Abstract

We consider a class of system of nonlinear difference equations arising from mathematical models describing a discrete epidemic model. Sufficient conditions are established that guarantee 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. The obtained results are new and they complement previously known results.

1. Introduction

Over the past decades, there has been an increasing interest in studying nonlinear and rational difference equations arising in applied sciences, and hence there are many papers and books concerning theory and applications of difference equations; see, for example, [119] and the references therein. The existence of positive solutions for difference equations has also received a great deal of attention in the literature. To identify a few, we refer the reader to Baštinec et al. [1, 2], Diblík and Hlavičková [6], and the references therein. In [11], the authors posed some research projects about the behavior of solutions of the following deterministic epidemic models:and, generally,The above equations are special cases of epidemic models which were derived in [3, 4]. For some other biological models, see, for example, [7, 10, 11] and the references therein. Recently, the dynamics of (2) and some related models have been discussed in [13, 15, 19].

Our aim in this paper is to 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 following system of nonlinear difference equations: where the constants and the initial values are positive numbers which satisfy the relations

System (3) can be viewed as an extension to two dimensions of (2). In fact, if , , , and , then system (3) reduces to (2) for .

The rest of the paper is organized as follows. In Section 2, we establish the existence of positive solutions and the convergence to the unique zero equilibrium for the positive solutions of (3). In Section 3, we present conditions that imply the convergence of the positive solutions to the unique positive equilibrium for (3).

2. Existence and Convergence to the Unique Zero Equilibrium for the Positive Solutions of (3)

This section deals with the existence of positive solutions and the convergence to the unique zero equilibrium for the positive solutions of (3). We begin with the next lemma which furnishes some useful inequalities.

Lemma 1. Consider the following functions:whereThen the following statements hold:(i)(ii)(iii)equation (resp., ) has a unique solution (resp., ), such that(iv)

Proof. (i) From (5), we have SetThenIf , thenSetIt is easy to prove that is a decreasing function for and hence In view of , is a decreasing function if , which implies thatNow, suppose thatSince is a decreasing function for , thenHence, by (13), we have that is a decreasing function and therefore, by (17), we get and obviouslyFinally, suppose thatSince is a decreasing function for , we haveNow by combining (13), (15), and (22) with (23) it follows that for every there exists a such thatWe claim thatAfter some calculations, in order to prove (25) it is sufficient to prove that which is true, since if we set it is easy to prove that is an increasing function for every , such that (23) holds, and . Relations (24) and (25) imply thatTherefore, by (17), (20), and (28) we have that relation (7) is true. This completes the proof of statement (i).
(ii) Since is an increasing function with respect to , for and , thenWe claim thatIndeed, by (11) we haveHence, equation has a unique solution in the interval . Using Newton’s method we can see that this solution is . So after some calculations we have which implies that our claim (30) is true.
In addition, consider the function It is easy to prove that is a decreasing function for and and hence, by (5), (29), (30), and (33), we have thatwhich imply that (8) is true. This completes the proof of statement (ii).
(iii) By the above statement (ii), we get that equation has a unique solution and obviously, . From (5) we haveSince is the unique solution of equation , it follows from (31) thatIn addition, from (35) and (36) it follows thatFrom (5) we getand hence, by (37), equation has a unique solution in the interval . Using Newton’s method, we can prove that this solution is . Obviously, . This completes the proof of statement (iii).
(iv) Consider the function and thenwhere from (9) and (31) From (30), (31), and (35) we haveSince is the unique solution of equation , it follows from (9) and (37) that Since , , and , we get that equation has a unique solution in the interval . Using Newton’s method, we can prove that this solution is . Since , then . This completes the proof of statement (iv).

Theorem 2. Consider system (3) with the constants , satisfying Let be a solution of (3) with initial values satisfying (4). Then

Proof. From (3), we haveIt follows from (4), (44), and (46) thatThen from (3) we get By (4) and (46), we takewhich, together with (5) and (48), imply thatFrom (4) we get , and then which together with (3) and (4) yields , and hence, by (3) and (44), we have . Similarly we can prove that . We next prove that Now, by (3), (4), and (44), we have We consider the function , for , , , as follows: Since is an increasing function with respect to , we obtainwhere , . Set the function as follows: where , . Then, we take the system of equationswhere , . System (57) is equivalent to systemWe now consider the functionwhere . From (58) we getFrom (59) and (60) we deduce that equation has a unique solution , such that . Using Newton’s method we have that In addition, by (57) we obtainThe derivatives of at the point () provideFrom (63), we have ,By (63) and (64), we can get that is the maximum point of and the unique solution of system (57), and then Hence, from (53)–(56) and (65), we haveSimilarly, we can prove that Thus, from (66) and (67) it can be shown that (52) are satisfied for . Following this fashion, we can prove that (52) are true for all . Then it is obvious that (45) are satisfied and thus the proof is complete.

Theorem 3. Let be a solution of (3) such that (4) are satisfied. Consider the system of algebraic equations:If and , then the following statements hold.(i)System (68) has a unique zero solution .(ii) tends to the zero equilibrium of (3) as .

Proof. (i) Consider the functionsThen which imply that and hence , . Therefore, is the unique zero solution for system (68).
(ii) By (3), (4), (44), (45), and (52), we haveSimilarly, , which imply that are decreasing sequences. Therefore, there exist constants such thatUsing (45), (52), and (73), we get . Relations (72) and (73) imply that and, hence,Further, from (3), (73), and (75) we have First, suppose that , then from (76) we get , and, hence, by using , , and Lemma 2.1 of [19] we get . Similarly, when , we get . Now, suppose that and then from (76) we get thatIfthen sincewe deduce that and thus is an increasing function. Hence, by , , (73), and (78), we have . From (75), we get . Therefore, there exist the and . By statement (i) of Theorem 3, we have that , which contradicts the suppose. Hence, and so . This completes the proof of the theorem.

3. Convergence to the Unique Positive Equilibrium for the Positive Solutions of (3)

The next theorem obtains sufficient conditions that imply the convergence of the positive solutions to the unique positive equilibrium for (3).

Theorem 4. Let be a solution of (3) such that (4) are satisfied. Consider the system of algebraic equations:If and , then the following statements hold.(i)System (82) has a unique positive solution , .(ii)Suppose there exists an such that for either orhold. Then tends to the unique positive equilibrium of (3) as .

Proof. (i) Setwhere was defined in (5). From and , we have that Further,and so has a unique solution such that Moreover, we haveFurther,Relations (91) and (92) imply that equation has a solution . It follows from (86) that and so is a solution of system (82) such that .
We prove now that is the unique solution of system (82). Indeed, by (5) we have and hence, by (6), (86), and (89), we get Firstly, assume that In view of (87), (89), (96), and (97), and are positive and decreasing functions for . And by (91) we get that . Therefore, is a decreasing function for . Further, by combining (89), (91) with (97) it follows that Hence, by (92) we have that there exists a unique such that Thus, is the unique solution of equation , such that and so is the unique solution of system (82), such that .
Secondly, assume that there exists an such that . In view of (96) and , there exist exactly two real numbers such that Hence, by (96) we have thatSince , by (87), (89), and (101), and are positive and decreasing functions for . Then and it follows that is a decreasing function. Since we deduce that there exists an such thatIn addition, since , it follows from (89), (91), and (101) thatWe also claim thatFirst, we prove that and , where are defined in statement (iii) of Lemma 1. Since is the unique solution of equation for , and (87) holds, in order to prove it is sufficient to prove that for any . From statements (i) and (iii) of Lemma 1 we have , and hence, .
Moreover, from (89) and (101), in order to prove , it is sufficient to prove that and for any and . From (12) and (20), we get that and since, from statement (iii) of Lemma 1, , it follows that and , , which imply that .
In addition, from statements (ii) and (iii) of Lemma 1 we have and, thus, . Now, by combining relations (89), (101) and statements (i) and (ii) of Lemma 1 with , it follows that Hence,which proves our claim.
Finally, since , by (89), (104), and (105), we have In summary, sinceand , , we deduce that is the unique solution of equation such that , and is the unique solution of system (82) such that .
(ii) Assuming that there exists an such that, for ,thenUsing Lemma 2.7 of [19], we havewhich lead to Since and, similarly, , then are decreasing sequences. Therefore, there exist constants such that Since are the positive solutions of (3), we get Relations (116) imply that and, hence,Further, by using (3) and (118), we have Thus, relations (121) and (122) imply that Then we getIfthen sincewe getwhich implies that is an increasing function, and we take . Then from (121) it is obvious that . Since from statement (i) of Theorem 4 () is the unique positive equilibrium of (3), and thus . This completes the proof of the theorem.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and the Fundamental Research Funds for the Central Universities of Hunan University (Grant no. 531107040013).