New results on the existences of solutions of the Dirichlet problem with respect to the Schrödinger-prey operator and their applications

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1 Chen Zhang Journal of Inequalities Applications :143 DOI /s R E S E A R C H Open Access New results on the existences of solutions of the Dirichlet problem with respect to the Schrödinger-prey operator their applications Xu Chen 1 Lei Zhang 2* * Correspondence: l.zhan.rail@qq.com 2 Department of Railway Building Hunan Technical College of Railway High-Speed Hengyang China Full list of author information is available at the end of the article Abstract In this paper by using the Beurling-Nevanlinna type inequality we obtain new results on the existence of solutions of the Dirichlet problem with respect to the Schrödinger-prey operator. Meanwhile the local stability of the Schrödingerean equilibrium endemic equilibrium of the model are also discussed. We specially analyze the existence stability of the Schrödingerean Hopf bifurcation by using the center manifold theorem bifurcation theory. As applications theoretic analysis numerical simulation show that the Schrödinger-prey system with latent period has a very rich dynamic characteristics. Keywords: existence; Beurling-Nevanlinna type inequality; Dirichlet problem; Schrödinger-prey operator 1 Introduction The role of mathematical modeling has been intensively growing in the study of epidemiology. Various epidemic models have been proposed explored extensively great progress has been achieved in the studies of disease control prevention. Many authors have investigated the autonomous epidemic models. May Odter [1]proposedatime- periodic reaction-diffusion epidemic model which incorporates a simple demographic structure the latent period of an infectious disease. Guckenheimer Holmes [2] examined an SIR epidemic model with a non-monotonic incidence rate they also analyzed the dynamical behavior of the model derived the stability conditions for the disease-free the endemic equilibrium. Berryman Millstein [3] investigated an SVEIS epidemic model for an infectious disease that spreads in the host population through horizontal transmission they have shown that the model exhibits two equilibria namely the disease-free equilibrium the endemic equilibrium. Hassell et al. [4] presented four discrete epidemic models with the nonlinear incidence rate by using the forward Euler backward Euler methods they discussed the effect of two discretizations on the stability of the endemic equilibrium for these models. Shilnikov et al. [5] proposed a VEISV network worm attack model derived the global stability of a worm-free state local stability of a unique worm-epidemic state by using the reproduc- The Authors This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use distribution reproduction in any medium provided you give appropriate credit to the original authors the source provide a link to the Creative Commons license indicate if changes were made.

2 Chen ZhangJournal of Inequalities Applications :143 Page 2 of 10 tion rate. Robinson Holmes [6] discussed the dynamical behaviors of a Schrödingerprey system showed that the model undergoes a flip bifurcation a Hopf bifurcation by using the center manifold theorem bifurcation theory. Bacaër Dads [7] investigated an SVEIS epidemic model for an infectious disease that spreads in the host population through horizontal transmission. Recently Yan et al. [8] Xue [9]Wan[10] discussed the threshold dynamics of a timeperiodic reaction-diffusion epidemic model with latent period. In this paper we will study the existence of the disease-free equilibrium endemic equilibrium the stability of the disease-free equilibrium the endemic equilibrium for this system. Conditions will be derived for the existence of a flip bifurcation a Hopf bifurcation by using bifurcation theory [11 12] the center manifold theorem [13]. The rest of this paper is organized as follows. A discrete SIR epidemic model with latent period is established in Section 2. InSection3 we obtain the main results: the existence local stability of fixed points for this system. We show that this system goes through a flip bifurcation a Hopf bifurcation by choosing a bifurcation parameter in Section 4. A brief discussion is given in Section 5. 2 Model formulation In 2015 Yan et al. [9] discussed the threshold dynamics of a time-periodic reactiondiffusion epidemic model with latent period. We consider the following continuous-time SIR epidemic model described by the Schröding-prey equations: ds dt = βstit di = βstit γit 1 dt dr dt = γ It St It Rt denote the sizes of the susceptible infected removed individuals respectively the constant β is the transmission coefficient γ is the recovery rate. Let S 0 = S0 be the density of the population at the beginning of the epidemic with everyone susceptible. It is well known that the basic reproduction number R 0 = βs 0 /γ completely determines the transmission dynamics an epidemic occurs if only if R 0 >1; see also [8]. It should be emphasized that system 1 has no vital dynamics births deaths because it was usually used to describe the transmission dynamics of a disease within a short outbreak period. However for an endemic disease we should incorporate the demographic structure into the epidemic model. The classical endemic model is the following SIR model with vital dynamics: ds dt di dt = βstit N = μn μst βstit N dr dt = γ It μit γ It μit which is almost the same as the SIR epidemic model 2aboveexceptthatithasaninflow of newborns into the susceptible class at rate μn deathsintheclassesatratesμn μi μrn is a positive constant denotes the total population size. For this model the basic reproduction number is given by 2 R 0 = βs 0 γ + μ

3 Chen ZhangJournal of Inequalities Applications :143 Page 3 of 10 which is the contact rate times the average death-adjusted infectious period 1 γ +μ.ifr 0 1 then the disease-free equilibrium E 0 N00ofmodel2 is defined as follows: S n+1 = S n + hμn μs n βs ni n N I n+1 = I n + h βs ni n N γ I n μi n 3 R n+1 = R n + hγ I n μi n h N μ β γ are all defined as in 2. 3 Main results We firstly discuss the existence of the equilibria of model 2. If we take two eigenvalues of JE 1 ω 1 =1 hμ ω 2 =1+hβ hγ + μ then we have the following results. Theorem 1 Let R 0 be the basic reproductive rate such that R 0 <1.Then: 1 If } 2 0<h < min μ 2 γ + μ β then E 1 N0is asymptotically stable. 2 If or } 2 h > max μ 2 γ + μ β 2 γ + μ β < h < 2 μ then E 1 N0is unstable. 3 If h = 2 μ or h = 2 γ + μ β then E 1 N0is non-hyperbolic. The Jacobian matrix of model 2atE 2 S I is JE 2 = which gives 1 hμβ hμ γ +μ or hγ + μ γ +μ β γ μ 1 2 μ < h < 2 γ + μ β Fω=ω 2 tr JE 2 ω + det JE 2 4

4 Chen ZhangJournal of Inequalities Applications :143 Page 4 of 10 tr JE 2 =2 hμβ γ + μ 5 det JE 2 =1 hμβ γ + μ + h2[ μβ μγ + μ ]. 6 Two eigenvalues of JE 2 are ω 12 = hμβ γ + μ ± μr [ μβ μγ + μ ]. 7 Next we obtain the following result as regards E 2 S I. Theorem 2 Let R 0 be the basic reproductive rate such that R 0 1. Then: 1 Put A h > h μr 0 2 4[μβ μγ + μ] 0 B h > h μr 0 2 4[μβ μγ + μ] 0. If one of the above conditions holds then we know that E 2 S I is asymptotically stable. 2 Put A h h μr 0 2 4[μβ μγ + μ] < 0 B h h μr 0 2 4[μβ μγ + μ] 0 C h h μr 0 2 4[μβ μγ + μ] < 0. If one of the above conditions holds then E 2 S I is unstable. 3 Put A h > h or h < h μr 0 2 4[μβ μγ + μ] 0 B h h μr 0 2 4[μβ μγ + μ] < 0 h = μβ μγ + μ μr 0 2 4[μβ μγ + μ] γ + μ[μβ μγ + μ] h = μβ γ + μ[μβ μγ + μ] h = μβ + μγ + μ μr 0 2 4[μβ μγ + μ]. γ + μ[μβ μγ + μ] If one of the above conditions holds then E 2 S I is non-hyperbolic. By a simple calculation Conditions A in Theorem 2 can be written in the following form: μ N β h γ M 1 M 2

5 Chen ZhangJournal of Inequalities Applications :143 Page 5 of 10 M 1 = μ N β h γ :h = h N >0 0 R 0 >10<μ β γ <1 } M 2 = μ N β h γ :h = h N >0 0 R 0 >10<μ β γ <1 }. It is well known that if h varies in a small neighborhood of h or h μ N β h γ M 1 or μ N β h γ M 2 then there may be a flip bifurcation of equilibrium E 2 S I. 4 Bifurcation analysis If h varies in a neighborhood of h μ N β h γ M 1 thenwederivetheflipbifurcation of model 2atE 2 S I. In particular in the case that h changes in the neighborhood of h μ N β h γ M 2 we need to give a similar calculation. Set μ N β h γ =μ 1 β 1 h 1 γ 1 M 1. If we give the parameter h 1 aperturbationh model2 is considered as follows: Sn+m = S n +r + h 1 μ 1 μ 1 S n β 1S n I n I n+1 = I n +h + h 1 β 1S n I n γ 1 I n μ 1 I n 8 h 1. Put U n = S n+1 S V n = I n+1 I.Wehave U n+1 = a 11 U n + a 12 V n + a 13 U n V n + b 11 U n h + b 12 V n h + b 13 U n V n h V n+1 = a 21 U n + a 22 V n + a 23 U n V n + b 21 U n h + b 22 V n h + b 23 U n V n h 9 a 11 =1 h 1 μ 1 + β 1I b 11 = μ 1 + β 1I a 12 = h 1β 1 S a 13 = h 1β 1 b 12 = β 1S b 13 = β 1 a 21 = h 1β 1 I a 22 =1 a 23 = β 1h 1 b 21 = β 1I b 22 =0 b 23 = β 1. If we define the matrix T as follows: a 12 a 12 T = 1 a 11 ω 2 a 11

6 Chen ZhangJournal of Inequalities Applications :143 Page 6 of 10 then we know that T is invertible. If we use the transformation U n V n = T X n Y n then model 2becomes X n+1 = X n + FU n V n h Y n+1 = ω 2 Y n + GU n V n h. 10 Thus W c 0 0 = X n Y n :Y n = a 1 X 2 n + a 2X n h + o X n + h 2} o X n + h 2 is a transform function a 1 = a 13a 11 a 21 1 ω 2 +1 a 2 = b a 11 2 a 12 ω a 12b 12 + b a 11 ω Further we find that the manifold W c 0 0 has the following form: c 1 = a a 11 ω 2 a 11 + a 12 ω 2 +1 c 2 = b 11ω 2 a 11 a 12 b 21 ω 2 +1 b 12ω 2 a a 11 a 12 ω 2 +1 a 13 ω 2 2a 11 1ω 2 a 11 + a 12 b a 11 ω 2 a 11 + a 12 c 3 = a 2 ω 2 +1 c 4 =0c 5 = a 1a 13 ω 2 2a 11 1ω 2 a 11 + a 12. ω 2 +1 Therefore the map G with respect to W c 0 0 can be defined by G X n = X n + c 1 X 2 n + c 2X n h + c 3 X 2 n h + c 4 X n h 2 + c 5 X 3 n + o X n + h Inordertocalculatemap11 we need two quantities α 1 α 2 which are not zero α 1 = GX n h G h G X n X n α 2 = 6 G X n X n X n + 2 G X n X n. 00

7 Chen ZhangJournal of Inequalities Applications :143 Page 7 of 10 By a simply calculation we obtain α 1 = c 2 = 2 h 1 [ α 2 = c 5 + c 2 1 = h 1 β 1 2 h 1β 1 μ h 1 γ 1 ] ω 2 +1 γ 1 μ 1 c 1 = h 1β 1 μ 1 [ h1 γ 1 + μ 1 2 ] [ 2+h 1 γ 1 + μ 1 + h ] 1β 1 μ 1. γ 1 μ 1 γ 1 μ 1 Therefore we have the following result. Theorem 3 Let h change in a neighborhood of the origin. If α 3 >0then the model 9 has a flip bifurcation at E 2 S I. If α 2 0 then the period-2 points of that bifurcation from E 3 S I are stable. If α 3 0 then it is unstable. We further consider the bifurcation of E 3 S I ifh varies in a neighborhood of h. Taking the parameters μ N β h γ =μ 2 β 2 h 2 γ 2 N arbitrarily also giving h aperturbationh at h 2 thenmodel2 gets the following form: Sn+1 = S n +h + h 2 μ 2 μ 2 S n β 2S n I n I n+1 = I n +h + h 2 β 2S n I n γ 2 I n μ 2 I n. 12 Put U n = S n S V n = I n I. We change the equilibrium E 3 S I ofmodel9 have the following result: Un+1 = U n +h + h 2 μ 2 U n β 2 U n V n β 2 U n I β 2 V n S V n+1 = V n +h + h 2 β 2 U n V n γ 1 + μ 1 V n + β 2 U n I + β 2 V n S 13 which gives ω 2 + P h ω + Q h =0 2+P h = β 2μ 2 h 2 + h γ 2 μ 2 Q h =1 β 2μ 2 h 2 + h γ 2 μ 2 + h 2 + h 2[ μ2 β 2 μ 2 μ 2 + γ 2 ]. It is easy to see that ω 12 = Ph ± Ph 2 4Qh 2

8 Chen ZhangJournal of Inequalities Applications :143 Page 8 of 10 which gives ω 12 = Q h k = d ω 12 μ 2 β 2 dh = h =0 2μ 2 + γ 2. We remark that μ 2 β 2 h 2 γ 2 N + <0thenwehave Thus μ 2 β 2 2 γ 2 + μ 2 2 [μ 2 β 2 μ 2 μ 2 + γ 2 ] <4. P0 = 2 + which means that μ 2 β 2 2 γ 2 + μ 2 2 [μ 2 β 2 μ 2 μ 2 + γ 2 ] ±2 μ 2 β 2 γ 2 + μ 2 2 [μ 2 β 2 μ 2 μ 2 + γ 2 ] jγ 2 + μ 2 j = μ 2 β 2 Hence the eigenvalues ω 12 of equilibrium 0 0 of model 14donotlieintheintersection when h =014holds. When h =0webegintostudythemodel14. Put α = μ 2 β 2 2 2γ 2 + μ 2 2 [μ 2 β 2 μ 2 μ 2 + γ 2 ] β = μ 2β 2 4[μ2 β 2 μ 2 μ 2 + γ 2 ] μ 2 β 2 2 2γ 2 + μ 2 [μ 2 β 2 μ 2 μ 2 + γ 2 ] 0 1 T = β α T is invertible. If we use the following transformation: U n V n = T X n Y n then the model 14 gets the following form: X n+1 = αx n βy n + FX n Y n Y n+1 = βx n + αy n + ḠX n Y n 15 FX n Y n = h 2β αβx n Y n + αy 2 n β

9 Chen ZhangJournal of Inequalities Applications :143 Page 9 of 10 ḠX n Y n = h 2β 2 βx n Y n + αy 2 n. Moreover F Xn X n =0 F Yn Y n = 2h 2β 2 α1 + α F Xn Y β n = h 2β α 2 F Xn X n X n = F Xn X n Y n = F Xn Y n Y n = F Yn Y n Y n =0 Ḡ Xn X n =0 Ḡ Yn Y n = 2h 2β 2 α Ḡ Xn Y n = h 2β 2 β Ḡ Xn X n X n = Ḡ Xn X n Y n = Ḡ Xn Y n Y n = Ḡ Yn Y n Y n =0. Thus we have [ ] 1 2 ω a = Re 1 ω ξ 11ξ ξ 11 2 ξ Re ωξ 21 ξ 02 = 1 8[ F Xn X n F Yn Y n 2Ḡ Xn Y n +Ḡ Xn X n Ḡ Yn Y n +2 F Xn Y n i ] ξ 11 = 1 4[ F Xn X n + F Yn Y n +Ḡ Xn X n + Ḡ Yn Y n i ] ξ 20 = 1 8[ F Xn X n F Yn Y n +2Ḡ Xn Y n +Ḡ Xn X n Ḡ Yn Y n 2 F Xn Y n i ] ξ 21 = 1 [ F Xn X 16 n X n + F Xn Y n Y n + Ḡ Xn X n Y n + Ḡ Yn Y n Y n ]. Therefore we have the following result. Theorem 4 Let a 0 h change in a neighborhood of h. If the condition 15 holds then model 13 undergoes a Hopf bifurcation at E 2 S I. If a >0then the repelling invariant closed curve bifurcates from E 2 for h <0.If a <0then an attracting invariant closed curve bifurcates from E 2 for h >0. 5 Conclusions The paper investigated the basic dynamic characteristics of a Schrödinger-prey system with latent period. First we applied the forward Euler scheme to a continuous-time SIR epidemic model obtained the Schrödinger-prey system. Then the existence local stability of the disease-free equilibrium endemic equilibrium of the model are discussed. In addition we chose h as the bifurcation parameter studied the existence stability of flip bifurcation Hopf bifurcation of this model by using the center manifold theorem the bifurcation theory. Numerical simulation results show that for the model 2 there occurs a flip bifurcation a Hopf bifurcation when the bifurcation

10 Chen ZhangJournal of Inequalities Applications :143 Page 10 of 10 parameter h passes through the respective critical values the direction stability of flip bifurcation Hopf bifurcation can be determined by the sign of α 2 arespectively. Apparently there are more interesting problems as regards this Schrödinger-prey system with latent period which deserve further investigation. Acknowledgements The authors would like to thank anonymous referees for their constructive comments which improve the readability of the paper. This work was supported by the Humanities Social Science Fund of Ministry of Education No HU-042. Competing interests The authors declare that they have no conflict of interest. Authors contributions LZ carried out the transformation process designed the solution methodology drafted the manuscript. XC participated in the design of the study helped to draft the manuscript. Both authors read approved the final manuscript. Author details 1 School of Mathematical Sciences Zhejiang University Hangzhou China. 2 Department of Railway Building Hunan Technical College of Railway High-Speed Hengyang China. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps institutional affiliations. Received: 8 February 2017 Accepted: 7 June 2017 References 1. May RM Odter GF: Bifurcations dynamic complexity in simple ecological models. Am. Nat Guckenheimer J Holmes P: Nonlinear Oscillations Dynamical Model Bifurcation of Vector Field. Springer New York Berryman AA Millstein JA: Are ecological systems chaotic - if not why not? Trends Ecol. Evol Hassell MP Comins HN May RM: Spatial structure chaos in insect population dynamics. Nature Shilnikov LP Shilnikov A Turaev D Chua L: Methods of Qualitative Theory in Nonlinear Dynamics. World Scientific Series on Nonlinear Science Series A vol. 4. World Scientific Singapore Robinson C Holmes P: Dynamical Models Stability Symbolic Dynamics Chaos 2nd edn. CRC Press Boca Raton Bacaër N Dads E: Genealogy with seasonality the basic reproduction number the influenza pemic. J. Math. Biol Guo Y Xue M: Characterizations of common fixed points of one-parameter nonexpansive semigroups. Fixed Point Theory Yan Z Zhang J Ye W: Rapid solution of 3-D oscillatory elastodynamics using the pfft accelerated BEM. Eng. Anal. Bound. Elem Wan L Xu R: Some generalized integral inequalities their applications. J. Math. Inequal Huang J: Well-posedness dynamics of the stochastic fractional magneto-hydrodynamic equations. Nonlinear Anal Jiang Z Sun S: On completely irreducible operators. Front. Math. China Li C: A free boundary problem for a ratio-dependent diffusion predator-prey system. Bound. Value Probl