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Similarity solutions for systems arising from an Aedes aegypti model
Commun Nonlinear Sci Numer Simulat 19 (2014) 872–879
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Similarity solutions for systems arising from an Aedes aegypti model Igor Leite Freire a, Mariano Torrisi a,b,⇑ a b
Centro de Matemática, Computação e Cognição, Universidade Federal do ABC – UFABC, Rua Santa Adélia, 166, Bairro Bangu, 09.210-170 Santo André, SP, Brazil Dipartimento di Matematica e Informatica, Università Degli Studi di Catania, Viale Andrea Doria, 6, 95125 Catania, Italy
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 30 December 2012 Received in revised form 11 August 2013 Accepted 12 August 2013 Available online 23 August 2013
In a recent paper a new model for the Aedes aegypti mosquito dispersal dynamics was proposed and its Lie point symmetries were investigated. According to the carried group classification, the maximal symmetry Lie algebra of the nonlinear cases is reached whenever the advection term vanishes. In this work we analyze the family of systems obtained when the wind effects on the proposed model are neglected. Wide new classes of solutions to the systems under consideration are obtained. Ó 2013 Elsevier B.V. All rights reserved.
Keywords: Aedes aegypti mosquito Symmetries Exact solutions
1. Introduction Aedes aegypti mosquito can be considered the primary vector for dengue and urban yellow fever. It is possible to find it around the world, not only in the tropical regions but also beyond them, reaching temperate climates. Because of its importance as a vector of deadly diseases, the significance of its distribution in urban areas and the possibility of breeding in laboratory facilities, Aedes aegypti is one of the best-focused and studied mosquitoes [1]. To the best of our knowledge, the first partial differential equation model for the Aedes aegypti was introduced in [2,3] by writing a semilinear system of two partial differential equations. In recent works, see [4,5], taking into account the modeling for Proteus mirabilis bacterial colonies (see [7]) we introduced the following quasilinear system for the dispersal dynamics of the Aedes aegypti,
(
ut ¼ ðup ux Þx 2muq ux þ ck v þ v t ¼ ku þ ðk l2 cÞv :
c k
l1 u;
ð1Þ
In this system, as well as in [2], u and v are, respectively, non-dimensional densities of winged population and aquatic population of mosquitoes and k; c; l1 ; l2 are non-dimensional, in general, positive parameters while m is a real parameter. Spe1 and k 2 , which are, respectively, the carrying capacity related to the amount of cifically k is the ratio between the constants k findable nutrients and the carrying capacity effect depending on the occupation of the available breeder, c denotes the specific rate of maturation of the aquatic form into winged female mosquitoes, while l1 ; l2 are, respectively, the mortality of winged population and the mortality of aquatic population. Finally m denotes the component of constant wind speed along the direction x due to wind currents. In general, such currents generate an advection motion of large masses of the winged mosquito population and, consequently, can improve a fast advance of the infestation. ⇑ Corresponding author at: Dipartimento di Matematica e Informatica, Università Degli Studi di Catania, Viale Andrea Doria, 6, 95125 Catania, Italy. E-mail addresses: [email protected], [email protected] (I.L. Freire), [email protected], [email protected] (M. Torrisi). 1007-5704/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2013.08.006
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The first equation of (1) gives the time rate of the change of the mosquito density as sum of the growth rate term ck v , the per capita death rate ck l1 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 growthrate term ku of aquatic population, due to the new egg depositions of female mosquitoes, with per capita death rate k l2 v of aquatic population. The term cv represents the loss due to the change of the aquatic into winged form. We would like to stress that the interaction between winged and aquatic populations is very weak. As a consequence of this fact and as a new proposal we removed the terms ck v ð1 uÞ and kð1 v Þu of logistic type, appearing in [2], and substituted them, respectively, with ck ðv þ uÞ and kðu þ v Þ. Then the interaction is given by the terms ck v and ku appearing respectively in the first equation and in the second equation. Differently from [2] we assume here a nonlinear diffusive-advective flux. In this paper we consider the quasilinear systems obtained from (1) by putting m ¼ 0. This case arose in [4] as the nonlinear case admitting the maximal Lie symmetry algebra. However, from biological point of view these systems can be considered cases of the general systems (1) where the wind effects can be neglected. Moreover we stress that even, in agreement with [6], this hypothesis can be considered realistic for several situations, the knowledge of mathematical features of the system we have got could be useful both for itself and also as a preliminary study of m – 0. The search for solutions is performed via group method techniques as they represent a tool offering a methodological way to obtain solutions of nonlinear systems (see, e.g. [8–12]). Moreover they could show mathematical suggestions about the constitutive parameters appearing in the system under consideration. The plan of the paper is as it follows. In the next section we show the symmetries of the systems under consideration. The most general reduction and exact solutions to the system (1), with m ¼ 0 and p ¼ 4=3 are found in the Section 3. Next, in the Section 4, it is discussed the case p – 4=3 and m ¼ 0. At the same section we not only find several exact solutions to the considered cases, but we also present some remarks on boundness of them. Conclusions are given in Section 5. 2. The symmetries for m ¼ 0 System (1) was studied from the point of view of Lie symmetry theory in [4] and there it was completely classified with respect to the constitutive parameters p; q; m. For m ¼ 0, apart from the case p ¼ 0, the largest Lie symmetry algebra is reached when p ¼ 4=3. In this case the system (1) reads
(
ut ¼ ðu3 ux Þx þ ck v þ ðck l1 Þu; 4
v t ¼ ku þ ðk l2 cÞv ;
ð2Þ
whose Lie algebra, taking the results obtained in [4] into account, is spanned by
X1 ¼ @x;
X2 ¼ @t ;
X p ¼ px@ x þ 2u@ u þ 2v @ v ;
ð3Þ
with p ¼ 4=3 in X p , and by the additional generator
X 3 ¼ x2 @ x 3xu@ u 3xv @ v : For
m ¼ 0 and for all p R f4=3; 0g, (1) assumes the form (
ut ¼ ðup ux Þx þ ck v þ ck l1 u; v t ¼ ku þ ðk l2 cÞv :
ð4Þ
The symmetries of (4) generates a 3-dimensional Lie algebra spanned by the generators (3). 3. Reduction and exact solutions of the system ð2Þ Now we consider the linear combination
X ¼ ðc1 þ 2xc4 þ c3 x2 Þ@ x þ c2 @ t ð3xuc3 þ 3uc4 Þ@ u ð3xv c3 þ 3v c4 Þ@ v
ð5Þ
of the basis of the Lie symmetry algebra for the system (2). Then in order to determine the invariant solutions, we must solve the characteristic system
dt dx du ¼ ¼ ; c2 c3 x2 þ 2xc4 þ c1 3xuc3 þ 3uc4
ð6Þ
dt dx dv ¼ ¼ ; c2 c3 x2 þ 2xc4 þ c1 3xv c3 þ 3v c4
ð7Þ
obtained from the invariant surface conditions.
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One can assume, without lost of generality, that c3 ¼ 1. Then, the solutions of the last equation of (6) and (7) are, respectively,
u¼
Að/Þ
v¼
;
3=2
ðx2 þ 2c4 x þ c1 Þ
Bð/Þ ðx2 þ 2c4 x þ c1 Þ
3=2
ð8Þ
;
while
8 c2 t þ xþc ; if c24 c1 ¼ 0; > > 4 > > > > > Z < dx c2 xþc4 pffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffi ; if c24 c1 < 0; arctan t / ¼ t c2 ¼ c1 c24 c1 c24 x2 þ 2c4 x þ c1 > > > > > xþc4 pffiffiffiffiffiffiffiffiffi > c24 c1 > 2 ; if c2 c1 > 0: p ffiffiffiffiffiffiffiffiffi : t pcffiffiffiffiffiffiffiffiffi ln 4 2 2 2
c4 c1
xþc4 þ
ð9Þ
c4 c1
In order to use the transformations (8) and (9) to construct exact solutions of (2), we avoid from (5) time translations substituting c2 ¼ 0 into (9). Therefore, after setting c2 ¼ 0 in (9), substituting it into (8) and, in the following, into (2), it is obtained the following ordinary differential equation system for A and B: 1
A0 ¼ 3A3 ðc1 c24 Þ þ ck B þ
c k
l1 A;
ð10Þ
0
B ¼ kA þ ðk l2 cÞB:
If we consider the case c1 ¼ c24 , by isolating B in the first equation of (10) and by substituting into the second one, the following linear second order ODE is obtained 0
A00 ðtÞ þ bA ðtÞ þ cAðtÞ ¼ 0;
ð11Þ
where
c
b ¼ l1 þ l2 þ c k ; k
c ¼ c þ
c k
l1 ðk l2 cÞ:
ð12Þ
The cases c2 – 0 and the cases c1 – c24 of the system (10) will not be considered in this paper. The solution of (11) is bt
AðtÞ ¼ a1 e 2 cosh
pffiffiffiffi ! pffiffiffiffi ! Dt Dt bt þ a2 e 2 sinh ; 2 2
ð13Þ
where a1 ; a2 are arbitrary constants and
D ¼ 4c þ
h
l1
c k
þ ðk l2 cÞ
i2
> 0:
Moreover, substituting (13) into (10), we conclude that BðtÞ is given by
BðtÞ ¼ a1
" " pffiffiffiffi ! pffiffiffiffi !# pffiffiffiffi ! pffiffiffiffi !# bt bt e 2 pffiffiffiffi Dt Dt e 2 pffiffiffiffi Dt Dt b cosh þ a2 b sinh : D sinh D cosh 2 2 2 2 2 2
ð14Þ
Going back to the original variables, taking (9) into account, we are able to write uðx; tÞ and v ðx; tÞ. 4. Reductions and exact solutions of the system ð4Þ 4.1. Reductions We now consider the system (4) with the following general infinitesimal operator
X ¼ pc1 X 1 þ c2 X 2 þ X p
ð15Þ
where c1 and c2 are arbitrary real constants. By applying the invariant surface conditions we get that the invariant solutions, with respect to the transformations generated by the operator X, must be of the following form: 2
uðx; tÞ ¼ AðrÞðx þ c1 Þp ; where
r¼
et c2
ðc1 þ xÞ p
2 p
v ðx; tÞ ¼ BðrÞðx þ c1 Þ ;
I.L. Freire, M. Torrisi / Commun Nonlinear Sci Numer Simulat 19 (2014) 872–879
875
is the so-called similarity variable and with AðrÞ and BðrÞ satisfying the following general reduced system of ODEs
8 h i < A0 r ¼ ð2A A0 rÞ2 þ c B þ c l A þ Ap 4 2 A þ A0 r ð1 c Þ 1 þ 2 þ c2 A00 r2 ; 2 1 p p k k p p p2 : B0 r ¼ kA þ ðk l cÞB:
ð16Þ
2
As in the previous case, by neglecting traslations in time, the reduced system (16) can be simplified. The similarity variable becomes r ¼ t, while the solutions are separable variable solutions of the form: 2
2 p
v ðx; tÞ ¼ BðtÞðx þ c1 Þ ;
uðx; tÞ ¼ AðtÞðx þ c1 Þp ;
ð17Þ
where AðtÞ and BðtÞ satisfy the following first order system of ODEs
(
A0 ðtÞ ¼ 4þ2p AðtÞpþ1 þ ck BðtÞ þ p2
c k
l1 AðtÞ;
B0 ðtÞ ¼ kAðtÞ þ ðk l2 cÞBðtÞ:
ð18Þ
It is easy to ascertain that by a simple calculations it is possible to reduce the search for solutions of (18) to the search or solutions of the following second order ODE:
c ðp þ 1Þð4 þ 2pÞ p A ðtÞ þ l þ k l c A0 ðtÞ 1 2 p2 k c 4 þ 2p AðtÞp c þ ðk l2 cÞ l1 AðtÞ ¼ 0; þ ðk l2 cÞ 2 p k
A00 ðtÞ
ð19Þ
that is a family of Lienard equations. After having put
a¼
ðp þ 1Þð4 þ 2pÞ ; p2
d ¼ ðk l2 cÞ
4 þ 2p ; p2
ð20Þ
we are able to write (19) as
A00 ðtÞ a Ap b A0 þ d Ap þ c A ¼ 0;
ð21Þ
where b and c are given by (12). Some classes of solutions of this family will be investigated in the following. 4.2. Solutions for p ¼ 2 Substituting p ¼ 2 into (20), and taking (12) and (21) into account, we obtain 0
A00 ðtÞ þ bA ðtÞ þ cAðtÞ ¼ 0: This equation is identical to Eq. (11) whose solutions have been shown in the previous section. In particular, if the parameters are related by (see [4])
c¼
a2 4
;
l1 ¼
a2 a2 ; k ¼ q; l2 ¼ q ; 4q 4
where a and q are positive constants, the solutions obtained in Section 3 are specialized as it follows
uðx; tÞ ¼
1 at ; sinh ðx þ c1 Þ 2
v ðx; tÞ ¼
2q at cosh aðx þ c1 Þ 2
uðx; tÞ ¼
1 at ; cosh ðx þ c1 Þ 2
v ðx; tÞ ¼
2q at : sinh aðx þ c1 Þ 2
or
4.3. Solutions for p ¼ 1 In this case (21) becomes 0
A00 ðtÞ þ bA ðtÞ þ cAðtÞ þ d ¼ 0; whose general solutions are
d bt AðtÞ ¼ þ a1 e 2 cosh c
pffiffiffiffi ! pffiffiffiffi ! Dt Dt bt þ a2 e 2 sinh ; 2 2
ð22Þ
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where
d 2ðk l2 cÞ ¼ : c c þ ck l1 ðk l2 cÞ After substituting (22) into the second equation of the system (18) we can easily obtain the function BðtÞ. 4.4. Solutions to Eq. (21) with p – 1; 2 In this subsection, once considered the previous special cases, the analysis of Eq. (21) is concerned with the study of a Lienard equation family. After having studied some special cases, where we have been able to get solutions in a closed form, in this subsection we show some special solutions concerned with this type of equations. 4.4.1. Solutions with c ¼ 0 Assuming c ¼ 0 into (21), we get
A00 ðtÞ a Ap b A0 þ d Apþ1 ¼ 0:
ð23Þ
0
Then, after putting wðAÞ ¼ A ðtÞ our equation becomes
ww0 a Ap b w þ d Apþ1 ¼ 0;
ð24Þ
that can be solved with respect to wðAÞ. For this equation we look for solutions of the type
w¼
d p1 A m
from where, by substituting into (24) we get
p 1 p2 dA aAp þ b þ mA2 ¼ 0; m which is satisfied for p ¼ 2 and
d þ b ¼ 0; m
m a ¼ 0;
that implies
ab þ d ¼ 0: 4.4.2. Additional exact solutions In the framework of this subsection, for further developments and taking into account the numerical values for the adimensional constitutive parameters c; k; l1 ; l2 , appearing in Takahashi et. al. [2], we consider
c ¼ 2:5 101 ; k ¼ 6:66 103 ; l1 ¼ 1:33 103 ; l2 ¼ 3:33 104 :
ð25Þ
We observe that for values of the constitutive parameters of the same order of magnitude of those given by (25) it is possible to assume that
a 6 0 8p 2 ½2; 1;
b < 0;
c < 0;
d 6 0 8p 2 ½2; þ1Þ:
ð26Þ
0
Let w be a function such that wðAÞ ¼ A ðtÞ. Then Eq. (21) becomes
ww0 a Ap b w þ d Ap þ c A ¼ 0:
ð27Þ
It is a simple matter to ascertain that the Eq. (27) admits the solutions w ¼ mA; m 2 R; provided that the following conditions are satisfied
ma d ¼ 0;
m2 þ bm þ c ¼ 0:
These conditions bring us to
2 d d þ b þ c ¼ 0: a a
ð28Þ
Taking into account the expressions for a; b; c; d and the relations (20), we try to solve (28) with respect to p, then from the previous data and the consequent inequalities for the quantities a; b; c and d we can affirm that exist two real solutions (one negative and one positive) for p:
I.L. Freire, M. Torrisi / Commun Nonlinear Sci Numer Simulat 19 (2014) 872–879
p1;2
877
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 bðk l2 cÞ jk l2 cj b 4c ¼ 1 þ : 2c
Therefore, the Lienard equations
! c ðp1;2 þ 1Þð4 þ 2p1;2 Þ p1;2 A0 ðtÞ A ðtÞ þ l þ k l c 1 2 k p21;2 ! c 4 þ 2p1;2 p1;2 AðtÞ c þ ðk l2 cÞ l1 AðtÞ ¼ 0 þ ðk l2 cÞ k p21;2
A00 ðtÞ
admit the solutions
A1;2 ¼ A0 em1;2 t ; where A0 is a constant and
m1;2 ¼
d : a p¼p1;2
Going back to the original variable we get 2
u1;2 ðx; tÞ ¼ A0 em1;2 t ðx þ c1 Þp1;2 : In the same way after substituting A1;2 in the second equation of the system (18) it is a simple matter to solve it and get the corresponding B1;2 ðtÞ. 4.4.3. About the solutions of Eq. (21) According to [13], after putting
z¼
Z
p a A b dA;
Eq. (27) becomes an Abel equation of the second kind in the canonical form pþ1
ww0 ¼ w þ
cz þ dz ; b azp
ð29Þ
where w ¼ wðzÞ. We observe that Eq. (29) is always defined because we are in the case p – 1; 2 which implies a – 0. In the case b ¼ 0, that is when
k l2 c ¼
c k
l1 ;
the Eq. (29) specializes as
ww0 w ¼ gz þ hz da
1p
ð30Þ
;
ac
with g ¼ and h ¼ . Once Eq. (29) is solved, we can determine the function A and, consequently, system (18) is solved. The solutions of (30) are, usually, determined in a parametric form and, not rarely, they depend on special functions, such as Elliptic Weierstrass function and Bessel function. A large number of equations of the type (30) are discussed in [13]. 4.5. Critical points of the system (4) Let us now find the values of the functions AðtÞ and BðtÞ at their critical points. By solving (18) after having put A0 ¼ B0 ¼ 0, we obtain
" #1p 2 p2 ððkl1 cÞðc þ l2 Þ k l1 Þ A0 ¼ ; kð4 þ 2pÞðc þ l2 kÞ
" B0 ¼
p2 k
p1
ððkl1 cÞðc þ l2 Þ k pþ1
ð4 þ 2pÞðc þ l2 kÞ
2
l1 Þ
#1p ;
for p – 2 and provided that there is not loss of meaning due to some values of p and/or the other constitutive parameters. For p ¼ 2 the search for the critical points bring to a linear algebraic homogeneous system. Then in order to admit nontrivial solutions the following additional compatibility condition holds
ðc þ l2 Þ
l1
c k
l1 k ¼ 0:
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I.L. Freire, M. Torrisi / Commun Nonlinear Sci Numer Simulat 19 (2014) 872–879
Therefore
A0 ¼ q
c k
c B0 ¼ q l1 k
ð31Þ
with q non zero arbitrary constant. 4.6. Some remarks about the solutions of the system (4) Taking into account the biological origin of u and v, we assume that u; v P 0. Moreover, solutions of the form (17) imply that AðtÞ > 0 and BðtÞ P 0, provided that c1 > 0 and ðx; tÞ 2 ½0; 1Þ ½0; 1Þ. Concerning with the first equation of (18), in order to avoid the infestation, we must have A0 ðtÞ 6 0 so that AðtÞ is a bounded function. Then a necessary condition to the function A be bounded is that at least one of the coefficients of A and Apþ1 in the first equation of the system (18) is negative, that is:
c k
l1 < 0;
ð32Þ
or
p < 2:
ð33Þ
If we take into account the values of the parameters given in (25) (see also [2]), we can easily conclude that condition (32) cannot be verified. Instead, such condition would only be verified if we assume c=k smaller than l1 . This fact could occurs whenever 1 =k 2 is big, that means that the carrying capacity for aquatic population is smaller than that one related with winged k¼k population. Whenever (33) holds, condition (32) is a sufficient condition to get bounded solutions since the coefficient of AðtÞpþ1 is negative. For p P 2, condition (32) is necessary, but not sufficient. On the other hand if we assume that the solutions of the system (18) are bounded, from the second equation of (18) it follows that a necessary condition to B is bounded is given by
k l2 c < 0: The biological condition for the existence of the mosquito population, see in [14,6], is c l1 ðl2 þ cÞ > 0 which is easily obtained once it is assumed
k<
c ; k > l2 þ c: l1
These assumptions imply that A0 ðtÞ > 0 and B0 ðtÞ > 0, for any p > 2. Such conditions occur when there is an infestation of mosquitos. 5. Conclusions In this paper we recall a model concerned with Aedes aegypti introduced in [4] and look for similarity solutions, in absence of wind effects, by using the symmetries found in [4]. The classification performed there shows that the system considered admits a four dimensional algebra for p ¼ 43 and a three-dimensional algebra for all p – 43 ; 0. After having found the most general reduction for both systems we look for separable variable solutions. Several classes of exact solutions have been found. In particular for p – 43 ; 0 the solution of reduced system is brought to the search for solutions of a class of Lienard equations, which was widely discussed. Finally some remarks on the behavior of invariant solutions of the system (4) and its biological compatibility are done. In particular we observed that, as expected, the boundness of solutions, as well as the biological condition (see [14,6]) for the existence of the mosquito population c l1 ðl2 þ cÞ > 0, are strictly related to the features of the per capita death rate ck l1 in adult phase and to the features of the per capita death rate k l2 in aquatic phase. We wish to stress that the solutions found here, as well as those obtained in [4], are, until now the only exact solutions concerned with the considered mathematical model for Aedes aegypti dispersal dynamics. These special invariant solutions we derived here, in general, do not satisfy arbitrary initial or boundary conditions prescribed for a given problem. However they might be useful for a benchmark test for larger numerical schemes devised to solve our system in a realistic case. Moreover, as subject of further investigations, it will be useful to study and look for additionl solutions of the Lienard Eqs. (19), Abel Eqs. (24) and Abel equations of the second kind (29). We plan to continue in the aforesaid researches together with the search, concerned with model (1), for additional exact solutions and additional news about the structure of its constitutive functions. Acknowledgements The authors 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
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879
thanks the support from University of Catania through PRA and from MIUR through PRIN: Modelli cinetici e macroscopici per il transporto di particelle in semiconductori: aspetti modellistici, analitici e computazionali. We would like to thank the referees for their useful comments and suggestions which have considerably improved the paper. References [1] Arunachalam N, Tana S, Espino F, Kittayapong P, Abeyewickreme W, et al. Eco-bio-social determinants of dengue vector breeding: a multicountry study in urban and periurban Asia. Bull World Health Organ 2010;88:173–84. [2] Takahashi LT, Maidana NA, Ferreira Jr WC, Pulino P, Yang HM. Mathematical models for the Aedes aegypti dispersal dynamics: travelling waves by wing and wind. Bull Math Biol 2005;67:509–28. [3] Takahashi LT. Modelos matemáticos de epidemiologia com vetores: simulação da propagação urbana e geográfica da dengue. Ph.D. Thesis, Univ. Estadual de Campinas-UNICAMP, Campinas, Brazil, 2004. [4] Freire IL, Torrisi M. Symmetry methods in mathematical modeling Aedes aegypti dispersal dynamics. Nonlinear Anal RWA 2013;14:1300–7. http:// dx.doi.org/10.1016/ j.nonrwa.2012.09.018. [5] Freire IL, Torrisi M. On a model for Aedes aegypti, Numerical Analysis and Applied Mathematics. In: ICNAAM 2012 International Conference of Numerical Analysis and Applied Mathematics, AIP Conf Proc., vol. 1479, p. 1373–1376.. [6] Maidana NA, Yang HM. Describing the geographic spread of dengue disease by traveling waves. Math Biosci 2008;215:64–77. [7] Torrisi M, Tracinà R. Exact solutions of a reaction-difusion system for Proteus Mirabilis bacterial colonies. Nonlinear Anal: Real World Appl 2011;12:1865–74. [8] Ovsiannikov LV. Group analysis of differential equations. Academic Press; 1982. [9] Olver PJ. Applications of Lie groups to differential equations. New York: Springer; 1986. [10] Bluman GW, Kumei S. Symmetries and differential equations. Applied mathematical sciences, vol. 81. New York: Springer; 1989. [11] Ibragimov NH. Transformation groups applied to mathematical physics, translated from the russian mathematics and its applications (soviet series). Dordrecht: D. Reidel Publishing Co.; 1985. [12] Ibragimov NH. CRC handbook of Lie group analysis of differential equations, vol. 1. CRC Press; 1994. [13] Polyanin AD, Zaitsev VF. Handbook of exact solutions for ordinary differential equations. second ed. Boca Raton: Chapman & Hall/CRC; 2003. [14] Maidana NA, Yang HM. A spatial model to describe the dengue propagation. TEMA Tend Mat Apl Comput 2007;8(1):83–92.
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Similarity solutions for systems arising from an Aedes aegypti model Igor Leite Freire a, Mariano Torrisi a,b,⇑ a b
Centro de Matemática, Computação e Cognição, Universidade Federal do ABC – UFABC, Rua Santa Adélia, 166, Bairro Bangu, 09.210-170 Santo André, SP, Brazil Dipartimento di Matematica e Informatica, Università Degli Studi di Catania, Viale Andrea Doria, 6, 95125 Catania, Italy
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 30 December 2012 Received in revised form 11 August 2013 Accepted 12 August 2013 Available online 23 August 2013
In a recent paper a new model for the Aedes aegypti mosquito dispersal dynamics was proposed and its Lie point symmetries were investigated. According to the carried group classification, the maximal symmetry Lie algebra of the nonlinear cases is reached whenever the advection term vanishes. In this work we analyze the family of systems obtained when the wind effects on the proposed model are neglected. Wide new classes of solutions to the systems under consideration are obtained. Ó 2013 Elsevier B.V. All rights reserved.
Keywords: Aedes aegypti mosquito Symmetries Exact solutions
1. Introduction Aedes aegypti mosquito can be considered the primary vector for dengue and urban yellow fever. It is possible to find it around the world, not only in the tropical regions but also beyond them, reaching temperate climates. Because of its importance as a vector of deadly diseases, the significance of its distribution in urban areas and the possibility of breeding in laboratory facilities, Aedes aegypti is one of the best-focused and studied mosquitoes [1]. To the best of our knowledge, the first partial differential equation model for the Aedes aegypti was introduced in [2,3] by writing a semilinear system of two partial differential equations. In recent works, see [4,5], taking into account the modeling for Proteus mirabilis bacterial colonies (see [7]) we introduced the following quasilinear system for the dispersal dynamics of the Aedes aegypti,
(
ut ¼ ðup ux Þx 2muq ux þ ck v þ v t ¼ ku þ ðk l2 cÞv :
c k
l1 u;
ð1Þ
In this system, as well as in [2], u and v are, respectively, non-dimensional densities of winged population and aquatic population of mosquitoes and k; c; l1 ; l2 are non-dimensional, in general, positive parameters while m is a real parameter. Spe1 and k 2 , which are, respectively, the carrying capacity related to the amount of cifically k is the ratio between the constants k findable nutrients and the carrying capacity effect depending on the occupation of the available breeder, c denotes the specific rate of maturation of the aquatic form into winged female mosquitoes, while l1 ; l2 are, respectively, the mortality of winged population and the mortality of aquatic population. Finally m denotes the component of constant wind speed along the direction x due to wind currents. In general, such currents generate an advection motion of large masses of the winged mosquito population and, consequently, can improve a fast advance of the infestation. ⇑ Corresponding author at: Dipartimento di Matematica e Informatica, Università Degli Studi di Catania, Viale Andrea Doria, 6, 95125 Catania, Italy. E-mail addresses: [email protected], [email protected] (I.L. Freire), [email protected], [email protected] (M. Torrisi). 1007-5704/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2013.08.006
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The first equation of (1) gives the time rate of the change of the mosquito density as sum of the growth rate term ck v , the per capita death rate ck l1 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 growthrate term ku of aquatic population, due to the new egg depositions of female mosquitoes, with per capita death rate k l2 v of aquatic population. The term cv represents the loss due to the change of the aquatic into winged form. We would like to stress that the interaction between winged and aquatic populations is very weak. As a consequence of this fact and as a new proposal we removed the terms ck v ð1 uÞ and kð1 v Þu of logistic type, appearing in [2], and substituted them, respectively, with ck ðv þ uÞ and kðu þ v Þ. Then the interaction is given by the terms ck v and ku appearing respectively in the first equation and in the second equation. Differently from [2] we assume here a nonlinear diffusive-advective flux. In this paper we consider the quasilinear systems obtained from (1) by putting m ¼ 0. This case arose in [4] as the nonlinear case admitting the maximal Lie symmetry algebra. However, from biological point of view these systems can be considered cases of the general systems (1) where the wind effects can be neglected. Moreover we stress that even, in agreement with [6], this hypothesis can be considered realistic for several situations, the knowledge of mathematical features of the system we have got could be useful both for itself and also as a preliminary study of m – 0. The search for solutions is performed via group method techniques as they represent a tool offering a methodological way to obtain solutions of nonlinear systems (see, e.g. [8–12]). Moreover they could show mathematical suggestions about the constitutive parameters appearing in the system under consideration. The plan of the paper is as it follows. In the next section we show the symmetries of the systems under consideration. The most general reduction and exact solutions to the system (1), with m ¼ 0 and p ¼ 4=3 are found in the Section 3. Next, in the Section 4, it is discussed the case p – 4=3 and m ¼ 0. At the same section we not only find several exact solutions to the considered cases, but we also present some remarks on boundness of them. Conclusions are given in Section 5. 2. The symmetries for m ¼ 0 System (1) was studied from the point of view of Lie symmetry theory in [4] and there it was completely classified with respect to the constitutive parameters p; q; m. For m ¼ 0, apart from the case p ¼ 0, the largest Lie symmetry algebra is reached when p ¼ 4=3. In this case the system (1) reads
(
ut ¼ ðu3 ux Þx þ ck v þ ðck l1 Þu; 4
v t ¼ ku þ ðk l2 cÞv ;
ð2Þ
whose Lie algebra, taking the results obtained in [4] into account, is spanned by
X1 ¼ @x;
X2 ¼ @t ;
X p ¼ px@ x þ 2u@ u þ 2v @ v ;
ð3Þ
with p ¼ 4=3 in X p , and by the additional generator
X 3 ¼ x2 @ x 3xu@ u 3xv @ v : For
m ¼ 0 and for all p R f4=3; 0g, (1) assumes the form (
ut ¼ ðup ux Þx þ ck v þ ck l1 u; v t ¼ ku þ ðk l2 cÞv :
ð4Þ
The symmetries of (4) generates a 3-dimensional Lie algebra spanned by the generators (3). 3. Reduction and exact solutions of the system ð2Þ Now we consider the linear combination
X ¼ ðc1 þ 2xc4 þ c3 x2 Þ@ x þ c2 @ t ð3xuc3 þ 3uc4 Þ@ u ð3xv c3 þ 3v c4 Þ@ v
ð5Þ
of the basis of the Lie symmetry algebra for the system (2). Then in order to determine the invariant solutions, we must solve the characteristic system
dt dx du ¼ ¼ ; c2 c3 x2 þ 2xc4 þ c1 3xuc3 þ 3uc4
ð6Þ
dt dx dv ¼ ¼ ; c2 c3 x2 þ 2xc4 þ c1 3xv c3 þ 3v c4
ð7Þ
obtained from the invariant surface conditions.
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One can assume, without lost of generality, that c3 ¼ 1. Then, the solutions of the last equation of (6) and (7) are, respectively,
u¼
Að/Þ
v¼
;
3=2
ðx2 þ 2c4 x þ c1 Þ
Bð/Þ ðx2 þ 2c4 x þ c1 Þ
3=2
ð8Þ
;
while
8 c2 t þ xþc ; if c24 c1 ¼ 0; > > 4 > > > > > Z < dx c2 xþc4 pffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffi ; if c24 c1 < 0; arctan t / ¼ t c2 ¼ c1 c24 c1 c24 x2 þ 2c4 x þ c1 > > > > > xþc4 pffiffiffiffiffiffiffiffiffi > c24 c1 > 2 ; if c2 c1 > 0: p ffiffiffiffiffiffiffiffiffi : t pcffiffiffiffiffiffiffiffiffi ln 4 2 2 2
c4 c1
xþc4 þ
ð9Þ
c4 c1
In order to use the transformations (8) and (9) to construct exact solutions of (2), we avoid from (5) time translations substituting c2 ¼ 0 into (9). Therefore, after setting c2 ¼ 0 in (9), substituting it into (8) and, in the following, into (2), it is obtained the following ordinary differential equation system for A and B: 1
A0 ¼ 3A3 ðc1 c24 Þ þ ck B þ
c k
l1 A;
ð10Þ
0
B ¼ kA þ ðk l2 cÞB:
If we consider the case c1 ¼ c24 , by isolating B in the first equation of (10) and by substituting into the second one, the following linear second order ODE is obtained 0
A00 ðtÞ þ bA ðtÞ þ cAðtÞ ¼ 0;
ð11Þ
where
c
b ¼ l1 þ l2 þ c k ; k
c ¼ c þ
c k
l1 ðk l2 cÞ:
ð12Þ
The cases c2 – 0 and the cases c1 – c24 of the system (10) will not be considered in this paper. The solution of (11) is bt
AðtÞ ¼ a1 e 2 cosh
pffiffiffiffi ! pffiffiffiffi ! Dt Dt bt þ a2 e 2 sinh ; 2 2
ð13Þ
where a1 ; a2 are arbitrary constants and
D ¼ 4c þ
h
l1
c k
þ ðk l2 cÞ
i2
> 0:
Moreover, substituting (13) into (10), we conclude that BðtÞ is given by
BðtÞ ¼ a1
" " pffiffiffiffi ! pffiffiffiffi !# pffiffiffiffi ! pffiffiffiffi !# bt bt e 2 pffiffiffiffi Dt Dt e 2 pffiffiffiffi Dt Dt b cosh þ a2 b sinh : D sinh D cosh 2 2 2 2 2 2
ð14Þ
Going back to the original variables, taking (9) into account, we are able to write uðx; tÞ and v ðx; tÞ. 4. Reductions and exact solutions of the system ð4Þ 4.1. Reductions We now consider the system (4) with the following general infinitesimal operator
X ¼ pc1 X 1 þ c2 X 2 þ X p
ð15Þ
where c1 and c2 are arbitrary real constants. By applying the invariant surface conditions we get that the invariant solutions, with respect to the transformations generated by the operator X, must be of the following form: 2
uðx; tÞ ¼ AðrÞðx þ c1 Þp ; where
r¼
et c2
ðc1 þ xÞ p
2 p
v ðx; tÞ ¼ BðrÞðx þ c1 Þ ;
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is the so-called similarity variable and with AðrÞ and BðrÞ satisfying the following general reduced system of ODEs
8 h i < A0 r ¼ ð2A A0 rÞ2 þ c B þ c l A þ Ap 4 2 A þ A0 r ð1 c Þ 1 þ 2 þ c2 A00 r2 ; 2 1 p p k k p p p2 : B0 r ¼ kA þ ðk l cÞB:
ð16Þ
2
As in the previous case, by neglecting traslations in time, the reduced system (16) can be simplified. The similarity variable becomes r ¼ t, while the solutions are separable variable solutions of the form: 2
2 p
v ðx; tÞ ¼ BðtÞðx þ c1 Þ ;
uðx; tÞ ¼ AðtÞðx þ c1 Þp ;
ð17Þ
where AðtÞ and BðtÞ satisfy the following first order system of ODEs
(
A0 ðtÞ ¼ 4þ2p AðtÞpþ1 þ ck BðtÞ þ p2
c k
l1 AðtÞ;
B0 ðtÞ ¼ kAðtÞ þ ðk l2 cÞBðtÞ:
ð18Þ
It is easy to ascertain that by a simple calculations it is possible to reduce the search for solutions of (18) to the search or solutions of the following second order ODE:
c ðp þ 1Þð4 þ 2pÞ p A ðtÞ þ l þ k l c A0 ðtÞ 1 2 p2 k c 4 þ 2p AðtÞp c þ ðk l2 cÞ l1 AðtÞ ¼ 0; þ ðk l2 cÞ 2 p k
A00 ðtÞ
ð19Þ
that is a family of Lienard equations. After having put
a¼
ðp þ 1Þð4 þ 2pÞ ; p2
d ¼ ðk l2 cÞ
4 þ 2p ; p2
ð20Þ
we are able to write (19) as
A00 ðtÞ a Ap b A0 þ d Ap þ c A ¼ 0;
ð21Þ
where b and c are given by (12). Some classes of solutions of this family will be investigated in the following. 4.2. Solutions for p ¼ 2 Substituting p ¼ 2 into (20), and taking (12) and (21) into account, we obtain 0
A00 ðtÞ þ bA ðtÞ þ cAðtÞ ¼ 0: This equation is identical to Eq. (11) whose solutions have been shown in the previous section. In particular, if the parameters are related by (see [4])
c¼
a2 4
;
l1 ¼
a2 a2 ; k ¼ q; l2 ¼ q ; 4q 4
where a and q are positive constants, the solutions obtained in Section 3 are specialized as it follows
uðx; tÞ ¼
1 at ; sinh ðx þ c1 Þ 2
v ðx; tÞ ¼
2q at cosh aðx þ c1 Þ 2
uðx; tÞ ¼
1 at ; cosh ðx þ c1 Þ 2
v ðx; tÞ ¼
2q at : sinh aðx þ c1 Þ 2
or
4.3. Solutions for p ¼ 1 In this case (21) becomes 0
A00 ðtÞ þ bA ðtÞ þ cAðtÞ þ d ¼ 0; whose general solutions are
d bt AðtÞ ¼ þ a1 e 2 cosh c
pffiffiffiffi ! pffiffiffiffi ! Dt Dt bt þ a2 e 2 sinh ; 2 2
ð22Þ
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where
d 2ðk l2 cÞ ¼ : c c þ ck l1 ðk l2 cÞ After substituting (22) into the second equation of the system (18) we can easily obtain the function BðtÞ. 4.4. Solutions to Eq. (21) with p – 1; 2 In this subsection, once considered the previous special cases, the analysis of Eq. (21) is concerned with the study of a Lienard equation family. After having studied some special cases, where we have been able to get solutions in a closed form, in this subsection we show some special solutions concerned with this type of equations. 4.4.1. Solutions with c ¼ 0 Assuming c ¼ 0 into (21), we get
A00 ðtÞ a Ap b A0 þ d Apþ1 ¼ 0:
ð23Þ
0
Then, after putting wðAÞ ¼ A ðtÞ our equation becomes
ww0 a Ap b w þ d Apþ1 ¼ 0;
ð24Þ
that can be solved with respect to wðAÞ. For this equation we look for solutions of the type
w¼
d p1 A m
from where, by substituting into (24) we get
p 1 p2 dA aAp þ b þ mA2 ¼ 0; m which is satisfied for p ¼ 2 and
d þ b ¼ 0; m
m a ¼ 0;
that implies
ab þ d ¼ 0: 4.4.2. Additional exact solutions In the framework of this subsection, for further developments and taking into account the numerical values for the adimensional constitutive parameters c; k; l1 ; l2 , appearing in Takahashi et. al. [2], we consider
c ¼ 2:5 101 ; k ¼ 6:66 103 ; l1 ¼ 1:33 103 ; l2 ¼ 3:33 104 :
ð25Þ
We observe that for values of the constitutive parameters of the same order of magnitude of those given by (25) it is possible to assume that
a 6 0 8p 2 ½2; 1;
b < 0;
c < 0;
d 6 0 8p 2 ½2; þ1Þ:
ð26Þ
0
Let w be a function such that wðAÞ ¼ A ðtÞ. Then Eq. (21) becomes
ww0 a Ap b w þ d Ap þ c A ¼ 0:
ð27Þ
It is a simple matter to ascertain that the Eq. (27) admits the solutions w ¼ mA; m 2 R; provided that the following conditions are satisfied
ma d ¼ 0;
m2 þ bm þ c ¼ 0:
These conditions bring us to
2 d d þ b þ c ¼ 0: a a
ð28Þ
Taking into account the expressions for a; b; c; d and the relations (20), we try to solve (28) with respect to p, then from the previous data and the consequent inequalities for the quantities a; b; c and d we can affirm that exist two real solutions (one negative and one positive) for p:
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p1;2
877
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 bðk l2 cÞ jk l2 cj b 4c ¼ 1 þ : 2c
Therefore, the Lienard equations
! c ðp1;2 þ 1Þð4 þ 2p1;2 Þ p1;2 A0 ðtÞ A ðtÞ þ l þ k l c 1 2 k p21;2 ! c 4 þ 2p1;2 p1;2 AðtÞ c þ ðk l2 cÞ l1 AðtÞ ¼ 0 þ ðk l2 cÞ k p21;2
A00 ðtÞ
admit the solutions
A1;2 ¼ A0 em1;2 t ; where A0 is a constant and
m1;2 ¼
d : a p¼p1;2
Going back to the original variable we get 2
u1;2 ðx; tÞ ¼ A0 em1;2 t ðx þ c1 Þp1;2 : In the same way after substituting A1;2 in the second equation of the system (18) it is a simple matter to solve it and get the corresponding B1;2 ðtÞ. 4.4.3. About the solutions of Eq. (21) According to [13], after putting
z¼
Z
p a A b dA;
Eq. (27) becomes an Abel equation of the second kind in the canonical form pþ1
ww0 ¼ w þ
cz þ dz ; b azp
ð29Þ
where w ¼ wðzÞ. We observe that Eq. (29) is always defined because we are in the case p – 1; 2 which implies a – 0. In the case b ¼ 0, that is when
k l2 c ¼
c k
l1 ;
the Eq. (29) specializes as
ww0 w ¼ gz þ hz da
1p
ð30Þ
;
ac
with g ¼ and h ¼ . Once Eq. (29) is solved, we can determine the function A and, consequently, system (18) is solved. The solutions of (30) are, usually, determined in a parametric form and, not rarely, they depend on special functions, such as Elliptic Weierstrass function and Bessel function. A large number of equations of the type (30) are discussed in [13]. 4.5. Critical points of the system (4) Let us now find the values of the functions AðtÞ and BðtÞ at their critical points. By solving (18) after having put A0 ¼ B0 ¼ 0, we obtain
" #1p 2 p2 ððkl1 cÞðc þ l2 Þ k l1 Þ A0 ¼ ; kð4 þ 2pÞðc þ l2 kÞ
" B0 ¼
p2 k
p1
ððkl1 cÞðc þ l2 Þ k pþ1
ð4 þ 2pÞðc þ l2 kÞ
2
l1 Þ
#1p ;
for p – 2 and provided that there is not loss of meaning due to some values of p and/or the other constitutive parameters. For p ¼ 2 the search for the critical points bring to a linear algebraic homogeneous system. Then in order to admit nontrivial solutions the following additional compatibility condition holds
ðc þ l2 Þ
l1
c k
l1 k ¼ 0:
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Therefore
A0 ¼ q
c k
c B0 ¼ q l1 k
ð31Þ
with q non zero arbitrary constant. 4.6. Some remarks about the solutions of the system (4) Taking into account the biological origin of u and v, we assume that u; v P 0. Moreover, solutions of the form (17) imply that AðtÞ > 0 and BðtÞ P 0, provided that c1 > 0 and ðx; tÞ 2 ½0; 1Þ ½0; 1Þ. Concerning with the first equation of (18), in order to avoid the infestation, we must have A0 ðtÞ 6 0 so that AðtÞ is a bounded function. Then a necessary condition to the function A be bounded is that at least one of the coefficients of A and Apþ1 in the first equation of the system (18) is negative, that is:
c k
l1 < 0;
ð32Þ
or
p < 2:
ð33Þ
If we take into account the values of the parameters given in (25) (see also [2]), we can easily conclude that condition (32) cannot be verified. Instead, such condition would only be verified if we assume c=k smaller than l1 . This fact could occurs whenever 1 =k 2 is big, that means that the carrying capacity for aquatic population is smaller than that one related with winged k¼k population. Whenever (33) holds, condition (32) is a sufficient condition to get bounded solutions since the coefficient of AðtÞpþ1 is negative. For p P 2, condition (32) is necessary, but not sufficient. On the other hand if we assume that the solutions of the system (18) are bounded, from the second equation of (18) it follows that a necessary condition to B is bounded is given by
k l2 c < 0: The biological condition for the existence of the mosquito population, see in [14,6], is c l1 ðl2 þ cÞ > 0 which is easily obtained once it is assumed
k<
c ; k > l2 þ c: l1
These assumptions imply that A0 ðtÞ > 0 and B0 ðtÞ > 0, for any p > 2. Such conditions occur when there is an infestation of mosquitos. 5. Conclusions In this paper we recall a model concerned with Aedes aegypti introduced in [4] and look for similarity solutions, in absence of wind effects, by using the symmetries found in [4]. The classification performed there shows that the system considered admits a four dimensional algebra for p ¼ 43 and a three-dimensional algebra for all p – 43 ; 0. After having found the most general reduction for both systems we look for separable variable solutions. Several classes of exact solutions have been found. In particular for p – 43 ; 0 the solution of reduced system is brought to the search for solutions of a class of Lienard equations, which was widely discussed. Finally some remarks on the behavior of invariant solutions of the system (4) and its biological compatibility are done. In particular we observed that, as expected, the boundness of solutions, as well as the biological condition (see [14,6]) for the existence of the mosquito population c l1 ðl2 þ cÞ > 0, are strictly related to the features of the per capita death rate ck l1 in adult phase and to the features of the per capita death rate k l2 in aquatic phase. We wish to stress that the solutions found here, as well as those obtained in [4], are, until now the only exact solutions concerned with the considered mathematical model for Aedes aegypti dispersal dynamics. These special invariant solutions we derived here, in general, do not satisfy arbitrary initial or boundary conditions prescribed for a given problem. However they might be useful for a benchmark test for larger numerical schemes devised to solve our system in a realistic case. Moreover, as subject of further investigations, it will be useful to study and look for additionl solutions of the Lienard Eqs. (19), Abel Eqs. (24) and Abel equations of the second kind (29). We plan to continue in the aforesaid researches together with the search, concerned with model (1), for additional exact solutions and additional news about the structure of its constitutive functions. Acknowledgements The authors 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
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thanks the support from University of Catania through PRA and from MIUR through PRIN: Modelli cinetici e macroscopici per il transporto di particelle in semiconductori: aspetti modellistici, analitici e computazionali. We would like to thank the referees for their useful comments and suggestions which have considerably improved the paper. References [1] Arunachalam N, Tana S, Espino F, Kittayapong P, Abeyewickreme W, et al. Eco-bio-social determinants of dengue vector breeding: a multicountry study in urban and periurban Asia. Bull World Health Organ 2010;88:173–84. [2] Takahashi LT, Maidana NA, Ferreira Jr WC, Pulino P, Yang HM. Mathematical models for the Aedes aegypti dispersal dynamics: travelling waves by wing and wind. Bull Math Biol 2005;67:509–28. [3] Takahashi LT. Modelos matemáticos de epidemiologia com vetores: simulação da propagação urbana e geográfica da dengue. Ph.D. Thesis, Univ. Estadual de Campinas-UNICAMP, Campinas, Brazil, 2004. [4] Freire IL, Torrisi M. Symmetry methods in mathematical modeling Aedes aegypti dispersal dynamics. Nonlinear Anal RWA 2013;14:1300–7. http:// dx.doi.org/10.1016/ j.nonrwa.2012.09.018. [5] Freire IL, Torrisi M. On a model for Aedes aegypti, Numerical Analysis and Applied Mathematics. In: ICNAAM 2012 International Conference of Numerical Analysis and Applied Mathematics, AIP Conf Proc., vol. 1479, p. 1373–1376.