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On a modification of a discrete epidemic model. (English) Zbl 1197.39010
Summary: In this paper under some conditions on the constants A,B0,) we study the existence of positive solutions, the existence of a unique nonnegative equilibrium and the convergence of the positive solutions to the nonnegative equilibrium of the system of difference equations x n+1 =(1-y n -y n-1 )(1-e -Ay n ),y n+1 =(1-x n -x n-1 )(1-e -Bx n ) where A,B(0,) and the initial values x -1 ,x 0 ,y -1 ,y 0 are positive numbers which satisfy the relations x 0 +x 1 <1,y 0 +y 1 <1·1-y0>(1-x 0 -x -1 )(1-e Bx 0 ,1-x 0 >(1-y 0 -y 1 )(1-e -Ay 0 ).
MSC:
39A30Stability theory (difference equations)
92D30Epidemiology
References:
[1]Kocic, V. L.; Ladas, G.: Global behavior of nonlinear difference equations of higher order with applications, (1993)
[2]Beddington, J. R.; Free, C. A.; Lawton, J. H.: Dynamic complexity in predator prey models framed in difference equations, Nature 255, 58-60 (1975)
[3]Cooke, K. L.; Calef, D. F.; Level, E. V.: Stability or chaos in discrete epidemic models, , 73-93 (1977)
[4]Zhang, D. C.; Shi, B.: Oscillation and global asymptotic stability in a disctete epidemic model, J. math. Anal. appl. 278, 194-202 (2003) · Zbl 1025.39013 · doi:10.1016/S0022-247X(02)00717-5
[5]Stevic, S.: On a discrete epidemic model, Discrete dynamics in nature and society 2007, 10 (2007)