Dynamics of Vertically and Horizontally Transmitted Parasites: Continuous vs Discrete Models

International Journal of Difference Equations ISSN 973-669 Volume 13 Number 2 pp 139 156 218) http://campusmstedu/ijde Dynamics of Vertically and Horizontally Transmitted Parasites: Continuous vs Discrete Models Priyanka Saha and Nandadulal Bairagi Jadavpur University Department of Mathematics Centre for Mathematical Biology and Ecology olkata-732 India priyankasaha392@gmailcom nbairagimath@jadavpuruniversityin Abstract In this paper we analyze a continuous time epidemic model and its discrete counterpart where infection spreads both horizontally and vertically We consider three cases: model with horizontal and imperfect vertical transmissions model with horizontal and perfect vertical transmissions and model with perfect vertical and no horizontal transmissions Stability of different equilibrium points of both the continuous and discrete systems in all cases are determined It is shown that the stability criteria are identical for continuous and discrete systems The dynamics of the discrete system have also shown to be independent of the step size Numerical computations are presented to illustrate analytical results of both the systems and their subsystems AMS Subject Classifications: 37N25 39A3 92B5 92D25 92D4 eywords: Epidemic model perfect and imperfect vertical transmissions horizontal transmission basic reproduction number stability 1 Introduction Mathematical models play significant role in understanding the dynamics of biological phenomena System of nonlinear differential equations are frequently used to describe these biological models Unfortunately nonlinear differential equations in general can not be solved analytically For this reason we go for numerical computations of the Received April 28 218; Accepted July 18 218 Communicated by Irena Jadlovská

14 Saha and Bairagi model system and discretization of the continuous model is essential in this process Here we shall eplore and compare the dynamics of a continuous time epidemic model and its subsystems with their corresponding discrete models Lipsitch et al [1] have investigated the dynamics of vertically and horizontally transmitted parasites of the following population model where the state variables X and Y represent respectively the densities of uninfected and infected hosts at time t: dx dy = = [b 1 X + Y ) } ] u βy X + e [ 1 X + Y ) } ] u y + βx Y 1 X + Y ) } Y 11) This model demonstrates the rate equations of a density dependent aseual host populations where infection spreads through imperfect vertical transmission as well as horizontal transmission Horizontal transmission of infection follows mass action law with β as the proportionality constant Vertical transmission is imperfect because infected hosts not only give birth of infected hosts at a rate but also produce uninfected offspring at a rate e In case of perfect vertical transmission however infected hosts give birth of infected hosts only and in that case e = Parasites may affect the fecundity and morbidity rates of its host population [23] It is assumed here that the death rate of infected hosts is higher than that of susceptible hosts ie u y > u and the birth rate of susceptible hosts is higher than that of infected hosts ie b + e Standard finite difference schemes such as Euler method Runge utta method etc are frequently used for numerical solutions of both ordinary and partial differential equations But the behavior of standard finite difference schemes depend heavily on the step size They fail to preserve positivity of the solutions for all step size These conventional discretized models also show numerical instability and ehibit spurious behaviors like chaos which are not observed in the corresponding continuous models In other words these discrete models are dynamically inconsistent So it becomes important to construct discrete models which will preserve all the properties of its constituent continuous models without any restriction on the step size Mickens in 1989 first proposed such nonstandard finite difference NSFD) scheme [4] and was shown to have identical dynamics with its corresponding continuous model It was also demonstrated that the dynamics is completely independent of step size Successful application of this technique in subsequent time is observed in different biological models [5 12] Here we will discretize a continuous time population model in which parasite transmitted both vertically and horizontally following dynamics preserving nonstandard finite difference NSFD) method introduced by Mickens [4] We present the local stability analysis of both the continuous and discrete systems and prove that the dynamic behaviour of both systems are identical with same parameter restrictions Moreover we prove that the proposed discrete models are positive for all step size and dynamically consistent The paper is organized as follows We present the analysis of the continuous time model and its subsystems in the net section Section 3 contains the corresponding

Dynamics of Vertically and Horizontally Transmitted Parasites 141 analysis in discrete system In Section 4 we present etensive numerical simulations in favour of our theoretical results Finally a summary is presented in Section 5 2 Analysis of Continuous Time Models Lipsitch et al [1] analyzed the system 11) with perfect vertical transmissions the case e = ) and horizontal transmission as well as the system with perfect vertical transmission but no horizontal transmission the case β = e = ) The general case when e β ) was analyzed numerically and its importance in the prevalence of infection was discussed Here we first give stability analysis of equilibrium points of the general model 11) and deduce the results of subcases whenever applicable The continuous system 11) has two boundary equilibrium points E = ) E 1 = X ) where X = 1 u ) and one interior equilibrium point E = X Y ) b where the equilibrium densities of susceptible and infected hosts are given by X = B + B 2 4AC 2A and Y = β )X + u y ) with A = β b b 2 y b e) + β + e)} y B = b u ) + b + β + e) u y ) + 2e β ) u y ) b 2 y C = e2 u y )u y b 2 y eβ ) 21) The trivial equilibrium E eists for all parameter values but the infection free equilibrium E 1 eists if b > u The coeisting equilibrium point E eists if b > u > u y > β and X > β u y We have the following theorem for the stability of different equilibrium points Theorem 21 System 11) is locally asymptotically stable around the equilibrium point i) E if b u and R < 1 where R = V + H with V = u H = b u y β 1 u ) and it is unstable whenever R > 1 u y b

142 Saha and Bairagi iii) E if b > u > u y > β and X > β u y Proof Local stability of the system around an equilibrium point is performed following linearization technique For it we compute the variational matri of system 11) at an arbitrary fied point X Y ) as ) a11 a V X Y ) = 12 22) a 21 a 22 where a 11 = b 1 X + Y ) a 12 = b X } b X u βy ey βx + e 1 X + Y ) } ey a 21 = Y + βy a 22 = 1 X + Y ) } Y u y + βx 23) At the trivial fied point E the variational matri is ) b u V E ) = e u y Corresponding eigenvalues are given by λ 1 = b u and λ 2 = u y E will be locally asymptotically stable if λ 1 = b u < and λ 2 = u y < ; ie if b < u and < u y Thus if the birth rates of susceptible hosts and infected hosts are less than their respective death rates then both populations goes to etinction and the trivial equilibrium will be stable At the aial equilibrium point E 1 the variational matri is computed as b 1 u ) b + β) 1 u ) + eu V E 1 ) = b b b u u y + β 1 u ) b b The corresponding eigenvalues are λ 1 = b 1 u ) and λ 2 = u u y + b b β 1 u ) It is to be noted that λ 1 < whenever E 1 eists The other eigenvalue b by u can be rearranged as λ 2 = u y + β 1 u ) } 1 Thus λ 2 < whenever b u y u y b ) ) by u R < 1 where R = V + H Note that V = is the basic reproduction b u y

Dynamics of Vertically and Horizontally Transmitted Parasites 143 number due to vertical transmission and H = β u y X is the basic reproduction number due to horizontal transmission At the interior equilibrium point E the variational matri is where a 11 = ey X V E ) = a 11 a 12 a 21 a 22 1 X + Y ) a 12 = b X βx + e a 21 = Y + βy a 22 = Y ) 24) } b X ey 1 X + Y ) } ey 25) E will be stable if and only if tracev E )) < and detv E )) > From the eistence condition of E one can observe that both a 11 and a 22 are negative Thus tracev E )) < After some simple algebraic manipulation one gets detv E )) = eu yy X 1 u y ) + βx Y b e) + β 2 X Y + eβ2 X Y Thus whenever E eists and b + e we have detv E )) > and E becomes locally asymptotically stable This completes the proof 21 Model with Horizontal and Perfect Vertical Transmissions The vertical transmission is perfect if infected hosts give birth to infected offspring only In this case e = and the system 11) becomes dx = b X 1 X + Y ) } u X βxy 26) dy = Y 1 X + Y ) } u y Y + βxy The continuous system 26) has four equilibrium points viz E H = ) E1 H = X ) E2 H = Ȳ ) and interior equilibrium point E H = XH YH) where X = 1 u ) b Ȳ = 1 u ) y and XH = b u y u β u y ) YH = u b u y + βb u ) ββ + b ) ββ + b )

144 Saha and Bairagi The trivial equilibrium point E H always eists E1 H eists if b > u E2 H eists if > u y and the interior fied point EH b u y eists if b > u > u y > u + β 1 u ) y and R > 1 where R = V + H with V = u H = b u y β u y 1 u b ) The following results are known [1] Theorem 22 System 26) is locally asymptotically stable around the equilibrium point i) E H if b u and R < 1 iii) E H 2 if > u y and b u y iv) E H if b > u > u y b u y u + β 1 u ) y 1 u ) y and R > 1 22 Model with Perfect Vertical Transmission and no Horizontal Transmission In this case e = β = and the system 11) reduces to dx = b X 1 X + Y ) } u X 27) dy = Y 1 X + Y ) } u y Y The continuous system 27) has three equilibrium points viz E V = ) E1 V = X ) and E2 V = Ȳ ) where X = 1 u ) and Ȳ = 1 u ) y The eistence conditions for E V 1 and E V 2 are b > u and > u y respectively It is to be noted that no interior equilibrium does eist here The following results are known [1] Theorem 23 System 27) is locally asymptotically stable around the equilibrium point i) E V if b u and u y < b u iii) The equilibrium point E V 2 is always unstable

Dynamics of Vertically and Horizontally Transmitted Parasites 145 3 Discrete Models In this section we construct three discrete models corresponding to the continuous models 11) 26) and 27) following nonstandard finite difference method The objective is to show that all the discrete models have the same dynamic properties corresponding to its continuous counterpart and the dynamics does not depend on the step size The NSFD procedures are based on just two fundamental rules [13 15]: i) The discrete first derivative has the representation d k+1 ψh) k h = t φh) where φh) ψh) satisfy the conditions ψh) = 1 + Oh 2 ) φh) = h + Oh 2 ); ii) Both linear and nonlinear terms may require a nonlocal representation on the discrete computational lattice For convenience we first epress the continuous system 11) as follows: dx dy = b X b X 2 b XY u X βxy + ey exy ey 2 = Y XY Y 2 u yy + βxy 31) We now employ the following nonlocal approimations term wise for the system 31) dx X n+1 X n dy φ 1 h) Y n+1 Y n φ 2 h) b X b X n Y Y n b X 2 b X n X n+1 XY X n Y n+1 XY X n+1 Y n Y 2 Y n Y n+1 u X u X n+1 u y Y u y Y n+1 ey ey n βxy βx n Y n ey 2 e X n+1yn 2 X n where h > ) is the step size and denominator functions are chosen as φ 1 h) = 1 ep βuy )} h Note that φ i h) i = 12 are positive for all h > 32) βu y φ 2 h) = h 33)

146 Saha and Bairagi By these transformations the continuous system 31) is converted to X n+1 X n φ 1 h) Y n+1 Y n φ 2 h) = b X n b X nx n+1 b X n+1y n u X n+1 βx n+1 Y n + ey n e X n+1y n e X n+1 Y 2 n X n = Y n X ny n+1 Y ny n+1 u y Y n+1 + βx n Y n 34) System 34) can be rearranged to obtain X n+1 = Y n+1 = X n 1 + φ 1 h)b ) + φ 1 h)ey n ) b 1 + φ 1 h) X n + b Y n + u + βy n + e Y n + e Yn 2 X n Y n 1 + φ 2 h) + βx n )} ) 35) 1 + φ 2 h) by X n + by Y n + u y where φ 1 h) and φ 2 h) are given in 33) The model 35) is our required discrete model corresponding to the continuous model 11) It is to be noted that all terms in the right hand side of 35) are positive so solutions of the system 35) will remain positive if they start with positive initial value Therefore the system 35) is said to be positive [16] The fied points of 35) can be calculated by setting X n+1 = X n = X and Y n+1 = Y n = Y One thus get the fied points as E = ) E 1 = X ) where X = 1 u ) and E = X Y ) Note that the equilibrium values and the eistence b conditions remain same as in the continuous system The variational matri of system 35) evaluated at an arbitrary fied point X Y ) is given by a11 a JX Y ) = 12 a 21 a 22 ) 36)

Dynamics of Vertically and Horizontally Transmitted Parasites 147 where a 11 = 1 + φ 1 h)b ) Y 2 X 1 + φ 1 h) b X + b Y + u + βy + e Y + e ) b X1 + φ 1 h)b ) + φ 1 h)ey }φ 1 h) e Y 2 X 2 1 + φ1 h) b X + b Y + u + βy + e Y + )} e Y 2 2 X φ 1 h)e a 12 = 1 + φ 1 h) b X + b Y + u + βy + e Y + ) e Y 2 X X1 + φ 1h)b ) + φ 1 h)ey }φ 1 h) b + β + e + ) 2e Y X 1 + φ1 h) b X + b Y + u + βy + e Y + )} e Y 2 2 X φ 2 h)βy a 21 = ) Y 1 + φ 2h) + βx)}φ 2 h) by 1 + φ 2 h) by X + by Y + u )} 2 y 1 + φ 2 h) by X + by Y + u y a 22 = 1 + φ 2 h) + βx) ) Y 1 + φ 2h) + βx)}φ 2 h) by 1 + φ 2 h) by X + by Y + u )} 2 y 1 + φ 2 h) by X + by Y + u y Let λ 1 and λ 2 be the eigenvalues of the variational matri 36) and we have the following definition [18] in relation to the stability of the system 35) Definition 31 A fied point y) of the system 35) is called stable if λ 1 < 1 λ 2 < 1 and a source if λ 1 > 1 λ 2 > 1 It is called a saddle if λ 1 < 1 λ 2 > 1 or λ 1 > 1 λ 2 < 1 and a nonhyperbolic fied point if either λ 1 = 1 or λ 2 = 1 Lemma 32 See [17 18]) Let λ 1 and λ 2 be the eigenvalues of the variational matri 36) Then λ 1 < 1 and λ 2 < 1 iff i) 1 detj) > ii) 1 tracej) + detj) > and iii) < a 11 < 1 < a 22 < 1 One can then prove the following theorem Theorem 33 System 35) is locally asymptotically stable around the fied point i) E if b u and R < 1 where R = V + H with V = u H = b u y β 1 u ) and it is unstable whenever R > 1 u y b iii) E if b > u > u y > β and X > β u y

148 Saha and Bairagi Proof At the fied point E the variational matri is given by 1 + φ 1 h)b φ 1 h)e 1 + φ 1 h)u 1 + φ 1 h)u JE ) = 1 + φ 2 h) 1 + φ 2 h)u y The corresponding eigenvalues are λ 1 = 1 + φ 1h)b and λ 2 = 1 + φ 2h) Clearly 1 + φ 1 h)u 1 + φ 2 h)u y λ 1 < 1 if b Therefore E will be stable if b < u and < u y hold simultaneously One can similarly compute the eigenvalues corresponding to the fied point E 1 as λ 1 = 1 + φ 1h)u 1 + φ 1 h)b and λ 2 = 1 + φ 2 h) 1 + φ 2 h) 1 u b )} + β ) byu b + u y Note that λ 1 < 1 whenever E 1 eists and λ 2 < 1 whenever R < 1 Thus E 1 is stable if b > u and R < 1 At the interior fied point E the variational matri is given by ) a JE ) = 11 a 12 a 21 a 22 where a 11 a 12 a 21 a 22 = 1 X φ 1 h) b X G + ey ) 1 X + Y + ey } X = φ ) 1h)X e 1 X + Y b X G βx ey } = φ 2h)Y β H = 1 φ 2h) Y H φ 2h)Y H with G = X 1 + φ 1 h)b ) + φ 1 h)ey and H = 1 + φ 2 h) + βx )

Dynamics of Vertically and Horizontally Transmitted Parasites 149 One can easily verify that < a 11 < 1 and < a 22 < 1 Net 1 detje )) = φ 1h)X b X + ey ) } 1 X + Y + ey GH X + φ 1h)φ 2 h)x Y b X + e ) 1 X + Y 2 + e GH Y X + βb X 2 + 2eβ ) ) 1 X + Y + eβx + b Y 1 βx + ey ) )} 1 βx + βx + β 1 X + Y X + φ 2h) X Y 1 φ ) 1h)βu y GH 1 traceje )) + detje )) = φ 1h)φ 2 h)x Y ) GH e 1 X + Y 1 u ) ) y +βx b 1 + β2 X + eβy } X ) From the eistence condition we have 1 X + Y = u X + βx Y > b X + ey Thus X + ) Y < ie X < Also from > β we have > βx and 1 βx > It is easy to observe that φ 1 h) < Thus 1 detje )) > βu y and 1 traceje )) + detje )) > Hence E is locally asymptotically stable whenever it eists This completes the proof Remark 34 It is interesting to note that the dynamic properties of the discrete system 35) are identical with its continuous counterpart 11) So the discrete model is dynamically consistent The stability of the fied points also does not depend on the step size Since all solutions of the discrete model 35) remain positive when starts with positive initial value there is no possibility of numerical instabilities and the model will not show any spurious dynamics 31 Discrete Model for Horizontal and Perfect Vertical Transmissions Here we rewrite the continuous model 26) as dx = b X b X 2 b XY u X βxy

15 Saha and Bairagi dy = Y XY Y 2 u yy + βxy 37) Now we employ the same nonlocal approimations 32) with e = term wise to have the following system: X n+1 X n φ 1 h) Y n+1 Y n φ 2 h) = b X n b X nx n+1 b X n+1y n u X n+1 βx n+1 Y n = Y n X ny n+1 Y ny n+1 u y Y n+1 + βx n Y n 38) The required discrete model is obtained after simplification as follows: X n+1 = Y n+1 = X n 1 + φ 1 h)b ) 1 + φ 1 h) b X n + b Y ) n + u + βy n Y n 1 + φ 2 h) + βx n )} ) 39) 1 + φ 2 h) by X n + by Y n + u y where φ 1 h) and φ 2 h) have the same epression as in 33) It is worth mentioning that the discrete model 39) is positive One can find the same four fied points of 39) as it were in the continuous case The stability properties of each fied point are presented in the following theorem Theorem 35 The system 39) is stable around the fied point i) E H = ) if b u and R < 1 where X = 1 u ) b = Ȳ ) if > u y and b u y Ȳ = and u > u y b u y > u + β 1 u ) and R > 1 b

Dynamics of Vertically and Horizontally Transmitted Parasites 151 32 Discrete Model for Perfect Vertical and no Horizontal Transmission For convenience we first epress the continuous system 27) as dx dy = b X b X 2 b XY u X 31) = Y XY Y 2 u yy In this case we consider the nonlocal approimations 32) with e = β = Note that here φ 1 h) h when β Then the system 31) reads X n+1 X n h Y n+1 Y n h = b X n b X nx n+1 b X n+1y n u X n+1 311) = Y n X ny n+1 Y ny n+1 u y Y n+1 On simplifications we obtain our desired discrete model as X n+1 = Y n+1 = X n 1 + hb ) 1 + h b X n + b Y ) 312) n + u Y n 1 + h ) ) 1 + h by X n + by Y n + u y This system also does not contain any negative terms so solutions remain positive for all step size as long as initial values are positive As in the continuous system 27) the discrete system 312) has same three fied points The stability of each fied point can be proved similarly and has been summarized in the following theorem Theorem 36 The system 312) is stable around the fied point i) E V = ) if b u and u y < b u iii) The fied point E V 2 = Ȳ ) is always unstable

152 Saha and Bairagi 4 Numerical Simulations 6 a) 6 b) 5 5 4 3 2 4 3 2 1 1 2 4 6 8 E 1 12 1 6 c) 2 4 6 8 E 1 12 1 6 d) 5 4 3 2 E * 5 4 3 2 E * 1 1 2 4 6 8 1 12 2 4 6 8 1 12 Figure 41: Phase portraits of the continuous system 11) left panel) and discrete system 35) right panel) Figs a) and b) show that all solutions converge to the disease free equilibrium point E 1 = 8333 ) for β = 1 Figs c) and d) depict that all solutions converge to the endemic equilibrium point E = 1818 4545) for β = 3 Other parameters are b = 6 = 4 u = 1 u y = 2 = 1 e = 2 as in [1] Step size for the discrete model is considered as h = 1 In this section we present some numerical simulations to validate the similar qualitative behavior of our discrete models with its corresponding continuous models For this we consider the same parameter set as in Lipsitch et al [1]: b = 6 = 4 u = 1 u y = 2 = 1 e = 2 We consider different initial values = 1 1) = 2 4) = 7 6) = 1 4) and = 12 15) for both continuous and discrete systems Step size h = 1 is kept fied in all simulations for the discrete systems If β takes the value 1 the parameter set satisfies conditions of Theorems 21ii) and 33ii) In this case all solutions starting from different initial points converge to the infection free point E 1 = 8333 ) in case of both the continuous system 11) Fig 41a)) and the discrete system 35) Fig 41b)) For β = 3 conditions

Dynamics of Vertically and Horizontally Transmitted Parasites 153 of Theorems 21iii) and 33iii) are satisfied and all solution trajectories reach to the coeistence equilibrium point E = 1818 4545) for both the systems as shown in Fig 41c) 41d) These figures indicate that the behavior of the continuous system 11) and the discrete system 35) are qualitatively similar 6 a) 6 b) 5 5 4 3 2 4 3 2 1 1 2 4 6 8 1 E 12 1 7 c) 2 4 6 8 1 E 12 1 7 d) 6 5 E * 6 5 E * 4 3 4 3 2 2 1 1 2 4 6 8 1 12 7 E 2 6 e) 2 4 6 8 1 12 7 E 2 6 f) 5 5 4 3 4 3 2 2 1 1 2 4 6 8 1 12 2 4 6 8 1 12 Figure 42: Phase portraits of the continuous system 26) left panel) and discrete system 39) right panel) Figs a) and b) show that all solutions converge to the disease free point E1 H = 1 ) for β = 1 Figs c) and d) depict that all solutions converge to the endemic point E = 476 5952) for β = 3 Figs e) and f) show that all solutions converge to the susceptible free point E2 H = 6) for β = 42 Other parameters are b = 6 = 4 u = 1 u y = 2 = 12 as in [1] Step size for the discrete model is considered as h = 1 To show dynamic consistency of the continuous system 26) and discrete system

154 Saha and Bairagi 39) we plotted the phase portraits of both systems in Fig 42 We considered the same initial points the same set of parameter values as in [1] with e = and the same step size as in Fig 41The conditions of Theorem 22ii) and Theorem 35ii) are satisfied when β = 1 In this case all solutions of both the systems converge to the point E1 H = 1 ) Figs 42a) 42b)) For β = 3 conditions of Theorem 22iv) and Theorem 35iv) are satisfied Consequently all solutions reach to the interior point EH = 476 5952) Figs 42c) 42d)) If we take β = 42 then all conditions of Theorem 22iii) and Theorem 35iii) are satisfied All solutions in this case converge to the susceptible free equilibrium point E2 H = 6) in both cases Figs 42e) 42f)) To observe dynamical consistency of the discrete system 312) with its corresponding continuous system 27) we plotted phase diagrams of both systems in Fig 43 The same parameter set as in [1] with e = β = was considered and the initial points step size remained unchanged Phase portraits of the continuous system Fig 43a)) and that of the discrete system Fig 43b)) show that all solutions reach to the infection free point E1 V = 1 ) indicating the dynamic consistency of both systems 6 a) 6 b) 5 5 4 3 2 4 3 2 1 1 2 4 6 8 1 E 12 1 2 4 6 8 1 E 12 1 Figure 43: Phase portrait of the continuous system 27) Fig a) and that of the discrete system 312) Fig b) indicate that all solutions converge to the infection free point E V 1 = 1 ) in each case The parameters are b = 6 = 4 u = 1 u y = 2 and = 12 as in [1] Step size for the discrete model is considered as h = 1 5 Summary We here considered a continuous time epidemic model where infection spreads through imperfect vertical transmission and horizontal transmission in a density dependent aseual host population Stability of different equilibrium points are presented with respect to the basic reproduction number and relative birth & death rates of susceptible & infected hosts A discrete version of the continuous system is constructed following nonlocal approimation technique and its dynamics has been shown to be identical with that

Dynamics of Vertically and Horizontally Transmitted Parasites 155 of the continuous system The proposed discrete model is shown to be positive implying that its solutions remains positive for all future time whenever it starts with positive initial value The dynamics of the discrete model have been shown to be independent of the step size Our simulation results also show dynamic consistency of the discrete models with its corresponding continuous model Two submodels of the general discrete model have also been shown to have the identical dynamics with their continuous counterparts Acknowledgements Research of P Saha is supported by CSIR; F No: 9/9699)/217-EMR-I and research of N Bairagi is supported by SERB DST; F No: MTR/217/32 References [1] Lipsitch M Nowak MA Ebert D and May RM 1995 The population dynamics of vertically and horizontally transmitted parasites Biological Sciences Vol 26 Issue 1359 321 327 [2] Holmes JC Bethel WM 1972 Modification of intermediate host behavior by parasites In: Canning EV Wright CA Eds) Behavioral Aspects of Parasite Transmission Suppl I to Zool f Linnean Soc 51 123 149 [3] Lafferty D Morris A 1996 Altered behaviour of parasitized killfish increases susceptibility to predation by bird final hosts Ecology 77 139 1397 [4] Mickens RE 1989 Eact solutions to a finite-difference model of a nonlinear reaction-advection equation: Implications for numerical analysis Numer Methods Partial Diff Eq5 313 325 [5] Moghadas SM Aleander ME Corbett BD and Gumel AB 23 A Positivity-preserving Mickens-type discretization of an epidemic model J Diff Equ Appl 9 137 151 [6] Biswas M and Bairagi N 217 Discretization of an eco-epidemiological model and its dynamic consistency J Diff Equ Appl 235) 86 877 [7] Sekiguchi M and Ishiwata E 21 Global dynamics of a discretized SIRS epidemic model with time delay J Math Anal Appl 371 195 22 [8] Biswas M and Bairagi N 216 Dynamic consistency in a predatorprey model with habitat compleity: Nonstandard versus standard finite difference methods Int J Diff Equ Appl 112) 139 162

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