vector - borne diseases are among the most complex infectious diseases to understand , model and 
 control . their complexity stems from the multi - staged life cycle of the vector , often influenced by 
 the climate , the intricate evolution of the parasite , developing both in the vector and the host , 
 and the fact that most vector - borne diseases have humans as well as non - human species as hosts . this article focuses on one 
 particular aspect , namely the evolution of the parasite . for instance , four species of the plasmodium parasite 
 cause malaria : p. falciparum , p. ovale , p. vivax , and p. malariae . however , within these large classification 
 groups , each pathogen can be antigenically diverse and may exist in multiple variants , called 
 strains . multi - strain models for vector - borne diseases have been considered before ( see [ 6 , 8 , 9 ] and the references therein ) , but in most cases strains are assumed to interact 
 through cross - immunity or other mechanism which generates coexistence . for directly transmitted infections , it is well known that in the simplest case when no trade off 
 mechanism is present , the only possible outcome of the 
 competition of n strains is competitive exclusion . in this case , the strain with 
 the largest reproduction number outcompetes and eliminates the remaining strains , a result first 
 established by bremermann and thieme . in the present 
 article because 
 the dynamics of the vector is subjected to a significant influence from the incubation period of the 
 pathogen within the vector , called extrinsic incubation period , and the incubation period within the 
 host , called intrinsic incubation period , we incorporate distributed delay both in the vector and 
 the host to account for these delays . natural delays occurring in the dynamics of vector - borne 
 diseases are typically incorporated as discrete delays or age - structure . as such they lead to 
 backward bifurcation , oscillations [ 1 , 7 , 1618 , 29 , 34 ] 
 and even chaos . in this article , we consider the outcome of the competition in the resulting multi - strain 
 vector - borne model with distributed delay . originally our hypothesis was that , analogously to 
 directly transmitted diseases , the strain with the maximal reproduction number eliminates all the 
 rest . however , the analysis revealed that we need a stronger condition for competitive dominance in 
 the case of vector - borne diseases . in particular , we notice that the basic reproduction number for a 
 strain i ( 0 ) is written as a 
 product of the reproduction number of the vector ( v ) 
 and the reproduction number of the host ( h ) , i.e. 
 0 = 
 hv . we find that for 
 vector - borne diseases the competitive exclusion principle can be formulated as follows : the 
 strain that maximizes both the vector and the human reproduction numbers dominates in the 
 population . we should 
 mention , however , that our competitive exclusion principle for vector - borne diseases is not 
 complete . it would not give the outcome of the competition of the strains when different strains 
 maximize the vector v and the human 
 h reproduction numbers . for instance , if strain one 
 maximizes the human reproduction number , that is , h 
 > h for all i  1 , 
 but strain two maximizes the vector reproduction number , 
 v > 
 v for all i  2 , then our 
 competitive exclusion principle does not imply which strain will dominate in the population . we 
 hypothesize that the strain with the maximal basic reproduction number will dominate but with a 
 different approach than the one we have applied here may be necessary to establish this result . this paper is structured as follows . in section 2 , we introduce a multi - strain version of the 
 vector - borne disease model with two distributed delays . in section 3 , we discuss the equilibria and 
 establish their local stabilities . in section 4 , we establish global stability of the disease - free 
 equilibrium . in section 5 , we verify competitive exclusion principle under the assumption that 
 strain one maximizes both the human and the vector reproduction numbers . as in the introduction , we assume the pathogen causes the vector  host transmission disease 
 existence through multiple strains . the multi - strain model of the vector host interaction 
 divides the vector population under consideration into the following groups : susceptible vectors , 
 denoted by sv , and infected vectors , which are divided into 
 n subgroups : those infected with a strain i , denoted by 
 ivi , ( i = 1 , 2 ,  , 
 n ) . if we let nv(t ) be the total 
 number of vectors at time t , we have 
 nv(t ) = 
 sv(t ) + 
 i=1 ivi(t ) . let 
 nh(t ) be the density of the total host population at 
 time t , which is composed of the number of susceptible individuals 
 sh(t ) , the number of infective individuals 
 ihi(t ) with a strain i , 
 ( i = 1 , 2 ,  , n ) , and the number of recovered or 
 immune individuals rh(t ) . thus , 
 nh(t ) = 
 sh(t ) + 
 i=1 
 ihi(t ) + 
 rh(t ) . for the disease transmission , it is assumed that 
 a susceptible host can receive the infection through contact with a strain i 
 infective vectors . similarly , a susceptible vector can receive the infection through contact with 
 strain i infectious human . furthermore , it is assumed that a susceptible host and 
 susceptible mosquito become exposed when they receive the infection from an infective individual and 
 remain exposed for some period before becoming infectious . let h and v be the birth 
 rate of the host population and the vectors , respectively . b is the average rate of 
 biting on hosts by a single mosquito ( number of bites per unit time ) , and the hosts are always 
 sufficient in abundance ; so that it is reasonable to assume that the biting rate b 
 is constant . thus , the number of bites on hosts per unit time per host is 
 b / nh(t ) . if 
 hi is probability of infected bites with 
 strain i on host individuals that produce an infection , 
 ( b / nh(t))hish(t)ivi(t ) 
 is the infection force of the interaction between the infected mosquitos with strain 
 i infected vectors ivi(t ) 
 and the susceptible host sh(t ) . as pointed out in the 
 introduction , there exists extrinsic and intrinsic incubation period in mosquitos and hosts , 
 respectively . let  be the extrinsic incubation period of the parasite in the vector 
 population . since only a proportion of vectors survive this incubation period , the force of 
 infection on host individuals at the present time t is 
 ( b / nh(t))hish(t)ivi(t 
  
 )e , 
 where vi is the death rate of the vectors infected 
 with strain i. in reality , however , the incubation period is not a number but an 
 interval during which the maturation of the parasite occurs in different hosts and vectors . hence , 
 we assume that  is a distributed parameter ( see [ 3 , 
 26 ] ) . as a result , the force of infection becomes 
  where the kernel function 
 ghi (  ) represents the distribution of the 
 infectivity of the parasites in host population where the time taken to become infectious is 
 . similarly , let h be the rate of host individuals , and 
 hi be the recovery rate of infected host individuals 
 with strain i. 
 vi is the 
 probability that a susceptible vector bites on the infective host individual with strain 
 i that produces an infection . based on the above facts and that the average total 
 rate of contacts between host and vectors must be conserved [ 28 , p667 ] , 
 gives the incidence of new cases of infection for the vector at the present time t. 
 the kernel function gvi (  ) expresses the 
 infectivity on susceptible vectors during the intrinsic incubation period . here , v , h are , respectively , the upper limits of 
 the parasites incubation periods in vectors and hosts . the terms 
 e , 
 e(h + 
 hi) account for the probability of 
 survival as infectious of host individuals during the intrinsic incubation period . in a more general sense , there would be a possibility that the removed population is alive and 
 bitten again by vectors , or multiple infectious bites on infected people may affect the disease 
 progression . for simplicity , however , we assume that the recovered population will no longer be 
 involved in the transmission process , and multiple infectious bites on infected hosts do not play a 
 role in the progression of the disease . under the above assumption , our multi - strain model with 
 extrinsic and intrinsic distributed incubation periods can be formulated as follows : model ( 1 ) is equipped with the following initial conditions : all parameters in model ( 1 ) are non - negative . we define the following space of functions where , the banach space c([h , 0 ] , r ) of 
 continuous functions mapping the interval [ h , 0 ] into r is 
 equipped with the sup - norm |||| = 
 suph0 
 |()| . by the standard theory of functional differential equations , it can be verified that solutions of ( 1 ) with non - negative 
 initial conditions ( 2 ) has a unique solution ( sv ( t ) , 
 ivi(t ) , 
 sh(t ) , 
 ihi(t ) , 
 rh(t ) ) which remains non - negative for all 
 t  0 . moreover , we can show the solutions of system ( 1 ) are ultimately 
 uniformly bounded in x. in fact , it follows from the total vector population size 
 nv(t ) satisfying 
 nv ( t ) = 
 v vnv(t ) that so , it is assumed that the total vector and host population sizes 
 nv(t ) , 
 nh(t ) are asymptotically constant . it is easy to see that the set  is positively invariant for system ( 1 ) . we note that 
 rh(t ) can be removed from the equations of system ( 1 ) , 
 it is sufficient to analyse the dynamical behaviour of solutions to system ( 1 ) without the equation 
 of rh(t ) . for simplicity , we introduce the following 
 notation . furthermore , we impose the following assumptions : ( 1 ) it is assumed that gvi( ) , and 
 ghi( ) are continuous on [ 0 , 
 h ] , i = 1 , 2 ,  , n ; ( 2 ) 
 gvi( ) , and 
 ghi( ) satisfy ( 3 ) gvi( )  0 , 
 ghi( )  0 for 0 n. here 
 h = max{h , 
 v}. in the next section , we investigate the dynamics of the following system from the above discussion , we have always the following proposition for system ( 3 ) there exists a unique solution 
 ( sv(t ) , 
 ivi(t ) , 
 sh(t ) , 
 ihi(t ) ) of system ( 3 ) 
 with initial conditions ( 2 ) . this solution is non - negative for all 
 t  0 . moreover , all the solutions of system ( 3 ) are 
 ultimately uniformly bounded in . system ( 3 ) always has a unique disease - free equilibrium 0 , which is given 
 by where 0 = ( 0 ,  , 0 ) is an n - dimensional vector of 
 zeroes . according to the definition of the basic reproduction number , which is the expected number of secondary cases produced in an entirely susceptible 
 population by a typical infected individual during its entire infectious period , we define the 
 reproduction number of strain i : we notice that the disease cycle in vector - borne disease consists of two transmission pathways : 
 human - to - vector transmission and vector - to - human transmission . the reproductive number of human - to - vector 
 transmission of strain i is given by and gives the number of secondary infected vectors that one infectious host will produce in an 
 entirely susceptible vector population . similarly , the reproduction number of the vector - to - human 
 pathway of strain i is given by obviously , the reproduction number for a strain i in system ( 3 ) is a product of 
 the human and vector reproduction numbers : we define a reproduction number of the whole system ( 3 ) by direct calculation , we show that for each i , in system ( 3 ) , there is a 
 corresponding strain - i equilibrium i given 
 by where the non - zero components i*vi and 
 i*hi are in position i + 1 and 
 n + j + 1 , respectively . moreover , equilibrium 
 i exists if and only if 
 0 > 1 . the non - zero components of the 
 equilibrium i are given by now we investigate the stability of the equilibria in system ( 3 ) , let us first linearize system 
 ( 3 ) at disease - free equilibrium 0 . in particular , let 
 sv(t ) = 
 v/v + 
 xv(t ) , 
 ivi(t ) = 
 yvi(t ) , 
 sh(t ) = 
 h/h + 
 xh(t ) , 
 ihi(t ) = 
 yhi(t ) . we look for exponential solutions 
 ( xv(t ) = 
 xve , 
 yvi(t ) = 
 yvie , 
 xh(t ) = 
 xhe , 
 yhi(t ) = 
 yhie ) 
 of the time - dependent system for the perturbations . thus , we obtain the following eigenvalue 
 problems : solving the linear system ( 4 ) , it is easy to obtain that 
 v , h are two 
 negative characteristic roots of system ( 3 ) and other characteristic roots ( n 
  2 ) are determined by the following equation : we now show that all characteristic roots in equation ( 5 ) have a negative real part . note that equation ( 5 ) is continuous functions of  , and there is a continuity in the eigenvalues for 
   0 . using rouche 's theorem [ 5 , theorem 
 9.17.4 ] , we know that given the eigenvalues were all negative 
 in the non - delay case , introducing a delay can cause a change in stability if and only if for some 
 value of  , there exists a purely imaginary root of equation ( 5 ) , i.e. a pair of eigenvalues must cross the imaginary axis if they are to become 
 positive . for the equation ( 5 ) , we have the following 
 lemma : if 0 = max{0 ,  , 
 0 } < 1 , the characteristic 
 equation ( 5 ) has no purely imaginary roots set to proceed , we first consider equation ( 6 ) with 
  = 0 . it is easy to verify that all roots in equation ( 6 ) have negative real parts for 0 
 < 1 and  = 0 . for > 0 , we show that 
 q( ,  ) = 0 has no purely imaginary roots . set let  = i (  > 0 ) be a root of equation ( 6 ) . thus , we have thus , we have |f1 ( i , )|  
 ahiavi . from equation ( 6 ) , we have f2(i ) 
  f3f1 ( i ,  ) = 
 0 . therefore , we have that is , |f2(i)|  
 ahiavi|f3| . 
 thus , we have it follows from equation ( 7 ) that we obtain obviously , if 0 < 1 , i 
 = 1 , 2 ,  , n , it is impossible for equation ( 8) to hold . hence , the linearized system ( 5 ) can not have any 
 roots with a positive real part . therefore , from the above discussion , we have established the following result : if 0 = max{0 ,  , 
 0 } < 1 , then the disease - free 
 equilibrium 0 
 of system ( 3 ) is locally asymptotically stable . if 
 0 > 1 , it is unstable . now we shall establish the local stability of the single - strain equilibrium 
 i for a fixed i. we have the following 
 result : assume for a fixed i , 0 > 
 1.if 0 < 
 0 for all j  i , 
 then the strain - i equilibrium i is locally asymptotically 
 stable . otherwise , if there exists k0 
 such 0 > 
 0 , ( k0  
 i ) , then the single - strain equilibrium 
 i is unstable . to simplify the presentation , without loss of generality , we shall assume that i 
 = 1 , i.e. 0 > 1 , and 
 0 < 0 , for 
 i = 2 ,  , n. we linearize system ( 3 ) around the 
 strain - one equilibrium 1 . let sv(t ) 
 = s*vi + 
 xv1(t ) , 
 iv1(t ) = 
 i*v1 + 
 yv1 ( t ) , 
 sh(t ) = 
 s*h1 + 
 xh1(t ) , 
 ih1(t ) = 
 i*h1 + 
 yh1(t ) , 
 ivi(t ) = 
 yvi(t ) , 
 ihi(t ) = 
 yhi(t ) , for i = 2 , 
  , n. we obtain the following linearized system : we notice that the linearized equations for strains k = 2 , 3 ,  , 
 n can separate from the whole system . thus , by direct calculation , we obtain the 
 following characteristic equations : similar to proof in lemma 3.1 , it is easy to show that if 
 0 < 0 , 
 i = 2 , 3 ,  , n , equation ( 10 ) has no purely imaginary roots and all eigenvalues of 
 these equations have negative real parts . therefore , the stability of 1 depends 
 on the eigenvalues of the following system : by looking for exponential solutions of ( 11 ) , we obtain the following the characteristic 
 equation now we show that equation ( 12 ) has no purely 
 imaginary roots . in fact , we divide both sides by (  + 
 h ) , then we set if  is a root with  > 0 , it 
 follows from equation ( 13 ) that using the equilibrium point ( sv1 * , 
 iv * , 
 sh1 , 
 iv * ) satisfies the corresponding equation , we 
 obtain it is a contradiction . hence , using rouche 's theorem [ 5 , theorem 9.17.4 ] , we only show that equation ( 12 ) can not have any roots with a non - negative real part . therefore , the strain - one equilibrium 1 is locally asymptotically stable for 
 0 > 1 and 0 
 < 0 , i = 2 , 3 ,  , 
 n. otherwise 1 is unstable . in the previous section , we have shown that equilibria are locally stable , i.e. given the 
 conditions on the parameters , if the initial conditions are close enough to the equilibrium , the 
 solution will converge to that equilibrium . in this section , we have the following result : if r0 = max{0 ,  , 
 0 } < 1 , the disease - free 
 equilibrium 0 
 is globally asymptotical stable . from the first and the third equations of system ( 3 ) , we obtain it follows from the above inequalities that thus , from system ( 3 ) and inequalities ( 15 ) , we have from equation ( 16 ) , we obtain that thus , from the above expression , for i = 1 ,  , n , 
 we have similarly , for i = 1 ,  , n we also have from equations ( 17 ) and ( 18 ) , we have since 0 < 1 and 
 ivi(t ) , i = 1 , 
  , n is bounded , it follows from the above expression that similarly , from equation ( 18 ) , we have therefore , ( ivi(t ) , 
 ihi(t ) )  ( 0 , 0 ) as 
 t  . from the above discussion and theorem 3.1 , we can conclude 
 that the disease - free equilibrium 0 is globally asymptotically stable for 
 0 for directly transmitted diseases , bremermann and thieme   establish that in the simplest scenario when competitive exclusion is the only outcome , 
 the strain with the largest reproduction number persists , while the remaining strains die out . while 
 extending the bremermann and thieme   result to vector - borne 
 diseases remains an open problem ( in the case of vector - borne diseases , we seem to need a 
 stronger condition for competitive exclusion ) , we were able to show competitive exclusion 
 under the somewhat stronger condition that the winning strain maximizes both the human and the 
 vector reproduction numbers , not just the overall reproduction number . mathematically speaking , establishing the competitive exclusion principle means establishing the 
 global stability of the strain one equilibrium 1 . from theorem 3.2 , we obtain 
 that under some conditions the equilibrium 1 is locally asymptotically stable . in 
 this section , we are ready to establish the global stability of the strain one equilibrium 
 1 by constructing the lyapunov functional , similar to the one used in [ 10 , 11 , 14 , 15 , 1921 , 24 , 25 , 31 , 32 ] . we note that for 
 constructing the lyapunov function , such as f(x ) = 
 x  1  1n x , here 
 f(x ) is not defined if x = 0 . thus , in 
 order to construct our lyapunov function , we must first show that when 0 > 
 1 , the disease persists in the form of strain one . now we apply the persistence theory by hale and 
 waltmann   for infinite - dimensional systems to show the 
 uniform persistence of strain one in system ( 3 ) . the methods and techniques have been recently 
 employed by other authors ( see [ 27 , 30 , 33 ] ) . let x be a 
 complete metric space with metric d. suppose that t is a 
 continuous semiflow on x , that is , a continuous mapping t : [ 0 , 
  )  x  x with the following properties : where tt denotes the mapping from x to 
 x given by tt(x ) = 
 t(t , x ) . the distance d(x , y ) of 
 a point x  x to a subset y of 
 x is defined by recall that the positive orbit (x ) 
 through x is defined as 
 (x ) = 
 t > 
 0{t(t)x } , and its 
 limit set is . define by w(e ) the strong stable set of a compact 
 invariant set e as w(e ) = 
 { x : x  x , (x ) 
 =  , (x )  e}. 
 denote by t(t ) , t  0 , the family of 
 solution operators corresponding to equation ( 3 ) . the 
 following assumption must be satisfied ( h1 ) : let x0  x , x  
 x , x  x = . 
 x0 is open and dense in x. moreover , the 
 c0-semigroup t(t ) on 
 x satisfies let tb(t ) = 
 t(t)|x0 and 
 ab be the global attractor for 
 tb(t ) . the following result is discussed in [ 13 , theorem 4.2 ] : suppose that t(t ) satisfies ( h1 ) and 
 the following conditions : ( i)there is a t0 > 0 such that 
 t(t ) is compact for t > 
 t0;(ii)t(t ) is point dissipative in x;(iii)b = x  
 ab(x ) is isolated and has an 
 acyclic covering , where(iv)w(mi )  
 x0 =  for i = 1 , 2 ,  , 
 n. there is a t0 > 0 such that 
 t(t ) is compact for t > 
 t0 ; t(t ) is point dissipative in x ; b = x  
 ab(x ) is isolated and has an 
 acyclic covering , where w(mi ) , 
 n. then t(t ) is a uniform repeller with respect to 
 x0 , i.e. , there is an  > 0 such that for 
 any x  x , lim inft+ 
 d(t(t ) , x0 )  . we use the above lemma to establish the following result : assume that 0 > 1 . then the disease persists 
 in the population . if 0 > 1 and strains from 
 the 2 to n approach zero , that is then , for all initial conditions that belong to 0 , 
 strain one can persist . in particular , there exists a constant  > 0 
 such that under the assumptions of theorem 5.1 , system has two non - negative equilibrium 
 0 and 1 . let  be a vector of initial 
 conditions for system ( 3 ) . let where , the banach space c([h , 0 ] , 
  ) of continuous functions mapping the interval 
 [ h , 0 ] into  equipped with the sup - norm 
 |||| = 
 suph0 |()| . it suffices to show that there exists an  
 > 0 such that for any solution ut of system ( 3 ) initiating from 
 x , lim inft 
 d(ut , x0 )  . to this end , we 
 have to verity below that the conditions of lemma 5.1 are satisfied . clearly , x0 is relatively closed in 
 x. it is easy to verify that system ( 3 ) is point dissipative . set we now claim that m = 
 { ( v/v , 0 , 
 h/h , 0 ) } , where 
 0 = ( 0 ,  , 0 ) is an n  dimensional vector of 
 zeroes . assuming   m it suffices to show that 
 ivi(t ) = 
 ihi(t ) = 0 , i 
 = 1 ,  , n , t  0 . assume that on the 
 contrary , there exists t0 > 0 such that case ( i ) 
 for some strain - i0 , we have 
 ivi0(t0 ) 
 > 0 while ihi(t0 ) = 
 0 , i = 1 ,  , n or case ( ii ) for 
 some strain - j0 , we have 
 ihj(t0 ) > 0 while 
 ivi(t0 ) = 0 , 
 i = 1 ,  , n in case ( i ) , from the third equation of ( 3 ) , we have hence , there is a sufficiently small constant 0 such that 
 hi0(t ) > 0 , 
 t  ( t0 , t0 
 + 0 ) . on the other hand , from 
 ivi0(t0 ) > 0 , we obtain a positive 1 ( 0 < 1 < 
 0 ) such that 
 hi0(t ) > 0 , 
 t  ( t0 , t0 
 + 1 ) . thus , we obtain 
 ivi0(t ) 
 > 0 , 
 ihi0(t ) 
 > 0 , this is in contradiction with the assumption that 
 ( sv(t ) , 
 ivi(t ) , 
 sh(t ) , 
 ihi(t ) )  
 m , t  0 . let 2 = xy2 
 (x ) , where y2 is the global attractor of 
 t(t ) restricted to x . we now show that 
 2 = { 0}. in fact , it follows from 2 
  m and the first and third equations of ( 3 ) . we have 
 limt+ 
 sv(t ) = 
 v/v , 
 limt+ 
 sv(t ) = 
 v/v . thus , 
 { 0 } is the isolated invariant set in x. finally , we need to show that w(0 )  
 x =  . assume on the contrary that there exists a solution ut  
 x such that hence , for each  > 0 there exists a time t0 such that 
 for each t > t0 we have by shifting the dynamical system , we may assume that the above inequalities are satisfied for 
 t  0 . from the first and third equations of ( 3 ) and taking into account the 
 above inequalities , we have therefore , from equation ( 20 ) , we have thus , there exists a t1 > t0 such 
 that , for any t > t1 , and using the second 
 equation of system ( 3 ) and the inequality above , we have similarly , there exists a t2 > t0 , 
 such that for any t  t2 , we have by the mean value theorem for integrals , we know that for any time t , there is a 
 t such that therefore , from equations ( 21)(23 ) , we obtain notice that 0 = 
 ( ah1 
 av1 
 h1 
 v1 
 h 
 v/(h 
 + 
 h1)hv ) 
 > thus , we may assume we have chosen sufficiently small  in such a way that we 
 have since ( ivi(t ) , 
 ihi(t ) )  ( 0 , 0 ) , i 
 = 1 , , n as t   , then by a standard 
 comparison argument , the solution ( xvi(t ) , 
 yhi(t ) ) of the following equation with initial conditions 
 xv0(t1 ) = 
 ivi(t1 ) , 
 yn0(t2 ) = 
 ihi(t2 ) has to converge to ( 0 , 0 ) 
 as well for inequality ( 25 ) . on the other hand , for i = 1 ,  , n , if we set , obviously , it follows from the solution 
 ( xvi(t ) , 
 yhi(t ) )  ( 0 , 0 ) that 
 wi(t )  0 as t  
 . in particular , we have w1(t )  0 as 
 t  . however , by directly calculating the derivative in equation 
 ( 27 ) , we obtain thus , for the above expression ( 28 ) , if we consider i = 1 and use 
 equation ( 25 ) , then we have 
 w1(t )   as t  
 . this is a contradiction . therefore , we have w(0 )  
 x = . this completes the proof of theorem 5.1 . assume that 0 > 1 , 
 h < h 
 and v < v for 
 i = 2 ,  , n. then , equilibrium 1 
 is globally asymptotically stable . because of the complexity of the expressions , we define the lyapunov functional in components and 
 take the derivative of each component separately . set f(x ) 
 = x  1  ln x , x > 0 . the components 
 of the lyapunov 's functional are defined as follows : where , j( ) = 
  gvj( ) 
 d , j = 1 ,  , n. calculating the derivative of the above expressions in equation ( 29 ) along with system ( 3 ) , we obtain the above equations can be reduced to the following : differentiating the time derivative of 
 w+iv1(t ) 
 with respective to t , we obtain notice that limh 
 1( ) = 0 and 
 f(ih1(t 
  )/i*h1 ) is bounded . it follows that limh 
 1( ) 
 ( ih1(t  
 )/i*h1 ) = 0 hence , we have thus , by equations ( 30)(33 ) , we obtain where , j( ) = 
  
 ghj( ) d , j = 1 ,  , 
 n. calculating the derivative of the expressions in equation ( 36 ) , respectively , along system ( 3 ) , we obtain by equations ( 37)(39 ) and the system ( 3 ) , we obtain from equations ( 35 ) and ( 40 ) , we obtain it follows from equation ( 41 ) that using the equality 
 0 
 gh1( ) d = 
 ah1 , 0 
 gv1( ) d = 
 av1 , we obtain using f(x ) = x  1  ln 
 x , x > 0 and the expression of 
 h and 
 v , from equation ( 42 ) , we obtain since f(x )  0 for x > 0 , 
 h < 
 h and 
 v < v , 
 i  1 we have dw(t)/dt  
 0 . let we want to show that the largest invariant set in  is the singleton 
 { 1}. in fact , from equation ( 43 ) , 
 dw(t)/dt = 0 , and using the fact that 1 
  x + ln x  0 for all x 
 > 0 with equality holding iff x = 1 , imply for all   [ 0 , h ) and t > 0 . furthermore , 
 for each element of  , we have 
 sv(t ) = 
 s*v1 , 
 sh(t ) = 
 s*h1 and , since 
  is invariant , dsv / dt 
 = 0 , dsh / dt = 0 . using equations ( 44 ) and ( 45 ) , we obtain using the equilibrium 1 satisfies the relation 
 v  
 vs*v1 = 
 v1 
 av1 
 i*h1s*v1 , 
 we easily obtain iv(t ) = 
 i*v1 , for t  
 0 . similarly , it is easy to prove that ih(t ) = 
 i*h1 , for t  0 . thus , we show 
 dw(t)/dt = 0 if and only if 
 sv(t ) = 
 s*v1 , 
 i*v1(t ) = 
 i*v1 , 
 sh(t ) = 
 s*h1 
 i*h1(t ) = 
 i*h1 ,  ( t ) 
 = 0 , i*hi(t ) 
 = 0 , i = 2 , hence , the largest invariant set in  is the singleton 
 { 1}. by the lasalle invariance principle and theorem 3.2 , we show that 
 equilibrium 1 is globally asymptotically stable . in this paper , we introduce a multi - strain model of a vector - borne disease with distributed delay 
 in the vector and the host . we define the basic reproduction number of the disease as the maximum of 
 the reproduction numbers of each strain . we show that if 0 < 1 the 
 disease - free equilibrium is locally and globally stable , that is the number of infected with each 
 strain goes to zero . furthermore , we show that if 0 > 1 , then the disease 
 persists . our main objective with the article was to extend the competitive exclusion result established by 
 bremermann and thieme . this principle states that in the 
 basic ode model when multiple strains compete only the strain with the largest reproduction number 
 persists , if its reproduction number is larger than one , and the other strains die out . in this 
 case , coexistence is not possible outside of the trivial scenario when the reproduction numbers of 
 two different strains are equal . to prove the competitive exclusion result for the multi - strain vector - borne model with 
 distributed delay however , we were able to establish only the 
 following weaker result : it is well known that the reproduction number in vector - borne diseases 
 splits into a vector reproduction number and host reproduction number , namely 
 0 = h 
 v . our result states that the strain that maximizes both the 
 vector reproduction number and the human reproduction number will dominate in the population and 
 eliminate the rest . in particular , if we assume that h 
 < h and 
 v < 
 v for i = 2 , 
  , n then strain one will dominate and eliminate the rest . in this case , 
 coexistence does not occur . in fact , it can be shown that model ( 3 ) does not have a coexistence 
 equilibrium if all strain reproduction numbers are different . our result does not resolve the competition outcomein the case when different strains maximize 
 the human and the vector reproduction numbers . in this sense , the question whether complete 
 competitive exclusion holds in the vector we surmise 
 that because coexistence equilibrium is not present , competitive exclusion in this case also occurs 
 with the strain with maximal reproduction number eliminating the rest . establishing this result , , public health efforts will work best if directed to 
 monitoring the vector and host reproduction numbers of the multiple strains in vector - borne 
 diseases . control measures may need to be applied to the strain(s ) that maximize the vector and the 
 host reproduction numbers . at last , we should mention that , similar to paper , 
 our analysed approaches in this paper can be applied for a multi - strain model of a vector - borne 
 disease with discrete delays . the obtained results in this paper also hold for the multi - strain 
 model with discrete delays .
a multi - strain model of a vector - borne disease with distributed delay in the vector and the host 
 is investigated . 
 it is shown that if the reproduction number of the model 0 
 < 1 , the unique disease - free equilibrium is globally asymptotically stable . without loss of 
 generality , strain one 
 is assumed to have the largest reproduction number . in this case , the 
 dominance equilibrium of strain one is shown to be locally stable . the basic reproduction number for 
 a strain i ( 0i ) is written as a 
 product of the reproduction number of the vector 
 ( vi ) and the reproduction number of 
 the host ( ih ) , i.e. 
 i0 = 
 ihvi . 
 the competitive exclusion principle is derived under the somewhat stronger condition that if strain 
 one maximizes both the reproduction number of the host 
 hi < 
 h1 , i  1 and the 
 reproduction number of the vector vi 
 < v1 , i  1 , strain 
 one dominance equilibrium is globally asymptotically stable.ams subject classification : 92d30