Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 434537, 10 pages
http://dx.doi.org/10.1155/2015/434537

Research Article

Existence and Convergence of the Positive Solutions of a Discrete Epidemic Model

Zhijian Wei and Meitao Le

College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China

Received 11 March 2015; Accepted 20 May 2015

Academic Editor: Josef Diblík

Copyright © 2015 Zhijian Wei and Meitao Le. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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, [1–19] 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:

where

Then the following statements hold:

(i)

(ii)

(iii)equation (resp., ) has a unique solution (resp., ), such that

(iv)

Proof. (i) From (5), we have

Set

Then

If , then

Set

It is easy to prove that is a decreasing function for and hence

In view of , is a decreasing function if , which implies that

Now, suppose that

Since is a decreasing function for , then

Hence, by (13), we have that is a decreasing function and therefore, by (17), we get

and obviously

Finally, suppose that

Since is a decreasing function for , we have

Now by combining (13), (15), and (22) with (23) it follows that for every there exists a such that

We claim that

After 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 that

Therefore, 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 , then

We claim that

Indeed, by (11) we have

Hence, 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 that

which 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 have

Since is the unique solution of equation , it follows from (31) that

In addition, from (35) and (36) it follows that

From (5) we get

and 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 then

where from (9) and (31)

From (30), (31), and (35) we have

Since 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 have

It follows from (4), (44), and (46) that

Then from (3) we get

By (4) and (46), we take

which, together with (5) and (48), imply that

From (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 obtain

where , . Set the function as follows:

where , . Then, we take the system of equations

where , . System (57) is equivalent to system

We now consider the function

where . From (58) we get

From (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 obtain

The derivatives of at the point () provide

From (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 have

Similarly, 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 functions

Then

which imply that

and hence , . Therefore, is the unique zero solution for system (68).
(ii) By (3), (4), (44), (45), and (52), we have

Similarly, , which imply that are decreasing sequences. Therefore, there exist constants such that

Using (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 that

If

then since

we 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

or

hold. Then tends to the unique positive equilibrium of (3) as .

Proof. (i) Set

where was defined in (5). From and , we have that

Further,

and so has a unique solution such that

Moreover, we have

Further,

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 that

Since , 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 that

In addition, since , it follows from (89), (91), and (101) that

We also claim that

First, 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, since

and , , 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 ,

then

Using Lemma 2.7 of [19], we have

which 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 get

If

then since

we get

which 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).

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