-diffeomorphism, depending analytically on the parameter 𝜀 in a neighborhood of 𝜀=0, and reduces to the identity transformation for 𝜀=0.

Following [6, 11–14] (see also, e.g., [5, 15–17] ) we consider the infinitesimal generator of the equivalence transformations (8) of the systems (1) that reads as follows: 𝑌=𝜉¹𝜕_𝑥+𝜉²𝜕_𝑡+𝜂¹𝜕_𝑢+𝜂²𝜕_𝑣+𝜇¹𝜕_𝑓+𝜇²𝜕_𝑔+𝜇³𝜕_ℎ, (9) where the infinitesimal components 𝜉¹, 𝜉², 𝜂¹, and 𝜂² are sought depending on 𝑥, 𝑡, 𝑢, and 𝑣, while the infinitesimal components 𝜇^𝑖(𝑖=1,2,3) are sought, at least in principle, depending on 𝑥, 𝑡, 𝑢, 𝑣, 𝑢_𝑡, 𝑢_𝑥, 𝑢_𝑥, 𝑣_𝑡, 𝑣_𝑥, 𝑓, 𝑔, and ℎ. In order to obtain the determining system which allows us to get the infinitesimal coordinates 𝜉^𝑖, 𝜂^𝑖, and 𝜇^𝑗 (𝑖=1,2 and 𝑗=1,2,3), we apply the Lie-Ovsiannikov infinitesimal criterion by requiring the invariance, with respect to suitable prolongations 𝑌⁽¹⁾ and 𝑌⁽²⁾ of (9), of the following equations: 𝑢_𝑡−𝑓𝑢_𝑥𝑥𝑣−𝑔=0,_𝑡−ℎ=0, (10) together with the invariance of the auxiliary conditions [13, 14, 18, 19]: 𝑓_𝑡=𝑓_𝑥=𝑓_𝑣=𝑓_𝑢𝑥=𝑓_𝑢𝑡=𝑓_𝑣𝑥=𝑓_𝑣𝑡=𝑔_𝑡=𝑔_𝑥=𝑔_𝑢𝑡=𝑔_𝑣𝑡=𝑔_𝑣𝑥ℎ=0,_𝑡=ℎ_𝑥=ℎ_𝑢𝑥=ℎ_𝑢𝑡=ℎ_𝑣𝑥=ℎ_𝑣𝑡=0, (11) where 𝑢 and 𝑣 are (𝑡,𝑥) functions while 𝑓, 𝑔, and ℎ are considered as functions depending, a priori, on (𝑡,𝑥,𝑢,𝑣,𝑢_𝑡,𝑢_𝑥,𝑣_𝑡,𝑣_𝑥). All of these functions are assumed to be analytical. The constraints, given by (11), characterize the functional dependence of 𝑓, 𝑔, and ℎ.

In this paper, instead, in view of further applications and following [20], we modify the previous classical procedure by looking for equivalence transformations whose generators are got by solving the determining system obtained from the following invariance conditions: 𝑌⁽²⁾[𝑢_𝑡−𝑓(𝑢)𝑢_𝑥)_𝑥−𝑔𝑢,𝑣,𝑢_{𝑥𝑢𝑡−(𝑓(𝑢)𝑢𝑥)𝑥−𝑔(𝑢,𝑣,𝑢𝑥𝑣)=0,𝑡−ℎ(𝑢,𝑣)=0}𝑌=0,₍₁₎𝑣_𝑡−ℎ(𝑢,𝑣)_{𝑢𝑡−(𝑓(𝑢)𝑢𝑥)𝑥−𝑔(𝑢,𝑣,𝑢𝑥𝑣)=0,𝑡−ℎ(𝑢,𝑣)=0}=0. (12)(13) As the functional dependences of the parameters 𝑓, 𝑔, and ℎ are known a priori we do not require the invariance of the auxiliary conditions (11). In this way we work in a basic augmented space  𝐴≡{𝑡,𝑥,𝑢,𝑣,𝑢_𝑥,𝑓,𝑔,ℎ}. Therefore the 𝜇^𝑖 components must be sought, at least in principle, depending on 𝑥, 𝑡, 𝑢, 𝑣, 𝑢_𝑥, 𝑓, 𝑔, and ℎ.

The infinitesimal operators, obtained by following this shortening procedure, can generate transformations that map equations of our class into new equations of the same class where the transformed arbitrary functions may have new additional functional dependencies. Such transformations are called weak equivalence transformations [13, 14].

With respect to the application in biomathematical models, equivalence and weak equivalence transformations were applied not only to study of tumor models [21, 22] but also to the population dynamics in [20, 23, 24].

3. Calculation of Weak Equivalence Transformations

In order to avoid long formulas and write 𝑌⁽¹⁾ and 𝑌⁽²⁾ in a compact way, we put 𝑥=𝑥¹,𝑡=𝑥²,𝑢=𝑢¹,𝑣=𝑢²,𝑓=ℎ¹,𝑔=ℎ²,ℎ=ℎ³. (14) For this reason system (1) is rewritten as 𝑢¹_𝑥2−ℎ²−ℎ¹_𝑢1𝑢¹_𝑥1²−ℎ¹𝑢¹_𝑥1𝑥1𝑢=0,_2𝑥2−ℎ³=0, (15) while the equivalence generator assumes the following form: 𝑌=𝜉^𝑖𝜕_𝑥𝑖+𝜂^𝛼𝜕_𝑢𝛼+𝜇^𝐴,𝜕_ℎ^𝐴, (16) where 𝑖=1,2, 𝛼=1,2, and 𝐴=1,2,3. Here the summation over the repeated indices is presupposed.

After putting 𝑧¹,𝑧²,𝑧³,𝑧⁴,𝑧⁵=𝑥¹,𝑥²,𝑢¹,𝑢²,𝑢¹₁,𝜈¹,𝜈²,𝜈³,𝜈⁴,𝜈⁵=𝜉¹,𝜉²,𝜂¹,𝜂²,𝜁¹₁,𝑢^𝛼_𝑖=𝑢^𝛼_𝑥𝑖,ℎ^𝐴_𝑎=ℎ^𝐴_𝑧𝑎,𝐷^𝑒_𝑎=𝜕_𝑧𝑎+ℎ^𝐴_𝑎𝜕_ℎ^𝐴,𝐷^𝑒_𝑗=𝜕_𝑥𝑗+𝑢^𝛼_𝑗𝜕_𝑢𝛼+𝑢^𝛼_𝑖𝑗𝜕_𝑢^𝛼_𝑖+⋯, (17)(18)(19)(20) the prolongations 𝑌⁽¹⁾ and 𝑌⁽²⁾ assume the following form: 𝑌⁽¹⁾=𝑌+𝜁^𝛼_𝑗𝜕_𝑢^𝛼_𝑗+𝜔^𝐴_𝑎𝜕_ℎ^𝐴_𝑎,𝑌⁽²⁾=𝑌⁽¹⁾+𝜁¹₁₁𝜕_𝑢¹₁₁=𝑌+𝜁^𝛼_𝑗𝜕_𝑢^𝛼_𝑗+𝜔^𝐴_𝑎𝜕_ℎ^𝐴_𝑎+𝜁¹₁₁𝜕_𝑢¹₁₁, (21) where 𝜁^𝛼_𝑗=𝐷^𝑒_𝑗𝜂^𝛼−𝑢^𝛼_𝑘𝐷^𝑒_𝑗𝜉^𝑘,𝜁^𝛼_𝑖𝑗=𝐷^𝑒_𝑗𝜁^𝛼_𝑗−𝑢^𝛼_𝑖𝑘𝐷^𝑒_𝑗𝜉^𝑘,𝜔^𝐴_𝑎=𝐷^𝑒_𝑎𝜇^𝐴−ℎ^𝐴_𝑏𝐷^𝑒_𝑎𝜈^𝑏. (22)

The invariant conditions read 𝑌⁽²⁾𝐹₁=𝜁¹₂−𝜇²−2𝜁¹₁ℎ¹_𝑢1𝑢¹₁−𝜔¹₃𝑢¹₁²−𝜇¹𝑢¹₁₁−ℎ¹𝜁¹₁₁𝑌=0,₍₁₎𝐹₂=𝜁²₂−𝜇³=0, (23)(24) both under the constraints (15), which are after (18) 𝑢¹₂=ℎ²+ℎ¹_𝑢1𝑢¹₁²+ℎ¹𝑢¹₁₁,𝑢²₂=ℎ³. (25)(26)

The coefficients 𝜁¹₁, 𝜁¹₂, 𝜁²₂, and 𝜔¹₃ are given, respectively, by 𝜁¹₁=𝜂¹₁+𝜂¹_𝑢1−𝜉¹₁𝑢¹₁−𝜉²₁𝑢¹₂+𝜂¹_𝑢2𝑢²₁−𝜉¹_𝑢1𝑢¹₁²−𝜉¹_𝑢2𝑢¹₁𝑢²₁−𝜉²_𝑢1𝑢¹₁𝑢¹₂−𝜉²_𝑢2𝑢¹₂𝑢²₁,𝜁¹₂=𝜂¹₂−𝜉¹₂𝑢¹₁+𝜂¹_𝑢1−𝜉²₂𝑢¹₂+𝜂¹_𝑢2𝑢²₂−𝜉¹_𝑢1𝑢¹₁𝑢¹₂−𝜉¹_𝑢2𝑢¹₁𝑢²₂−𝜉²_𝑢1𝑢¹₂²−𝜉²_𝑢2𝑢¹₂𝑢²₂,𝜁²₂=𝜂²₂+𝜂²_𝑢1𝑢¹₂+𝜂²_𝑢2−𝜉²₂𝑢²₂−𝜉¹₂𝑢²₁−𝜉¹_𝑢1𝑢²₁𝑢¹₂−𝜉¹_𝑢2𝑢²₁𝑢²₂−𝜉²_𝑢1𝑢¹₂𝑢²₂−𝜉²_𝑢2𝑢²₂²,𝜔¹₃=𝜇¹_𝑢1+ℎ¹_𝑢1𝜇¹_ℎ1+ℎ²_𝑢1𝜇¹_ℎ2+ℎ³_𝑢1𝜇¹_ℎ3−ℎ¹_𝑢1𝜂¹_𝑢1. (27)

Taking into account (24) and (27), we can write 𝑌⁽¹⁾𝐹₂=𝜂²₂+𝜂²_𝑢1𝑢¹₂+𝜂²_𝑢2−𝜉²₂𝑢²₂−𝜉¹₂𝑢²₁−𝜉¹_𝑢1𝑢²₁𝑢¹₂−𝜉¹_𝑢2𝑢²₁𝑢²₂−𝜉²_𝑢1𝑢¹₂𝑢²₂−𝜉²_𝑢2𝑢²₂²−𝜇³=0, (28) with the constraints (25) and (26).

Then, substituting (25) and (26) into (28), we get 𝜂²₂+𝜂²_𝑢1ℎ²+ℎ¹_𝑢1𝑢¹₁²+ℎ¹𝑢¹₁₁+𝜂²_𝑢2−𝜉²₂ℎ³−𝜉¹₂𝑢²₁−𝜉¹_𝑢1𝑢²₁ℎ²+ℎ¹_𝑢1𝑢¹₁²+ℎ¹𝑢¹₁₁−𝜉¹_𝑢2𝑢²₁ℎ³−𝜉²_𝑢1ℎ²+ℎ¹_𝑢1𝑢¹₁²+ℎ¹𝑢¹₁₁ℎ³−𝜉²_𝑢2ℎ³²−𝜇³=0. (29)

From this condition we obtain the following determining equations: 𝜂²₂+𝜂²_𝑢2−𝜉²₂ℎ³−𝜉²_𝑢2ℎ³²−𝜇³𝜉=0,₁₂=0,𝜉¹_𝑢1=0,𝜉¹_𝑢2𝜂=0,_2𝑢1=0,𝜉²_𝑢1=0. (30) Then it follows that 𝜉¹=𝜉¹𝑥¹,𝜉²=𝜉²𝑥¹,𝑥²,𝑢²,𝜂²=𝜂²𝑥¹,𝑥²,𝑢²,𝜂¹=𝜂¹𝑥¹,𝑥²,𝑢¹,𝑢²,𝜇³=𝜂²₂+𝜂²_𝑢2−𝜉²₂ℎ³−𝜉²_𝑢2ℎ³² (31) and, consequently, 𝜇³_𝑢11=0.

Following the same procedure we can write the invariance condition (23) as 𝑌⁽²⁾𝐹₁=𝜂¹₂+𝜂¹_𝑢1−𝜉²₂ℎ²+ℎ¹_𝑢1𝑢¹₁²+ℎ¹𝑢¹₁₁+𝜂¹_𝑢2ℎ³−𝜉²_𝑢2ℎ²+ℎ¹_𝑢1𝑢¹₁²+ℎ¹𝑢¹₁₁ℎ³−𝜇²−2𝜂¹₁+𝜂¹_𝑢1−𝜉¹₁𝑢¹₁−𝜉²₁ℎ²+ℎ¹_𝑢1𝑢¹₁²+ℎ¹𝑢¹₁₁+𝜂¹_𝑢2𝑢²₁+−𝜉²_𝑢2ℎ²+ℎ¹_𝑢1𝑢¹₁²+ℎ¹𝑢¹₁₁𝑢²₁ℎ¹_𝑢1𝑢¹₁−𝜇¹_𝑢1+ℎ¹_𝑢1𝜇¹_ℎ1+ℎ²_𝑢1𝜇¹_ℎ2+ℎ³_𝑢1𝜇¹_ℎ3−ℎ¹_𝑢1𝜂¹_𝑢1×𝑢¹₁²−𝜇¹𝑢¹₁₁−ℎ¹𝐷^𝑒₁𝜂¹₁+𝐷^𝑒₁𝜂¹_𝑢1−𝜉¹₁𝑢¹₁+𝜂¹_𝑢1−𝜉¹₁𝑢¹₁₁+𝐷^𝑒₁𝜂¹_𝑢2𝑢²₁+𝜂¹_𝑢2𝑢²₁₁−ℎ¹−𝐷^𝑒₁𝜉²₁ℎ²+ℎ¹_𝑢1𝑢¹₁²+ℎ¹𝑢¹₁₁−𝜉²₁𝑢¹₂₁−𝐷^𝑒₁𝜉²_𝑢2ℎ²+ℎ¹_𝑢1𝑢¹₁²+ℎ¹𝑢¹₁₁𝑢²₁−𝜉²_𝑢2𝑢¹₂₁𝑢²₁−𝜉²_𝑢2ℎ²+ℎ¹_𝑢1𝑢¹₁²+ℎ¹𝑢¹₁₁𝑢²₁₁−𝑢¹₁₁𝜉¹₁=0. (32)

Collecting the terms with (ℎ¹_𝑢1)² and with ℎ¹_𝑢1 and equating their respective coefficients to zero it is obtained that, 𝜉²₁=0,𝜉²_𝑢2=0,𝜂¹₁𝜇=0,_1ℎ1=2𝜉¹₁−𝜉²₂,𝜂¹_𝑢2=0𝜂¹₁=0. (33)

Thus, from the coefficients of ℎ²_𝑢1 and ℎ³_𝑢1𝜇¹_ℎ2=𝜇¹_ℎ3=0. (34) Considering the coefficients of 𝑢¹₁₁ we get 𝜇¹=2𝜉¹₁−𝜉²₂ℎ¹=0⟹𝜇¹_𝑢1=0. (35) The remaining terms of (32) give the following form for the infinitesimal component of ℎ²: 𝜇²=𝜂¹₂+𝜂¹_𝑢1−𝜉²₂ℎ²+𝜉¹₁₁−𝜂¹_𝑢1𝑢1𝑢¹₁ℎ¹. (36) Therefore, once having taken into account all restrictions obtained we are finally able to write the following infinitesimal components for the weak equivalence generators: 𝜉¹=𝛼𝑥¹,𝜉²=𝛽𝑥²,𝜂¹=𝛿𝑥²,𝑢¹,𝜂²=𝜆𝑥¹,𝑥²,𝑢²,𝜇¹=2𝛼^′−𝛽^′ℎ¹,𝜇³=𝜆_𝑢2−𝛽^′ℎ³+𝜆_𝑥2, (37) while, from (36), we get 𝜇²=𝛿_𝑥2+𝛿_𝑢1−𝛽_𝑥2ℎ²+𝛼_𝑥1𝑥1−𝛿_𝑢1𝑢1𝑢¹_𝑥1ℎ¹, (38) where 𝛼(𝑥¹), 𝛽(𝑥²), 𝛿(𝑥²,𝑢¹), and 𝜆(𝑥¹,𝑥²,𝑢²) are arbitrary real functions of their arguments. Then, going back to the original variables, the most general operator of these continuous weak equivalence transformations reads 𝑌=𝛼(𝑥)𝜕_𝑥+𝛽(𝑡)𝜕_𝑡+𝛿(𝑡,𝑢)𝜕_𝑢+𝜆(𝑥,𝑡,𝑣)𝜕_𝑣+2𝛼^′−𝛽^′𝑓𝜕_𝑓+𝛿_𝑡+𝛿_𝑢−𝛽_𝑡𝑔+𝛼^′′−𝛿_𝑢𝑢𝑢_𝑥𝑓𝜕_𝑔+𝜆_𝑣−𝛽^′ℎ+𝜆_𝑡𝜕_ℎ. (39)

4. Symmetries for the System (1)

In the next sections in order to carry out symmetries for the system (1) we do not use the classical Lie approach. Instead of the mentioned method we apply the projection theorem, introduced in [25] and eventually reconsidered in [13, 14, 18, 19]. In agreement with these references, we can affirm the following.

Theorem 1. Let 𝑌=𝛼(𝑥)𝜕_𝑥+𝛽(𝑡)𝜕_𝑡+𝛿(𝑡,𝑢)𝜕_𝑢+𝜆(𝑥,𝑡,𝑣)𝜕_𝑣+2𝛼^′−𝛽^′𝑓𝜕_𝑓+𝛿_𝑡+𝛿_𝑢−𝛽_𝑡𝑔+𝛼^′′−𝛿_𝑢𝑢𝑢_𝑥𝑓𝜕_𝑔+𝜆_𝑣−𝛽^′ℎ+𝜆_𝑡𝜕_ℎ (40) be an infinitesimal equivalence generator for the system (1); then the operator 𝑋=𝛼(𝑥)𝜕_𝑥+𝛽(𝑡)𝜕_𝑡+𝛿(𝑡,𝑢)𝜕_𝑢+𝜆(𝑥,𝑡,𝑣)𝜕_𝑣, (41) which corresponds to the projection of  𝑌 on the space (𝑥,𝑡,𝑢,𝑣), is an infinitesimal symmetry generator of the system (1) if and only if the constitutive equations, specifying the forms of 𝑓, ℎ, and 𝑔, are invariant with respect to 𝑌.

For the system under consideration, in general, the constitutive equations whose invariance must be requested are 𝑓=𝐷(𝑢),𝑔=𝐺𝑢,𝑣,𝑢_𝑥,ℎ=𝐹(𝑢,𝑣). (42) The request of invariance 𝑌𝑓−𝐷(𝑢)₍₄₂₎=0,𝑌⁽¹⁾𝑔−𝐺𝑢,𝑣,𝑢_𝑥(42)𝑌((=0,ℎ−𝐹𝑢,𝑣))|₍₄₂₎=0 (43) brings us to the following equations: 𝜇¹−𝜂¹𝐷_𝑢𝜇=0,₂−𝜁¹₁𝐺_𝑢𝑥−𝜂¹𝐺_𝑢−𝜂²𝐺_𝑣𝜇=0,₃−𝜂¹𝐹_𝑢−𝜂²𝐹_𝑣=0, (44) under the restrictions (42).

Then substituting 𝜁¹₁=𝜂¹_𝑢1−𝜉¹₁𝑢¹₁=𝛿_𝑢−𝛼^′𝑢_𝑥 (45) and taking into account the constraints we can write (44) as 2𝛼^′−𝛽^′𝐷(𝑢)−𝛿(𝑡,𝑢)𝐷_𝑢𝛿=0,_𝑡+𝛿_𝑢−𝛽_𝑡𝐺−𝛿_𝑢𝑢𝑢²_𝑥−𝛼_𝑥𝑥𝑢_𝑥𝐷−𝛿_𝑢−𝛼^′𝑢_𝑥𝐺_𝑢𝑥−𝛿𝐺_𝑢−𝜆(𝑡,𝑥,𝑣)𝐺_𝑣𝜆=0,_𝑣−𝛽^′𝐹+𝜆_𝑡−𝛿𝐹_𝑢−𝜆(𝑡,𝑥,𝑣)𝐹_𝑣=0. (46)(47)(48) We recall here that the principal Lie algebra 𝐿_𝒫 [5, 12] is the Lie algebra of the principal Lie group, that is, the group of the all Lie point symmetries ()𝜕𝑋=𝜉𝑥,𝑡,𝑢,𝑣()𝜕𝜕𝑥+𝜏𝑥,𝑡,𝑢,𝑣𝜕𝑡+𝜂¹()𝜕𝑥,𝑡,𝑢,𝑣𝜕𝑢+𝜂²()𝜕𝑥,𝑡,𝑢,𝑣𝜕𝑣 (49) that leave the system (1) invariant for any form of the functions 𝐷(𝑢), 𝐺(𝑢,𝑣,𝑢_𝑥), and 𝐹(𝑢,𝑣). In other words we can remark that the principal Lie algebra is the subalgebra of the equivalence algebra such that any operator 𝑌 of this subalgebra leaves the equations 𝑓=𝐷(𝑢), 𝑔=𝐺(𝑢,𝑣,𝑢_𝑥), and ℎ=𝐹(𝑢,𝑣) invariant for any form of the functions 𝐷(𝑢), 𝐺(𝑢,𝑣,𝑢_𝑥), and 𝐹(𝑢,𝑣). Then we can say [5] the following.

Corollary 2. An equivalence operator for the system (1) belongs to the principal Lie algebra  𝐿_𝒫 if and only if 𝜂^𝑖=0, 𝜇^𝑗=0, 𝑖=1,2, and 𝑗=1,2,3.

Taking Corollary 2 into account from the previous equations (46)-(48) it is a simple matter to ascertain that the 𝐿_𝒫 [5, 12] is spanned by the following translation generators: 𝑋₀=𝜕_𝑡,𝑋₁=𝜕_𝑥. (50)

5. Some Extensions of 𝐿_𝒫

In order to show some extensions of the principal algebra, which could be of interest in biomathematics, we assume that the advection-reaction function is of the form 𝐺=𝜌𝑢^𝑟𝑢^𝑠_𝑥+Γ₁𝑢^𝑎+Γ₂𝑣^𝑏, (51) where the parameters 𝜌, Γ₁, Γ₂, 𝑟, 𝑠, 𝑎, and 𝑏 are constitutive parameters of the considered phenomena.

This form of 𝐺 is a generalization of 𝐺=−2𝜈𝑢^𝑞𝑢_𝑥+𝛾𝑘𝛾𝑣+𝑘−𝑚₁𝑢, (52) appearing in (3), where 𝜌=−2𝜈,Γ₁=𝛾𝑘−𝑚₁,Γ₂=𝛾𝑘 (53) Consequently in (51) we must consider Γ₂>0 and as limit cases Γ₁=0 and 𝑎=0. Moreover, in this section, we assume that the value 𝑠=0 will not be considered because in this case the advective effects disappear. We also assume that 𝑏≠0. This last restriction implies that the balance equation of the density 𝑢 depends on the density 𝑣. Finally for the sake of simplicity we omit the limit case Γ₁=0 and assume that the diffusion is only nonlinear; that is, 𝐷_𝑢≠0.

In the following we continue the discussion of invariance conditions written in the previous section.

From (46), by deriving with respect to 𝑥, we get 𝛼^′′=0. (54) Then 𝛼=𝛼₁𝑥+𝛼₀ (55) with 𝛼₁ and 𝛼₀ arbitrary constants, so (46) becomes 2𝛼₁−𝛽^′𝐷(𝑢)−𝛿(𝑡,𝑢)𝐷_𝑢=0 (56) while, after having taken (51) into account, (47) reads 𝛿_𝑡+𝛿_𝑢−𝛽_𝑡𝜌𝑢^𝑟𝑢^𝑠_𝑥+Γ₁𝑢^𝑎+Γ₂𝑣^𝑏−𝐷𝛿_𝑢𝑢𝑢²_𝑥−𝛿_𝑢−𝛼₁𝜌𝑠𝑢^𝑟𝑢^𝑠_𝑥+−𝛿𝜌𝑟𝑢^𝑟−1𝑢^𝑠_𝑥+Γ₁𝑎𝑢^𝑎−1−𝜆(𝑡,𝑥,𝑣)Γ₂𝑏𝑣^𝑏−1=0. (57) From (57) we get immediately 𝜆=𝜆(𝑡,𝑣). (58)

In the following we analyze separately the case 𝑠≠2 and the case 𝑠=2.

5.1. 𝑠≠2

From 𝑢²_𝑥 coefficient we get 𝛿_𝑢𝑢=0; that is, 𝛿=𝛿₁(𝑡)𝑢+𝛿₀(𝑡). (59)

Therefore, from the remaining terms we have 𝜌𝑢^𝑟−1𝑢^𝑠_𝑥𝑢(1−𝑠−𝑟)𝛿₁+𝑠𝛼₁−𝛽^′−𝑟𝛿₀+𝛿^′₀+𝛿^′₁𝑢+Γ₁(1−𝑎)𝛿₁−𝛽^′+−𝑎Γ₁𝛿₀𝑢^𝑎−1+Γ₂𝛿₁−𝛽^′𝑣^𝑏−Γ₂𝜆𝑏𝑣^𝑏−1=0. (60)

As we assumed 𝑠≠0, from the coefficient of 𝑢^𝑠_𝑥 in (60) we conclude that 𝛿₀(𝑡)=0 and 𝛽^′=(1−𝑠−𝑟)𝛿₁+𝑠𝛼₁. (61) Then, still from (60) we have the following constraints to consider: 𝛿^′₁𝑢+Γ₁𝑢^𝑎(1−𝑎)𝛿₁−𝛽^′𝛿=0,₁−𝛽^′𝑣^𝑏−𝜆𝑏𝑣^𝑏−1. (62)(63) From (62) two cases are obtained.

(i) Case 𝑎≠1. Then, from (62) we conclude that 𝛿₁=const and it follows that 𝛽^′=(1−𝑎)𝛿₁. (64) From (64) and (61) we obtain 𝛼₁=𝑠+𝑟−𝑎𝑠𝛿₁,𝛽=(1−𝑎)𝛿₁𝑡+𝛽₀, (65) with 𝛽₀ and 𝛿₁ arbitrary constants.

The analysis of (63) leads to the following two subcases.

(1) Consider 𝜆(𝑣)=𝜆₀𝑣, with 𝜆₀=𝑎𝑏𝛿₁. (66) Taking into account the previous results and going back to (56) and (48) we get that the system (3) with 𝐺 of the form (51) admits the 3-dimensional Lie algebra spanned by the translations in space and time and by the following additional generator: 𝑋₃=(1−𝑎)𝜕_𝑡+1𝑠(𝑠+𝑟−𝑎)𝑥𝜕_𝑥+𝑢𝜕_𝑢+𝑎𝑏𝑣𝜕_𝑣, (67) provided that 𝐷 and 𝐹 are solutions of the following differential equations: 𝑢𝐷_𝑢=(1+𝑎)+2𝑟−𝑎𝑠𝐷,𝑏𝑢𝐹_𝑢+𝑎𝑣𝐹_𝑣=(𝑎−𝑏(1−𝑎))𝐹. (68)

(2) Consider 𝜆(𝑣)=𝛿₁=0. In this case the only symmetries admitted are translations of the independent variables and the form of 𝐷 and 𝐹 is arbitrary, so there is not an extension of the principal Lie algebra.

(ii) Case 𝑎=1. In this case from (62) it follows that 𝛿^′₁−Γ₁𝛽^′=0. (69) After having substituted (69) into (61) we obtain 𝛽(𝑡)=−𝛽₀+𝑐₁𝑒^{(1−𝑠−𝑟)Γ1𝑡},𝛿₁(𝑡)=𝑠𝛼𝑠+𝑟−1¹+𝑐₁Γ₁𝑒^{(1−𝑠−𝑟)Γ1𝑡}, (70) with 𝛼₁, 𝛽₀, and 𝑐₁ arbitrary constants and once assumed 𝑠+𝑟−1≠0. Finally we analyze the contribution of (63) from where the following two subcases arise.

(1)  Consider 𝜆(𝑡,𝑣)=𝜆₀(𝑡)𝑣, with 𝜆₀(𝑡)=1𝑏𝛿₁(𝑡). (71) Then taking into account the previous results it reads 𝜆₀(𝑡)=1𝑏𝑠𝛼𝑠+𝑟−1¹+𝑐₁Γ₁𝑒^{(1−𝑠−𝑟)Γ1𝑡}. (72) Going to put the previous result in the condition (46) and by separating the variable we get 𝛽(𝑡)=−𝛽₀,𝛿₁(𝑡)=𝑠𝛼𝑠+𝑟−1¹,𝜆₀(𝑡)=1𝑏𝑠𝛼𝑠+𝑟−1¹, (73) with 𝛼₁ and 𝛽₀ arbitrary constants and provided that the diffusion coefficient 𝐷 is solution of 𝑢𝐷_𝑢=21+𝑟−1𝑠𝐷. (74) From condition (48) we do not get further restrictions on the infinitesimal components of the symmetry generator but only the following constraint on the reaction function 𝐹: 𝑏𝑢𝐹_𝑢+𝑣𝐹_𝑣=𝐹. (75) Taking into account the arbitrariness of 𝛼₀ in this case we have got a 3-dimensional Lie algebra. The additional generator is 𝑋₃=𝑥𝜕_𝑥+𝑠𝑠+𝑟−1𝑢𝜕_𝑢+1𝑏𝑠𝑠+𝑟−1𝑣𝜕_𝑣. (76) In particular, from (75), setting 𝐹=𝑘𝑢+(𝑘−𝑚₂−𝛾)𝑣, we obtain 𝑏=1, while from (75) we obtain the power function 𝐷=𝐷₀𝑢^{2((𝑟+𝑠−1)/𝑠)}.

Remark 3. By putting 𝑟=1, the corresponding system of class (1) admits still a 3-dimensional symmetry Lie algebra generated by translations in time and in space and by 𝑋₃. These results are in agreement with the ones obtained in [2].

By considering 𝑠+𝑟−1=0 we do not get extension of 𝐿_𝒫 for 𝐷 and 𝐹 arbitrary. We get, instead, the following extension: 𝑋₃=𝑠𝑡𝜕_𝑡+𝑥𝜕_𝑥+Γ₁𝑠𝑡𝑢𝜕_𝑢+Γ₁𝑠𝑡𝑏𝑣𝜕_𝑣, (77) provided that 𝐷=𝐷₀=const. and 𝐹=(Γ₁/𝑏)𝑣.

(2) Consider 𝜆(𝑡,𝑣)=𝛿₁(𝑡)=0. In this case the only symmetries admitted are the translations in space and time and the form of 𝐷 and 𝐹 is arbitrary.

5.2. 𝑠=2

We analyze this case by beginning with the discussion of (56) from which two cases arise.

(1)  𝐷(𝑢) Is Arbitrary.  It follows, of course, that 𝛿=0 and 𝛽^′=2𝛼₁ so 𝛽=2𝛼₁𝑡+𝛽₀ (78) with 𝛽₀ arbitrary constant.

Moreover (46) and (48) become −2𝛼₁𝐺+𝛼₁𝑢_𝑥𝐺_𝑢𝑥−𝜆(𝑣)𝐺_𝑣𝜆=0,_𝑣−2𝛼₁𝐹−𝜆(𝑣)𝐹_𝑣=0. (79)(80) But taking into account that 𝐺=𝜌𝑢^𝑟𝑢²_𝑥+Γ₁𝑢^𝑎+Γ₂𝑣^𝑏 (81) (79) becomes −2𝛼₁𝜌𝑢^𝑟𝑢²_𝑥+Γ₁𝑢^𝑎+Γ₂𝑣^𝑏+2𝛼₁𝜌𝑢^𝑟𝑢²_𝑥−𝜆(𝑣)𝑏Γ₂𝑣^𝑏−1=0 (82) and gives us the following conditions: 2𝛼₁Γ₁=0,2𝛼₁𝑣+𝜆(𝑣)𝑏=0 (83) from where, taking into account the work hypotheses at the beginning of this section, we get 𝛼₁=𝜆=0 with 𝐹 arbitrary function of 𝑢 and 𝑣.

In this case the only admitted symmetries are translations in time and space.

(2) Consider 𝐷_𝑢/𝐷=(2𝛼₁−𝛽^′)/𝛿(𝑡,𝑢). Then by requiring 𝜕𝜕𝑡2𝛼₁−𝛽^′𝛿(𝑡,𝑢)=0, (84) we get −𝛿(𝑡,𝑢)𝛽^′′−2𝛼₁−𝛽^′𝛿_𝑡=0 (85) from where we derive

(a) 𝛿=𝛿(𝑡,𝑢) arbitrary function and 2𝛼₁−𝛽^′=0 and then 𝛽=2𝛼₁𝑡+𝛽₀ (86) and 𝐷_𝑢=0, so we omit this case,

(b) 𝛿(𝑡,𝑢)=2𝛼₁−𝛽^′𝐴(𝑢). (87) In this case 𝐷_𝑢𝐷=1𝐴(𝑢), (88) which implies 𝐷(𝑢)=𝐷₀𝑒^{∫𝑑𝑢/𝐴(𝑢)}≡𝐷₀𝑒^𝑎(𝑢), (89) where 𝑎^′(𝑢)=1𝐴(𝑢). (90) Equation (47) after (54), (58), and (81) becomes −𝛽^′′𝐴(𝑢)+2𝛼₁−𝛽^′𝐴^′(𝑢)−𝛽_𝑡𝜌𝑢^𝑟𝑢²_𝑥+Γ₁𝑢^𝑎+Γ₂𝑣^𝑏−𝛿_𝑢𝑢𝑢²_𝑥𝐷₀𝑒^{∫𝑑𝑢/𝐴(𝑢)}+−2𝛼₁−𝛽^′𝐴^′(𝑢)−𝛼₁𝑢_𝑥2𝜌𝑢^𝑟𝑢_𝑥−2𝛼₁−𝛽^′𝐴(𝑢)𝑟𝜌𝑢^𝑟−1𝑢²_𝑥+𝑎Γ₁𝑢^𝑎−1−𝜆(𝑡,𝑣)𝑏Γ₂𝑣^𝑏−1=0. (91) From terms in 𝑣 we get the following condition: 2𝛼₁−𝛽^′𝐴^′(𝑢)−𝛽^′𝑣−𝜆(𝑡,𝑣)𝑏=0, (92) which gives us 𝐴^′′(𝑢)=0 and then 𝐴=𝐴₁𝑢+𝐴₀. Therefore, from (87) 𝛿(𝑡,𝑢)=2𝛼₁−𝛽^′𝐴₁𝑢+𝐴₀, (93) which implies that 𝛿_𝑢𝑢=0. (94)

From (92) arise two cases.

(i) Consider 𝜆(𝑡,𝑣)=0 and (2𝛼₁−𝛽^′)𝐴₁−𝛽^′=0. In this case after having derived from (91) the following additional conditions 𝐴₀=0,𝛽^′′𝐴₁=0,2𝐴₁2𝛼₁−𝛽^′(2−𝑟)−2𝛼₁=0,𝐴₁2𝛼₁−𝛽^′𝑎Γ₁=0, (95) it is a simple matter to ascertain that there does not exist extension of 𝐿_𝒫.

(ii) Consider 𝜆(𝑡,𝑣)=𝜆₀(𝑡)𝑣 with 𝜆₀(𝑡)=1𝑏2𝛼₁−𝛽^′𝐴₁−𝛽^′. (96)

Then (91) assumes the following form: −𝛽^′′𝐴₁𝑢+𝐴₀+2𝛼₁−𝛽^′𝐴₁−𝛽_𝑡𝜌𝑢^𝑟𝑢²_𝑥+Γ₁𝑢^𝑎+−2𝛼₁−𝛽^′𝐴₁−𝛼₁𝑢_𝑥2𝜌𝑢^𝑟𝑢_𝑥−2𝛼₁−𝛽^′𝐴₁𝑢+𝐴₀𝑟𝜌𝑢^𝑟−1𝑢²_𝑥+𝑎Γ₁𝑢^𝑎−1=0. (97)

For 𝑟≠1, from (97) we obtain the following: 𝐴₀=0,𝛽=𝛽₁𝑡+𝛽₀,2𝛼₁−𝛽₁1−𝐴₁()Γ1+𝑟=0,₁2𝛼₁−𝛽₁𝐴₁(1−𝑎)−𝛽₁=0. (98)

From the previous conditions we consider the following subclasses

(A) Consider 2𝛼₁−𝛽₁=0 and (2𝛼₁−𝛽₁)𝐴₁(1−𝑎)−𝛽₁=0. As a consequence we get 𝛽₁=𝛼₁=𝛿=𝜆=0, (99) and then there is no extension of 𝐿_𝒫.

(B) For 1−𝐴₁(1+𝑟)=0 and (2𝛼₁−𝛽₁)𝐴₁(1−𝑎)−𝛽₁=0 we get, for 𝑎≠1, 𝐴₁=1𝛽1+𝑟,𝛼=¹2(2−𝑎−𝑟)𝑥+𝛼₀,𝛽=𝛽₁𝑡+𝛽₀,𝛽𝛿=¹1−𝑎𝑢,𝜆=𝛽₁1𝑏𝑎𝑣1−𝑎 (100) and then in this subcase we got an extension by one of 𝐿_𝒫 given by 𝑋₃=𝑡𝜕_𝑡+2−𝑎−𝑟2𝑥𝜕_𝑥+𝑢𝜕1−𝑎^𝑡𝑢+𝑎𝑣()𝑏𝜕1−𝑎^𝑣 (101) provided that 𝐷(𝑣) and 𝐹(𝑢,𝑣) are solutions of the following equations: 𝑢𝐷1−𝑎^𝑢=()((1−𝑎−𝑟𝐷,𝑎−𝑏1−𝑎))𝐹−𝑏𝑢𝐹_𝑢−𝑎𝑣𝐹_𝑣=0. (102) For 𝑎=1, instead, we conclude that 𝛽₁=0,𝛿=2𝛼₁11+𝑟,𝜆=𝑏2𝛼₁1+𝑟𝑣. (103) Therefore, the extension is given by 𝑋₃=𝑥𝜕_𝑥+21+𝑟𝑢𝜕_𝑢+1𝑏2𝑣1+𝑟 (104) provided that 𝐷(𝑢) and 𝐹(𝑢,𝑣) are solutions of the following differential equations: 𝑢𝐷1+𝑟^𝑢=𝐷,𝐹−𝑏𝑢𝐹_𝑢−𝑣𝐹_𝑣=0. (105)

6. Conclusions

In this paper we have considered a class of reaction-diffusion systems with an additional advection term. The studied class includes, as particular cases, all partial differential equation models concerned with the Aedes aegypti mosquito that have been proposed until now. We have investigated such a system from the point of view of equivalence transformations in the spirit of the Lie-Ovsiannikov algorithm based on the Lie infinitesimal criterion. In agreement with some modifications of the Lie-Ovsiannikov algorithm, introduced in [20], we obtained a group of weak equivalence transformations. We have applied these transformations in order to obtain symmetry generators by using a projection theorem. In particular after having obtained the principal Lie algebra, we investigated a specific, but quite general, form for the advection-reaction term 𝐺 and we derived some extensions of the principal Lie algebra. The specializations of the results for the systems studied in [2] are in agreement with those ones obtained there.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank FAPESP for financial support (Grants 2011/20072-0 and 2011/19089-6). Mariano Torrisi would also like to thank CMCC-UFABC for its warm hospitality and GNFM (Gruppo Nazionale per Fisica-Matematica) for its support. He also thanks the support from the University of Catania through PRA.