Discrete Dynamics in Nature and Society
Volume 2009 (2009), Article ID 143019, 15 pages
doi:10.1155/2009/143019
Research Article

A General Discrete Time Model of Population Dynamics in the Presence of an Infection

Giuseppe Izzo,1 Yoshiaki Muroya,2 and Antonia Vecchio3

1Department of Mathematics and Applications “R. Caccioppoli“, University of Naples “Federico II“, Via Cintia, Monte S. Angelo, 80126 Napoli, Italy
2Department of Mathematical Sciences, Waseda University, 3-4-1 Ohkubo Shinjuku-ku, Tokyo 169-8555, Japan
3Istituto per le Applicazioni del Calcolo “M. Picone”, Via P. Castellino, 111, 80131 Napoli, Italy

Received 16 January 2009; Revised 12 March 2009; Accepted 17 March 2009

Academic Editor: Leonid Berezansky

Copyright © 2009 Giuseppe Izzo et al. 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 present a set of difference equations which generalizes that proposed in the work of G. Izzo and A. Vecchio (2007) and represents the discrete counterpart of a larger class of continuous model concerning the dynamics of an infection in an organism or in a host population. The limiting behavior of this new discrete model is studied and a threshold parameter playing the role of the basic reproduction number is derived.

1. Introduction

Consider the following set of difference equations:

𝑦 ( 𝑛 + 1 ) = 𝛼 + ( 1 − 𝛽 ) 𝑦 ( 𝑛 ) − 𝑚 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 𝜓 𝑖 ( 𝑦 ( 𝑛 + 1 ) ) 𝑥 𝑖 𝑥 ( 𝑛 + 1 ) , 𝑛 ≥ 0 , 1 ( 𝑛 + 1 ) = 1 − 𝑎 1 𝑥 1 ( 𝑛 ) + 𝜙 1 ( 𝑦 ( 𝑛 ) ) 𝑥 𝐿 ( 𝑥 𝑛 ) , 1 ≤ 𝐿 ≤ 𝑚 , 𝑖 ( 𝑛 + 1 ) = 1 − 𝑎 𝑖 𝑥 𝑖 ( 𝑛 ) + 𝜙 𝑖 ( 𝑦 ( 𝑛 ) ) 𝑥 𝑖 − 1 ( 𝑛 ) , 𝑖 = 2 , 3 , … , 𝑚 , ( 1 . 1 ) where 𝜓 𝑖 ( 𝑥 ) , 𝜙 𝑖 ( 𝑥 ) ∈ 𝐶 0 ( ℝ ) , 1 ≤ 𝑖 ≤ 𝑚 . Moreover 𝜙 𝑖 , 1 ≤ 𝑖 ≤ 𝑚 , and at least one of the 𝜓 𝑗  ( 1 ≤ 𝑗 ≤ 𝐿 ) is strictly monotone increasing functions and

𝑦 𝜓 𝑖 ( 𝑦 ) ≥ 0 , ∀ 𝑦 ∈ ℝ , 𝑖 = 1 , … , 𝑚 . ( 1 . 2 ) This difference system contains a very large class of population dynamics models in the presence of an infection involving typically at least two populations: susceptible individuals and infective ones. The former is represented in (1.1) by the sequence 𝑦 , while the latter is represented by one of the sequences 𝑥 𝑖 ; we name it 𝑥 𝐼 where 𝐼 is an integer between 1 and 𝐿 .

The paper is organized as follows. In the next section we report various examples of continuous models which can be discretized by (1.1). The correspondence among the sequences appearing in (1.1) and the dependent variables of the continuous problem is indicated for each example. In Section 3 we prove some basic properties of the solution of the proposed scheme such as positivity and boundedness, which makes it meaningful in the applications. Our main result is proved in Section 4 where the question of the asymptotic behavior of the solution is investigated. We prove a necessary and sufficient condition for the vanishing of the sequences { 𝑥 𝑖 ( 𝑛 ) } and we derive the expression of the basic reproduction number, a threshold parameter which allows to predict whether the infection develops or not. Such a parameter permit to check that, in all the examples quoted in Section 2, the asymptotic behavior of the discrete and continuous problem coincides; therefore, our discrete system incorporates the dynamical characteristics (such as positivity and steady states) of the continuous-time models.

2. Continuous Models

In this section we report different classes of continuous models which can be discretized by means of (1.1). In order to avoid the introduction of many different symbols and for the sake of brevity we do not always use the symbolism found in the literature and we indicate the specific references for the explanation of their meaning.

Example 2.1 (see [1]). This continuous model represents the spread of HIV-1 infection inside the human organism. Here 𝑆 ( 𝑡 ) represents the number of susceptible cells which are present at time 𝑡 in a unit of plasma. The process of infection of a cell is divided into several sequential stages; therefore, 𝑇 𝑗 ( 𝑡 ) is the number of infected cells at time 𝑡 at stage 𝑗 . The variable 𝑉 ( 𝑡 ) is the number of viruses at time 𝑡 . The meaning of the rest of symbols can be found in [1] 𝑆 ' 𝐼 ( 𝑡 ) = 𝑎 − 𝑏 𝑆 ( 𝑡 ) − 𝑐 𝑆 ( 𝑡 ) 𝑉 ( 𝑡 ) , ' 1 ( 𝑡 ) = 𝑐 𝑆 ( 𝑡 ) 𝑉 ( 𝑡 ) − 𝑘 𝐼 1 𝐼 ( 𝑡 ) , ' 𝑗 𝐼 ( 𝑡 ) = 𝑘 𝑗 − 1 ( 𝑡 ) − 𝐼 𝑗 𝑉 ( 𝑡 ) , 𝑗 = 2 , … , 5 , ' ( 𝑡 ) = 𝑝 𝐼 5 ( 𝑡 ) − 𝑞 𝑉 ( 𝑡 ) . ( 2 . 1 ) Rewrite (1.1) at a general time step 𝑡 the length of which is ℎ and put 𝑚 = 6 ; 𝐼 = 𝐿 = 6 ; 𝜓 𝑖 𝜓 ( 𝑦 ) = 0 , 𝑖 = 1 , … , 5 ; 6 ( 𝑦 ) = 𝜙 1 ( 𝑦 ) = 𝛾 𝑦 ; 𝜙 𝑖 ( 𝑦 ) = 𝑘 ℎ , 𝑖 = 2 , … , 5 ; 𝜙 6 𝑎 ( 𝑦 ) = 𝑝 ℎ ; 𝑖 = 𝑘 ℎ , 𝑖 = 1 , … , 5 ; 𝑎 6 = 𝑞 ℎ ; 𝛼 = 𝑎 ℎ ; 𝛽 = 𝑏 ℎ ; 𝛾 = 𝑐 ℎ . ( 2 . 2 ) We obtain 𝑦 ( 𝑡 + ℎ ) − 𝑦 ( 𝑡 ) = 𝑎 ℎ − 𝑏 ℎ 𝑦 ( 𝑡 ) − 𝑐 ℎ 𝑦 ( 𝑡 + ℎ ) 𝑥 6 𝑥 ( 𝑡 + ℎ ) , 1 ( 𝑡 + ℎ ) − 𝑥 ( 𝑡 ) = − 𝑘 ℎ 𝑥 1 ( 𝑡 ) + 𝑐 ℎ 𝑦 ( 𝑡 ) 𝑥 6 𝑥 ( 𝑡 ) , 𝑖 ( 𝑡 + ℎ ) − 𝑥 ( 𝑡 ) = − 𝑘 ℎ 𝑥 𝑖 ( 𝑡 ) + 𝑘 ℎ 𝑥 𝑖 − 1 𝑥 ( 𝑡 ) , 𝑖 = 2 , … , 5 , 6 ( 𝑡 + ℎ ) − 𝑥 ( 𝑡 ) = − 𝑞 ℎ 𝑥 6 ( 𝑡 ) + 𝑝 ℎ 𝑥 5 ( 𝑡 ) . ( 2 . 3 ) This can be easily seen (see, e.g., [2]) to be the discrete analogue of (2.1) by dividing each equation by ℎ .
In conclusion, by assuming ℎ = 1 , we have that (1.1) is the discrete counterpart of (2.1) provided that 𝑚 = 6 ; 𝐼 = 𝐿 = 6 ; 𝜙 1 ( 𝑦 ) = 𝑐 𝑦 ; 𝜙 𝑖 ( 𝑦 ) ≡ 𝑘 , 𝑖 = 2 , … , 5 , 𝜙 6 𝑎 ( 𝑦 ) ≡ 𝑝 ; 𝑖 = 𝑘 , 𝑖 = 1 , … , 5 ; 𝑎 6 = 𝑞 ; 𝛼 = 𝑎 ; 𝛽 = 𝑏 . ( 2 . 4 ) The role of 𝑦 , 𝑥 𝑖 , 𝑖 = 1 , … , 6 is related to that of the variable of (2.1) according to the following scheme: 𝑦 ↔ 𝑆 , 𝑥 1 ↔ 𝐼 1 , … , 𝑥 5 ↔ 𝐼 5 , 𝑥 6 ↔ 𝑉 .
Observe that 𝑦 ( 𝑛 ) and 𝑥 6 ( 𝑛 ) play the role of 𝑆 ( 𝑡 ) and 𝑉 ( 𝑡 ) , respectively, and therefore they correspond to the susceptible and infective populations as we mentioned in the introduction.

Example 2.2 (see [3]). This represents the spread of HTLV-I infection in a human organism we refer to [3] for the meaning of the symbols 𝑇 ' ( 𝑡 ) = 𝑎 − 𝑏 𝑇 ( 𝑡 ) − 𝑐 𝑇 𝐴 𝑇 ( 𝑡 ) 𝑇 ( 𝑡 ) , ' 𝐵 ( 𝑡 ) = 𝑐 𝑇 𝐴 ( 𝑡 ) 𝑇 ( 𝑡 ) − 𝑑 𝐵 𝑇 𝐿 𝑇 ( 𝑡 ) , ' 𝐴 ( 𝑡 ) = 𝑔 𝑇 𝐵 ( 𝑡 ) − 𝑑 𝐴 𝑇 𝐴 ( 𝑡 ) . ( 2 . 5 ) As in Example 2.1 we can see that (1.1) is the discrete counterpart of (2.5) provided that 𝑚 = 2 ; 𝐼 = 𝐿 = 2 ; 𝜓 1 ( 𝑦 ) = 0 ; 𝜙 1 ( 𝑦 ) = 𝜓 2 ( 𝑦 ) = 𝑐 𝑦 ; 𝜙 2 ( 𝑦 ) ≡ 𝑔 ; 𝑎 1 = 𝑑 𝐵 , 𝑎 2 = 𝑑 𝐴 ; 𝛼 = 𝑎 ; 𝛽 = 𝑏 . The correspondence between the variables of (1.1) and (2.5) is summarized by 𝑦 ↔ 𝑇 , 𝑥 1 ↔ 𝑇 𝐵 ; 𝑥 2 ↔ 𝑇 𝐴 .
Let us note that this model is mathematically equivalent to the classical SIR model [4, model (2.5)].

Example 2.3 (see [5]). This represents the spread of HIV-I infection in a human organism 𝑇 ' ( 𝑡 ) = 𝑎 − 𝑏 𝑇 ( 𝑡 ) − 𝑐 𝑉 𝐼 𝑉 ( 𝑡 ) 𝑇 ( 𝑡 ) , ' 𝐼 ( 𝑡 ) = 𝑔 1 𝑇 ∗ ( 𝑡 ) − 𝑑 1 𝑉 𝐼 𝑇 ( 𝑡 ) , ∗ ' ( 𝑡 ) = 𝑐 𝑉 𝐼 ( 𝑡 ) 𝑇 ( 𝑡 ) − 𝑑 2 𝑇 ∗ 𝑉 ( 𝑡 ) , ' 𝑁 𝐼 ( 𝑡 ) = 𝑔 2 𝑇 ∗ ( 𝑡 ) − 𝑑 1 𝑉 𝑁 𝐼 ( 𝑡 ) . ( 2 . 6 ) Here we have 𝑚 = 3 ; 𝐼 = 1 ; 𝐿 = 2 ; 𝜙 1 ( 𝑦 ) ≡ 𝑔 1 ; 𝜙 2 ( 𝑦 ) = 𝜓 1 ( 𝑦 ) = 𝑐 𝑦 ; 𝜙 3 ( 𝑦 ) ≡ 𝑔 2 ; 𝜓 2 ( 𝑦 ) = 𝜓 3 ( 𝑦 ) = 0 ; 𝑎 1 = 𝑑 1 , 𝑎 2 = 𝑑 2 ; 𝑎 3 = 𝑑 1 ; 𝛼 = 𝑎 ; 𝛽 = 𝑏 . 𝑦 ↔ 𝑇 , 𝑥 1 ↔ 𝑉 𝐼 ; 𝑥 2 ↔ 𝑇 ∗ ; 𝑥 3 ↔ 𝑉 𝑁 𝐼 .

It is worth to note that all the continuous models just proposed can be discretized by means of the discrete model proposed in [6]. The following example shows instead a continuous model that has not this property but can be discretized by means of (1.1).

Example 2.4 (see [5]). This represents the spread of HIV-I infection in a human organism, too 𝑇 ' ( 𝑡 ) = 𝜆 − 𝑑 𝑇 ( 𝑡 ) − ( 1 − 𝜀 ) 𝑘 𝑉 ( 𝑡 ) 𝑇 ( 𝑡 ) − 𝑞 𝑇 ∗ 𝑇 ( 𝑡 ) 𝑇 ( 𝑡 ) , ∗ ' ( 𝑡 ) = ( 1 − 𝜀 ) 𝑘 𝑉 ( 𝑡 ) 𝑇 ( 𝑡 ) − 𝛿 𝑇 ∗ 𝑉 ( 𝑡 ) , ' ( 𝑡 ) = 𝑁 𝑇 𝛿 𝑇 ∗ ( 𝑡 ) − 𝑐 𝑉 ( 𝑡 ) . ( 2 . 7 ) This continuous model cannot be discretized by means of the discrete model proposed in [6] because of the presence of the two nonlinear terms ( 𝑘 𝑉 ( 𝑡 ) 𝑇 ( 𝑡 ) and 𝑞 𝑇 ∗ ( 𝑡 ) 𝑇 ( 𝑡 ) ) in the first equation. Instead, that can be done by means of (1.1) and we have 𝑚 = 2 ; 𝐼 = 1 ; 𝐿 = 2 ; 𝛼 = 𝜆 ; 𝑎 1 = 𝛿 ; 𝑎 2 = 𝑐 ; 𝛽 = 𝑑 ; 𝜓 1 ( 𝑦 ) = 𝑞 𝑦 ; 𝜓 2 ( 𝑦 ) = 𝜙 1 ( 𝑦 ) = ( 1 − 𝜀 ) 𝑘 𝑦 ; 𝜙 2 ( 𝑦 ) = 𝑁 𝑇 𝑎 1 ; 𝑇 ↔ 𝑦 ; 𝑇 ∗ ↔ 𝑥 1 ; 𝑉 ↔ 𝑥 2 .

3. Basic Properties

Since functions 𝑦 and 𝑥 𝑖 ( 𝑖 = 1 , 2 , … , 𝑚 ) represent populations, at first, we can prove in the following two theorems their positivity and boundedness by using very natural hypotheses.

Theorem 3.1. Assume that (i) 𝛼 > 0 ;(ii) 𝛽 < 1 ; (iii) 𝑎 𝑖 < 1 , 𝑖 = 1 , … , 𝑚 ; (iv) 𝜙 𝑖 ( 𝑦 ) is not decreasing and 𝜙 𝑖 ( 0 ) ≥ 0 , 𝑖 = 1 , … , 𝑚 ; (v) 𝑦 𝜓 𝑖 ( 𝑦 ) ≥ 0 , ∀ 𝑦 ∈ ℝ , 𝑖 = 1 , … , 𝑚 ; (vi) ∃ s.t. 1 ≤ î ≤ 𝐿 and 𝜓 î is strictly increasing. Then 𝑦 ( 𝑛 ) > 0 , 𝑥 𝑖 ( 𝑛 ) > 0 , 𝑛 ≥ 0 , 𝑖 = 1 , … , 𝑚 .

Proof. From ( i i i ) , ( i v ) and the positivity of 𝑥 𝑖 ( 0 ) we have 𝑥 𝑖 ( 1 ) > 0 , 𝑖 = 1 , … , 𝑚 . Now assume 𝑦 ( 1 ) ≤ 0 . ( 3 . 1 ) From (v), (vi), we get ∑ 𝑚 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 𝜓 𝑖 ( 𝑦 ( 1 ) ) 𝑥 𝑖 ( 1 ) ≤ 0 and from (i), (ii) and the first of (1.1) we obtain 𝑦 ( 1 ) > 0 which contradicts (3.1). The rest of the theorem can be proved in the same way (by induction).

Theorem 3.2. Assume that (i) 𝛼 > 0 ;(ii) 0 < 𝛽 < 1 ;(iii) 0 < 𝑎 𝑖 < 1 , 𝑖 = 1 , … , 𝑚 ;(iv) 𝜙 𝑖 ( 𝑦 ) is not decreasing and 𝜙 𝑖 ( 0 ) ≥ 0 , 𝑖 = 1 , … , 𝑚 ;(v) 𝑦 𝜓 𝑖 ( 𝑦 ) ≥ 0 , ∀ 𝑦 ∈ ℝ , 𝑖 = 1 , … , 𝑚 ; (vi) ∃ î s.t. 1 ≤ î ≤ 𝐿 and 𝜓 î is strictly increasing;(vii) ∃ 𝑖 s.t. 1 ≤ 𝑖 ≤ 𝐿 and 𝜙 ∃ 𝑞 > 0 ∶ 1 ( 𝑦 ) ≤ 𝑞 𝜓 𝐿 ( 𝑦 ) , 𝑦 ≥ 0 i f 𝜙 𝑖 = 1 , 𝑖 ( 𝑦 ) ≤ 𝑞 𝜓 𝑖 − 1 ( 𝑦 ) , 𝑦 ≥ 0 i f 2 ≤ 𝑖 ≤ 𝐿 . ( 3 . 2 )
Then, the sequences { 𝑦 ( 𝑛 ) } , { 𝑥 𝑖 ( 𝑛 ) } , 𝑖 = 1 , … , 𝑚 are bounded.

Proof. In order to prove this theorem it is convenient to represent (1.1) in the form of the following system of Volterra difference equations (see, e.g., [7, 8]): 𝛼 𝑦 ( 𝑛 + 1 ) = 𝛽 1 − ( 1 − 𝛽 ) 𝑛 + 1 + ( 1 − 𝛽 ) 𝑛 + 1 𝑦 ( 0 ) − 𝑛 + 1 𝑙 = 1 0 𝑥 0 2 0 0 𝑑 ( 1 − 𝛽 ) 𝑚 𝑛 + 1 − 𝑙 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 𝜓 𝑖 ( 𝑦 ( 𝑙 ) ) 𝑥 𝑖 𝑥 ( 𝑙 ) , 1 ( 𝑛 + 1 ) = 1 − 𝑎 1 𝑛 + 1 𝑥 1 ( 0 ) + 𝑛 𝑙 = 0 0 𝑥 0 2 0 0 𝑑 1 − 𝑎 1 𝑛 − 𝑙 𝜙 1 ( 𝑦 ( 𝑙 ) ) 𝑥 𝐿 𝑥 ( 𝑙 ) , 1 ≤ 𝐿 ≤ 𝑚 , 𝑖 ( 𝑛 + 1 ) = 1 − 𝑎 𝑖 𝑛 + 1 𝑥 𝑖 ( 0 ) + 𝑛 𝑙 = 0 0 𝑥 0 2 0 0 𝑑 1 − 𝑎 𝑖 𝑛 − 𝑙 𝜙 𝑖 ( 𝑦 ( 𝑙 ) ) 𝑥 𝑖 − 1 ( 𝑙 ) , 𝑖 = 2 , … , 𝑚 . ( 3 . 3 ) Since the hypotheses of the previous theorem hold, positivity of the sequences { 𝑦 ( 𝑛 ) } , { 𝑥 𝑖 ( 𝑛 ) } is assured. From the first of (3.3) we obtain 𝛼 𝑦 ( 𝑛 + 1 ) ≤ 𝛽 1 − ( 1 − 𝛽 ) 𝑛 + 1 + ( 1 − 𝛽 ) 𝑛 + 1 𝛼 𝑦 ( 0 ) ≤ m a x 𝛽 , 𝑦 ( 0 ) = 𝑦 𝑀 ( 3 . 4 ) and so the boundedness of 𝑦 . From the first of (1.1) and (3.4) we also obtain for all 𝑗 = 1 , … , 𝑚 𝜓 𝑗 ( 𝑦 ( 𝑙 ) ) 𝑥 𝑗 ( 𝑙 ) ≤ 𝑚 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 𝜓 𝑖 ( 𝑦 ( 𝑙 ) ) 𝑥 𝑖 ( 𝑙 ) , = 𝛼 + ( 1 − 𝛽 ) 𝑦 ( 𝑙 − 1 ) − 𝑦 ( 𝑙 ) ≤ 𝛼 + ( 1 − 𝛽 ) 𝑦 𝑀 . ( 3 . 5 ) Assume that there exists 𝑞 > 0 such that 𝜙 1 ( 𝑦 ) ≤ 𝑞 𝜓 𝐿 ( 𝑦 ) , 𝑦 ≥ 0 . ( 3 . 6 ) From the second of (3.3), ( i i ) , ( i i i ) and (3.5) we have 𝑥 1 ( 𝑛 + 1 ) = 1 − 𝑎 1 𝑛 + 1 𝑥 1 ( 0 ) + 𝑛 𝑙 = 0 0 𝑥 0 2 0 0 𝑑 1 − 𝑎 1 𝑛 − 𝑙 𝜙 1 ( 𝑦 ( 𝑙 ) ) 𝑥 𝐿 ≤ ( 𝑙 ) 1 − 𝑎 1 𝑛 + 1 𝑥 1 ( 0 ) + 𝑞 𝑛 𝑙 = 0 0 𝑥 0 2 0 0 𝑑 1 − 𝑎 1 𝑛 − 𝑙 𝜓 𝐿 ( 𝑦 ( 𝑙 ) ) 𝑥 𝐿 ( 𝑙 ) ≤ 𝑥 1 ( 0 ) + 𝑞 𝛼 + ( 1 − 𝛽 ) 𝑦 𝑀 𝑎 1 = 𝑥 1 , 𝑀 . ( 3 . 7 )
Let us consider the third of (3.3) for 𝑖 = 2 . From the boundedness of 𝑥 1 and ( i v ) we have 𝑥 2 ( 𝑛 + 1 ) ≤ 𝑥 2 ( 0 ) + 𝑥 1 , 𝑀 𝜙 2 𝑦 𝑀 𝑛 𝑙 = 0 0 𝑥 0 2 0 0 𝑑 1 − 𝑎 2 𝑛 − 𝑙 ≤ 𝑥 2 𝑥 ( 0 ) + 1 , 𝑀 𝜙 2 𝑦 𝑀 𝑎 2 = 𝑥 2 , 𝑀 . ( 3 . 8 ) The boundedness of the remaining sequences can be proved in the same way.
If (3.6) does not hold then, from ( i ) and ( v i i ) , there exists 𝑞 > 0 and 𝑖 such that 2 ≤ 𝑖 ≤ 𝑚 and 𝜙 𝑖 ( 𝑦 ) ≤ 𝑞 𝜓 𝑖 − 1 ( 𝑦 ) , 𝑦 ≥ 0 . ( 3 . 9 ) Thus, by the third of (3.3), ( i i ) , ( i i i ) , and (3.5), the boundedness of 𝑥 𝑖 (and then the 𝑥 𝑗 sequences, 𝑗 > 𝑖 ) can be proved with the same argumentation used before for 𝑥 1 and 𝑥 2 . Since by 1 ≤ 𝑖 ≤ 𝐿 , 𝑥 𝑖 is bounded, we obtain the boundedness of the remaining sequences 𝑥 𝑗 , 1 ≤ 𝑗 ≤ 𝑖 .

In order to simplify the theorems' proofs of the remaining section, let us set

𝑥 0 = 𝑥 𝐿 = l i m i n f 𝑛 → ∞ 𝑥 𝐿 ( 𝑛 ) , 𝑥 0 = 𝑥 𝐿 l i m s u p 𝑛 → ∞ 𝑥 𝐿 ( 𝑛 ) ( 3 . 1 0 ) and introduce the following basic lemma.

Lemma 3.3. Let one assume that hypotheses of Theorem 3.2 hold. Then ∑ 𝛼 − 𝑚 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 𝜓 𝑖 𝑦 𝑥 𝑖 𝛽 ≤ 𝑦 ≤ ∑ 𝑦 ≤ 𝛼 − 𝑚 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 𝜓 𝑖 𝑦 𝑥 𝑖 𝛽 ≤ 𝛼 𝛽 , 𝜙 𝑖 𝑦 𝑎 𝑖 𝑥 𝑖 − 1 ≤ 𝑥 𝑖 ≤ 𝑥 𝑖 ≤ 𝜙 𝑖 𝑦 𝑎 𝑖 𝑥 𝑖 − 1 𝑥 , 𝑖 = 1 , 2 , … , 𝑚 , 𝑖 ≥ ⎛ ⎜ ⎜ ⎝ 𝐿 𝑗 = 1 𝜙 0 𝑥 0 2 0 0 𝑑 𝑗 𝑦 𝑎 𝑗 ⎞ ⎟ ⎟ ⎠ 𝑥 𝑖 , 𝑥 𝑖 ≤ 𝐿 𝑗 = 1 𝜙 0 𝑥 0 2 0 0 𝑑 𝑗 𝑦 𝑎 𝑗 𝑥 𝑖 , 𝑖 = 1 , 2 , … , 𝐿 . ( 3 . 1 1 )

Proof. From (1.1) and Theorems 3.1 and 3.2, we easily have that 𝑦 ≥ 𝛼 + ( 1 − 𝛽 ) 𝑦 − 𝑚 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 𝜓 𝑖 𝑦 𝑥 𝑖 ( 3 . 1 2 ) and then 𝑦 ≥ ∑ 𝛼 − 𝑚 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 𝜓 𝑖 𝑦 𝑥 𝑖 𝛽 . ( 3 . 1 3 ) In the same way it can be proved that ∑ 𝑦 ≤ 𝛼 − 𝑚 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 𝜓 𝑖 𝑦 𝑥 𝑖 𝛽 ≤ 𝛼 𝛽 . ( 3 . 1 4 )
Also, we easily have that 𝜙 𝑖 𝑦 𝑎 𝑖 𝑥 𝑖 − 1 ≤ 𝑥 𝑖 ≤ 𝑥 𝑖 ≤ 𝜙 𝑖 𝑦 𝑎 𝑖 𝑥 𝑖 − 1 , 𝑖 = 1 , 2 , … , 𝑚 , ( 3 . 1 5 ) and we have that 𝑥 𝐿 ≤ 𝐿 𝑗 = 1 𝜙 0 𝑥 0 2 0 0 𝑑 𝑗 𝑦 𝑎 𝑗 𝑥 𝐿 , 𝑥 𝑖 ≤ 𝜙 𝑖 𝑦 𝑎 𝑖 𝑥 𝑖 − 1 ≤ ⋯ ≤ 𝑖 𝑗 = 2 𝜙 0 𝑥 0 2 0 0 𝑑 𝑗 𝑦 𝑎 𝑗 𝑥 1 ≤ 𝑖 𝑗 = 1 𝜙 0 𝑥 0 2 0 0 𝑑 𝑗 𝑦 𝑎 𝑗 𝑥 𝐿 ≤ 𝑖 𝑗 = 1 𝜙 0 𝑥 0 2 0 0 𝑑 𝑗 𝑦 𝑎 𝑗 𝜙 𝐿 𝑦 𝑎 𝐿 𝑥 𝐿 − 1 ≤ ⋯ ≤ 𝑖 𝑗 = 1 𝜙 0 𝑥 0 2 0 0 𝑑 𝑗 𝑦 𝑎 𝑗 𝐿 𝑗 = 𝑖 + 1 𝜙 0 𝑥 0 2 0 0 𝑑 𝑗 𝑦 𝑎 𝑗 𝑥 𝑖 = 𝐿 𝑗 = 1 𝜙 0 𝑥 0 2 0 0 𝑑 𝑗 𝑦 𝑎 𝑗 𝑥 𝑖 , 1 ≤ 𝑖 ≤ 𝐿 , ( 3 . 1 6 ) and then 𝑥 𝑖 ≤ 𝐿 𝑗 = 1 𝜙 0 𝑥 0 2 0 0 𝑑 𝑗 𝑦 𝑎 𝑗 𝑥 𝑖 , 1 ≤ 𝑖 ≤ 𝐿 . ( 3 . 1 7 ) Similarly, we have that 𝑥 𝑖 ≥ ⎛ ⎜ ⎜ ⎝ 𝐿 𝑗 = 1 𝜙 0 𝑥 0 2 0 0 𝑑 𝑗 𝑦 𝑎 𝑗 ⎞ ⎟ ⎟ ⎠ 𝑥 𝑖 , 1 ≤ 𝑖 ≤ 𝐿 . ( 3 . 1 8 ) The remaining parts are obtained similarly.

Note that if there exists an integer 𝑖 ∈ { 1 , 2 , … , 𝑚 } such that 𝑥 𝑖 > 0 , then by the last inequalities of (3.11) in Lemma 3.3, we have that

𝐿 𝑗 = 1 𝜙 0 𝑥 0 2 0 0 𝑑 𝑗 𝑦 𝑎 𝑗 ≤ 1 ≤ 𝐿 𝑗 = 1 𝜙 0 𝑥 0 2 0 0 𝑑 𝑗 𝑦 𝑎 𝑗 . ( 3 . 1 9 )

4. Asymptotic Properties

We assume that hypotheses of Theorem 3.2 hold and

(viii) ∏ 𝑃 ( 𝜆 ) ≡ 𝐿 𝑗 = 1 0 𝑥 0 2 0 0 𝑑 ( 𝜙 𝑗 ( 𝜆 ) / 𝑎 𝑗 ) i s a s t r i c t l y i n c r e a s i n g p o s i t i v e c o n t i n u o u s f u n c t i o n o f 𝜆 on ( 0 , + ∞ ) and 0 ≤ 𝑃 ( 0 ) < 1 ,

and put 𝑅 0 𝛼 = 𝑃 𝛽 . ( 4 . 1 ) We have the following theorems.

Theorem 4.1. Let one assume that hypotheses of Theorem 3.2 and ( v i i i ) hold. Then 𝑅 0 ≤ 1 if and only if the desease free equilibrium point 𝐸 0 = ( 𝛼 / 𝛽 , 0 , … , 0 ) is global asymptotically stable. In this case l i m 𝑛 → ∞ 𝑦 𝛼 ( 𝑛 ) = 𝛽 , l i m 𝑛 → ∞ 𝑥 𝑖 ( 𝑛 ) = 0 , 𝑖 = 1 , 2 , … , 𝑚 . ( 4 . 2 )

Proof. Sufficient condition (" ⇒ ") is proved by Lemmas 4.4 and 4.5. Necessary condition (" ⇐ ") is proved by Lemma 4.7.

Theorem 4.2. Let one assume that hypotheses of Theorem 3.2 and ( v i i i ) hold. Then 𝑅 0 > 1 if and only if 𝛼 𝑦 < 𝛽 , 𝑥 𝑖 > 0 , 1 ≤ 𝑖 ≤ 𝑚 . ( 4 . 3 ) In this case, it holds that for 𝜆 ∗ defined in Lemma 4.6, 𝑦 ≤ 𝜆 ∗ ≤ 𝑦 , 𝑥 𝑖 ≤ 𝛼 − 𝛽 𝜆 ∗ ∑ 𝑚 𝑙 = 1 0 𝑥 0 2 0 0 𝑑 𝜓 𝑙 𝜆 ∗ ∏ 𝑙 𝑗 = 𝑖 + 1 𝜙 0 𝑥 0 2 0 0 𝑑 𝑗 𝜆 ∗ / 𝑎 𝑗 ≤ 𝑥 𝑖 , 𝑖 = 1 , 2 , … , 𝑚 . ( 4 . 4 )

Proof. By Theorem 4.1 and (3.11) in Lemma 3.3, we obtain (4.3). Moreover, by Lemma 4.6 and (3.19), we have 𝑦 ≤ 𝜆 ∗ ≤ 𝑦 and by (3.11), we have (4.4).

Theorem 4.3. Let one assume that hypotheses of Theorem 3.2 and ( v i i i ) hold and let one suppose that there exists a globally asymptotically stable endemic equilibrium point ( 𝑦 ∗ , 𝑥 ∗ 1 , … , 𝑥 ∗ 𝑚 ) , then 𝑅 0 > 1 if and only if there exists a unique solution 0 < 𝜆 = 𝜆 ∗ < 𝛼 / 𝛽 such that 𝑃 ( 𝜆 ) = 1 , and 𝑦 ∗ = 𝜆 ∗ < 𝛼 𝛽 , 𝑥 ∗ 𝑖 ≤ 𝛼 − 𝛽 𝜆 ∗ ∑ 𝑚 𝑙 = 1 0 𝑥 0 2 0 0 𝑑 𝜓 𝑙 𝜆 ∗ ∏ 𝑙 𝑗 = 𝑖 + 1 𝜙 0 𝑥 0 2 0 0 𝑑 𝑗 𝜆 ∗ / 𝑎 𝑗 , 𝑖 = 1 , 2 , … , 𝑚 . ( 4 . 5 )

Proof. By Lemmas 4.5 and 3.3, we obtain the thesis.

For the proofs of Theorems 4.1–4.3, we need the following lemmas.

Lemma 4.4. Let one assume that hypotheses of Theorem 3.2 and ( v i i i ) hold. Then, if 𝑅 0 < 1 , then l i m 𝑛 → ∞ 𝑦 𝛼 ( 𝑛 ) = 𝛽 , l i m 𝑛 → ∞ 𝑥 𝑖 ( 𝑛 ) = 0 , 𝑖 = 1 , 2 , … , 𝑚 . ( 4 . 6 ) If 𝑅 0 = 1 , then 𝛼 𝑦 = 𝛽 . ( 4 . 7 )

Proof. If 𝑃 ( ∏ 𝑦 ) = 𝐿 𝑗 = 1 0 𝑥 0 2 0 0 𝑑 ( 𝜙 𝑗 ( 𝑦 ) / 𝑎 𝑗 ) < 1 , then by (3.17) and Lemma 3.3, we have that 𝑥 𝑖 = 𝑥 𝑖 = 0 , 1 ≤ 𝑖 ≤ 𝑚 and 𝑦 = 𝑦 = 𝛼 / 𝛽 which implies (4.6). Therefore, if 𝑅 0 < 1 , then 𝑃 ( 𝑦 ) ≤ 𝑃 ( 𝛼 / 𝛽 ) = 𝑅 0 < 1 and hence, (4.6) holds.
If 𝑅 0 = 1 , then 𝑦 = 𝛼 / 𝛽 , because if 𝑦 < 𝛼 / 𝛽 then by the fact that 𝑃 ( 𝜆 ) is a strictly increasing positive function of 𝜆 on ( 0 , + ∞ ) , we have that 𝑃 ( 𝑦 ) < 𝑃 ( 𝛼 / 𝛽 ) = 𝑅 0 = 1 and by the above discussion, we obtain (4.6), which is a contradiction.

Lemma 4.5. Let one assume that hypotheses of Theorem 3.2 and ( v i i i ) hold. Then 𝑅 0 = 1 implies (4.6).

Proof. First let us assume 𝑦 𝛼 ( 0 ) ≤ 𝛽 . ( 4 . 8 ) From Lemma 4.4 we have 𝑦 = 𝛼 / 𝛽 , so there exists a sequence { 𝑛 𝑘 } ∞ 𝑘 = 1 such that l i m 𝑘 → ∞ 𝑦 ( 𝑛 𝑘 ) = 𝑦 . Let us define the two sets: 𝐸 𝑔 𝑡 = 𝑛 𝑘 𝑛 ∶ 𝑦 𝑘 𝑛 − 1 ≥ 𝑦 𝑘 , 𝐸 𝑙 𝑡 = 𝑛 𝑘 𝑛 ∶ 𝑦 𝑘 𝑛 − 1 ≤ 𝑦 𝑘 . ( 4 . 9 ) Let us consider two cases with respect to the cardinality of the set 𝐸 𝑔 𝑡 .
Case 1 ( 𝐸 𝑔 𝑡 = ∞ ). Let us consider the subsequence { 𝑛 𝑘 𝑗 } ∞ 𝑗 = 1 corresponding to indexes belonging to 𝐸 𝑔 𝑡 .
In this case we easily see (from Lemma 3.3   𝑦 ≤ 𝛼 / 𝛽 ) that l i m 𝑗 → ∞ 𝑦 ( 𝑛 𝑘 𝑗 − 1 ) = 𝑦 . From the first of (1.1) computed in 𝑛 𝑘 𝑗 , 𝑗 = 1 , 2 , … we have 𝑦 𝑛 𝑘 𝑗 𝑛 = 𝛼 + ( 1 − 𝛽 ) 𝑦 𝑘 𝑗 − − 1 𝑚 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 𝜓 𝑖 𝑦 𝑛 𝑘 𝑗 𝑥 𝑖 𝑛 𝑘 𝑗 , ( 4 . 1 0 ) and as 𝑗 goes to infinity: 𝑚 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 𝜓 𝑖 𝑦 l i m 𝑗 → ∞ 𝑥 𝑖 𝑛 𝑘 𝑗 = 0 . ( 4 . 1 1 ) We know that there exists î such that 𝜓 î is strictly increasing, so from positivity of 𝜓 î ( 𝑦 ) , we obtain: l i m 𝑗 → ∞ 𝑥 î 𝑛 𝑘 𝑗 = 0 . ( 4 . 1 2 )
Case 2 ( 𝐸 𝑔 𝑡 < ∞ ). In this case we have | 𝐸 𝑙 𝑡 | = ∞ . Let us consider the subsequence corresponding to indexes belonging to 𝐸 𝑙 𝑡 , name it { 𝑛 𝑘 𝑗 } ∞ 𝑗 = 1 again, so we have 𝑦 𝑛 𝑘 𝑗 𝑛 − 𝑦 𝑘 𝑗 − 1 ≥ 0 ∀ 𝑘 ∈ ℕ . ( 4 . 1 3 ) From the first of (1.1) we have 𝑛 𝛽 𝑦 𝑘 𝑗 𝑛 = 𝛼 + ( 1 − 𝛽 ) 𝑦 𝑘 𝑗 𝑛 − 1 − ( 1 − 𝛽 ) 𝑦 𝑘 𝑗 − 𝑚 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 𝜓 𝑖 𝑦 𝑛 𝑘 𝑗 𝑥 𝑖 𝑛 𝑘 𝑗 ( 4 . 1 4 ) and then 𝑦 𝑛 𝑘 𝑗 = 𝑦 𝑛 𝛼 − ( 1 − 𝛽 ) 𝑘 𝑗 𝑛 − 𝑦 𝑘 𝑗 − ∑ − 1 𝑚 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 𝜓 𝑖 𝑦 𝑛 𝑘 𝑗 𝑥 𝑖 𝑛 𝑘 𝑗 𝛽 ≤ 𝛼 𝛽 . ( 4 . 1 5 ) As 𝑗 goes to infinity, we obtain l i m 𝑗 → ∞ 𝑦 𝑛 𝛼 − ( 1 − 𝛽 ) 𝑘 𝑗 𝑛 − 𝑦 𝑘 𝑗 − ∑ − 1 𝑚 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 𝜓 𝑖 𝑦 𝑛 𝑘 𝑗 𝑥 𝑖 𝑛 𝑘 𝑗 𝛽 = 𝛼 𝛽 ( 4 . 1 6 ) and then l i m 𝑗 → ∞ 𝑦 𝑛 ( 1 − 𝛽 ) 𝑘 𝑗 𝑛 − 𝑦 𝑘 𝑗 + − 1 𝑚 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 𝜓 𝑖 𝑦 𝑛 𝑘 𝑗 𝑥 𝑖 𝑛 𝑘 𝑗 = 0 . ( 4 . 1 7 ) This leads (from (4.13) and positivity of ∑ 𝑚 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 𝜓 𝑖 ( 𝑦 ( 𝑛 𝑘 𝑗 ) ) 𝑥 𝑖 ( 𝑛 𝑘 𝑗 ) ) to l i m 𝑗 → ∞ 𝑦 𝑛 𝑘 𝑗 = l i m 𝑗 → ∞ 𝑦 𝑛 𝑘 𝑗 = − 1 𝛼 𝑦 = 𝛽 , l i m 𝑚 𝑗 → ∞ 𝑖 = 1 0 𝑥