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

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=1𝜓_𝑖(𝑦(𝑛+1))𝑥_𝑖𝑥(𝑛+1),𝑛≥0,₁(𝑛+1)=(1−𝑎₁)𝑥₁(𝑛)+𝜙₁(𝑦(𝑛))𝑥_𝐿𝑥(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)∈𝐶⁰(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

𝑦𝜓_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−𝑏𝑆(𝑡)−𝑐𝑆(𝑡)𝑉(𝑡),_'1(𝑡)=𝑐𝑆(𝑡)𝑉(𝑡)−𝑘𝐼₁𝐼(𝑡),_'𝑗(𝑡)=𝑘(𝐼_𝑗−1(𝑡)−𝐼_𝑗𝑉(𝑡)),𝑗=2,…,5,_'(𝑡)=𝑝𝐼₅(𝑡)−𝑞𝑉(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_𝑖𝜓(𝑦)=0,𝑖=1,…,5;₆(𝑦)=𝜙₁(𝑦)=𝛾𝑦;𝜙_𝑖(𝑦)=𝑘ℎ,𝑖=2,…,5;𝜙₆𝑎(𝑦)=𝑝ℎ;_𝑖=𝑘ℎ,𝑖=1,…,5;𝑎₆=𝑞ℎ;𝛼=𝑎ℎ;𝛽=𝑏ℎ;𝛾=𝑐ℎ. (2.2) We obtain 𝑦(𝑡+ℎ)−𝑦(𝑡)=𝑎ℎ−𝑏ℎ𝑦(𝑡)−𝑐ℎ𝑦(𝑡+ℎ)𝑥₆𝑥(𝑡+ℎ),₁(𝑡+ℎ)−𝑥(𝑡)=−𝑘ℎ𝑥₁(𝑡)+𝑐ℎ𝑦(𝑡)𝑥₆𝑥(𝑡),_𝑖(𝑡+ℎ)−𝑥(𝑡)=−𝑘ℎ𝑥_𝑖(𝑡)+𝑘ℎ𝑥_𝑖−1𝑥(𝑡),𝑖=2,…,5,₆(𝑡+ℎ)−𝑥(𝑡)=−𝑞ℎ𝑥₆(𝑡)+𝑝ℎ𝑥₅(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;𝜙₁(𝑦)=𝑐𝑦;𝜙_𝑖(𝑦)≡𝑘,𝑖=2,…,5,𝜙₆𝑎(𝑦)≡𝑝;_𝑖=𝑘,𝑖=1,…,5;𝑎₆=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,𝑥₁↔𝐼₁,…,𝑥₅↔𝐼₅,𝑥₆↔𝑉.
Observe that 𝑦(𝑛) and 𝑥₆(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−𝑏𝑇(𝑡)−𝑐𝑇_𝐴𝑇(𝑡)𝑇(𝑡),_'𝐵(𝑡)=𝑐𝑇_𝐴(𝑡)𝑇(𝑡)−𝑑_𝐵𝑇_𝐿𝑇(𝑡),_'𝐴(𝑡)=𝑔𝑇_𝐵(𝑡)−𝑑_𝐴𝑇_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;𝜓₁(𝑦)=0;𝜙₁(𝑦)=𝜓₂(𝑦)=𝑐𝑦;𝜙₂(𝑦)≡𝑔;𝑎₁=𝑑_𝐵,𝑎₂=𝑑_𝐴;𝛼=𝑎;𝛽=𝑏. The correspondence between the variables of (1.1) and (2.5) is summarized by 𝑦↔𝑇,𝑥₁↔𝑇_𝐵;𝑥₂↔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−𝑏𝑇(𝑡)−𝑐𝑉_𝐼𝑉(𝑡)𝑇(𝑡),_'𝐼(𝑡)=𝑔₁𝑇^∗(𝑡)−𝑑₁𝑉_𝐼𝑇(𝑡),_∗'(𝑡)=𝑐𝑉_𝐼(𝑡)𝑇(𝑡)−𝑑₂𝑇^∗𝑉(𝑡),_'𝑁𝐼(𝑡)=𝑔₂𝑇^∗(𝑡)−𝑑₁𝑉_𝑁𝐼(𝑡). (2.6) Here we have 𝑚=3;𝐼=1;𝐿=2;𝜙₁(𝑦)≡𝑔₁;𝜙₂(𝑦)=𝜓₁(𝑦)=𝑐𝑦;𝜙₃(𝑦)≡𝑔₂;𝜓₂(𝑦)=𝜓₃(𝑦)=0;𝑎₁=𝑑₁,𝑎₂=𝑑₂;𝑎₃=𝑑₁;𝛼=𝑎;𝛽=𝑏.𝑦↔𝑇,𝑥₁↔𝑉_𝐼;𝑥₂↔𝑇^∗;𝑥₃↔𝑉_𝑁𝐼.

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−𝑑𝑇(𝑡)−(1−𝜀)𝑘𝑉(𝑡)𝑇(𝑡)−𝑞𝑇^∗𝑇(𝑡)𝑇(𝑡),_∗'(𝑡)=(1−𝜀)𝑘𝑉(𝑡)𝑇(𝑡)−𝛿𝑇^∗𝑉(𝑡),_'(𝑡)=𝑁_𝑇𝛿𝑇^∗(𝑡)−𝑐𝑉(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 (𝑘𝑉(𝑡)𝑇(𝑡) and 𝑞𝑇^∗(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;𝑎₁=𝛿;𝑎₂=𝑐;𝛽=𝑑;𝜓₁(𝑦)=𝑞𝑦;𝜓₂(𝑦)=𝜙₁(𝑦)=(1−𝜀)𝑘𝑦;𝜙₂(𝑦)=𝑁_𝑇𝑎₁;𝑇↔𝑦;𝑇^∗↔𝑥₁;𝑉↔𝑥₂.

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)𝑦𝜓_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 (iii), (iv) 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=1𝜓_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)𝑦𝜓_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)≤𝑞𝜓_𝐿(𝑦),𝑦≥0if𝜙𝑖=1,_𝑖(𝑦)≤𝑞𝜓_𝑖−1(𝑦),𝑦≥0if2≤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]): 𝛼𝑦(𝑛+1)=𝛽[1−(1−𝛽)^𝑛+1]+(1−𝛽)^𝑛+1𝑦(0)−^𝑛+1_𝑙=1(1−𝛽)^{𝑚𝑛+1−𝑙}_𝑖=1𝜓_𝑖(𝑦(𝑙))𝑥_𝑖𝑥(𝑙),₁(𝑛+1)=(1−𝑎₁)^𝑛+1𝑥₁(0)+^𝑛_𝑙=0(1−𝑎₁)^𝑛−𝑙𝜙₁(𝑦(𝑙))𝑥_𝐿𝑥(𝑙),1≤𝐿≤𝑚,_𝑖(𝑛+1)=(1−𝑎_𝑖)^𝑛+1𝑥_𝑖(0)+^𝑛_𝑙=0(1−𝑎_𝑖)^𝑛−𝑙𝜙_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 𝛼𝑦(𝑛+1)≤𝛽[1−(1−𝛽)^𝑛+1]+(1−𝛽)^𝑛+1𝛼𝑦(0)≤max{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,…,𝑚𝜓_𝑗(𝑦(𝑙))𝑥_𝑗(𝑙)≤^𝑚_𝑖=1𝜓_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 𝜙₁(𝑦)≤𝑞𝜓_𝐿(𝑦),𝑦≥0. (3.6) From the second of (3.3), (ii), (iii) and (3.5) we have 𝑥₁(𝑛+1)=(1−𝑎₁)^𝑛+1𝑥₁(0)+^𝑛_𝑙=0(1−𝑎₁)^𝑛−𝑙𝜙₁(𝑦(𝑙))𝑥_𝐿(𝑙)≤(1−𝑎₁)^𝑛+1𝑥₁(0)+𝑞^𝑛_𝑙=0(1−𝑎₁)^𝑛−𝑙𝜓_𝐿(𝑦(𝑙))𝑥_𝐿(𝑙)≤𝑥₁(0)+𝑞𝛼+(1−𝛽)𝑦_𝑀𝑎₁=𝑥_1,𝑀. (3.7)
Let us consider the third of (3.3) for 𝑖=2. From the boundedness of 𝑥₁ and (iv) we have 𝑥₂(𝑛+1)≤𝑥₂(0)+𝑥_1,𝑀𝜙₂(𝑦_𝑀)^𝑛_𝑙=0(1−𝑎₂)^𝑛−𝑙≤𝑥₂𝑥(0)+^1,𝑀𝜙₂(𝑦_𝑀)𝑎₂=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 (vii), there exists 𝑞>0 and 𝑖 such that 2≤𝑖≤𝑚 and 𝜙_𝑖(𝑦)≤𝑞𝜓_𝑖−1(𝑦),𝑦≥0. (3.9) Thus, by the third of (3.3), (ii),(iii), 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 𝑥₁ and 𝑥₂. 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

𝑥₀=𝑥_𝐿=liminf_𝑛→∞𝑥_𝐿(𝑛),𝑥₀=𝑥_𝐿limsup_𝑛→∞𝑥_𝐿(𝑛) (3.10) and introduce the following basic lemma.

Lemma 3.3. Let one assume that hypotheses of Theorem 3.2 hold. Then 𝛼−∑^𝑚_𝑖=1𝜓_𝑖(𝑦)𝑥_𝑖𝛽≤𝑦≤𝑦≤𝛼−∑^𝑚_𝑖=1𝜓_𝑖(𝑦)𝑥_𝑖𝛽≤𝛼𝛽,𝜙_𝑖(𝑦)𝑎_𝑖𝑥_𝑖−1≤𝑥_𝑖≤𝑥_𝑖≤𝜙_𝑖(𝑦)𝑎_𝑖𝑥_𝑖−1𝑥,𝑖=1,2,…,𝑚,_𝑖≥(^𝐿_𝑗=1𝜙_𝑗(𝑦)𝑎_𝑗)𝑥_𝑖,𝑥_𝑖≤(^𝐿_𝑗=1𝜙_𝑗(𝑦)𝑎_𝑗)𝑥_𝑖,𝑖=1,2,…,𝐿. (3.11)

Proof. From (1.1) and Theorems 3.1 and 3.2, we easily have that 𝑦≥𝛼+(1−𝛽)𝑦−^𝑚_𝑖=1𝜓_𝑖(𝑦)𝑥_𝑖 (3.12) and then 𝑦≥𝛼−∑^𝑚_𝑖=1𝜓_𝑖(𝑦)𝑥_𝑖𝛽. (3.13) In the same way it can be proved that 𝑦≤𝛼−∑^𝑚_𝑖=1𝜓_𝑖(𝑦)𝑥_𝑖𝛽≤𝛼𝛽. (3.14)
Also, we easily have that 𝜙_𝑖(𝑦)𝑎_𝑖𝑥_𝑖−1≤𝑥_𝑖≤𝑥_𝑖≤𝜙_𝑖(𝑦)𝑎_𝑖𝑥_𝑖−1,𝑖=1,2,…,𝑚, (3.15) and we have that 𝑥_𝐿≤(^𝐿_𝑗=1𝜙_𝑗(𝑦)𝑎_𝑗)𝑥_𝐿,𝑥_𝑖≤𝜙_𝑖(𝑦)𝑎_𝑖𝑥_𝑖−1≤⋯≤(^𝑖_𝑗=2𝜙_𝑗(𝑦)𝑎_𝑗)𝑥₁≤(^𝑖_𝑗=1𝜙_𝑗(𝑦)𝑎_𝑗)𝑥_𝐿≤(^𝑖_𝑗=1𝜙_𝑗(𝑦)𝑎_𝑗)𝜙_𝐿(𝑦)𝑎_𝐿𝑥_𝐿−1≤⋯≤(^𝑖_𝑗=1𝜙_𝑗(𝑦)𝑎_𝑗)(^𝐿_𝑗=𝑖+1𝜙_𝑗(𝑦)𝑎_𝑗)𝑥_𝑖=(^𝐿_𝑗=1𝜙_𝑗(𝑦)𝑎_𝑗)𝑥_𝑖,1≤𝑖≤𝐿, (3.16) and then 𝑥_𝑖≤(^𝐿_𝑗=1𝜙_𝑗(𝑦)𝑎_𝑗)𝑥_𝑖,1≤𝑖≤𝐿. (3.17) Similarly, we have that 𝑥_𝑖≥(^𝐿_𝑗=1𝜙_𝑗(𝑦)𝑎_𝑗)𝑥_𝑖,1≤𝑖≤𝐿. (3.18) 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=1𝜙_𝑗(𝑦)𝑎_𝑗≤1≤^𝐿_𝑗=1𝜙_𝑗(𝑦)𝑎_𝑗. (3.19)

4. Asymptotic Properties

We assume that hypotheses of Theorem 3.2 hold and

(viii)𝑃(𝜆)≡∏^𝐿_𝑗=1(𝜙_𝑗(𝜆)/𝑎_𝑗)isastrictlyincreasingpositivecontinuousfunctionof𝜆 on (0,+∞)and 0≤𝑃(0)<1,

and put 𝑅₀𝛼=𝑃(𝛽). (4.1) We have the following theorems.

Theorem 4.1. Let one assume that hypotheses of Theorem 3.2 and (viii) hold. Then 𝑅₀≤1 if and only if the desease free equilibrium point 𝐸₀=(𝛼/𝛽,0,…,0) is global asymptotically stable. In this case lim_𝑛→∞𝛼𝑦(𝑛)=𝛽,lim_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 (viii) hold. Then 𝑅₀>1 if and only if 𝛼𝑦<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−𝛽𝜆^∗∑^𝑚_𝑙=1𝜓_𝑙(𝜆^∗)∏^𝑙_𝑗=𝑖+1(𝜙_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 (viii) hold and let one suppose that there exists a globally asymptotically stable endemic equilibrium point (𝑦^∗,𝑥^∗₁,…,𝑥^∗_𝑚), then 𝑅₀>1 if and only if there exists a unique solution 0<𝜆=𝜆^∗<𝛼/𝛽 such that 𝑃(𝜆)=1, and 𝑦^∗=𝜆^∗<𝛼𝛽,𝑥^∗_𝑖≤𝛼−𝛽𝜆^∗∑^𝑚_𝑙=1𝜓_𝑙(𝜆^∗)∏^𝑙_𝑗=𝑖+1(𝜙_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 (viii) hold. Then, if 𝑅₀<1, then lim_𝑛→∞𝛼𝑦(𝑛)=𝛽,lim_𝑛→∞𝑥_𝑖(𝑛)=0,𝑖=1,2,…,𝑚. (4.6) If 𝑅₀=1, then 𝛼𝑦=𝛽. (4.7)

Proof. If 𝑃(𝑦)=∏^𝐿_𝑗=1(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 𝑅₀<1, then 𝑃(𝑦)≤𝑃(𝛼/𝛽)=𝑅₀<1 and hence, (4.6) holds.
If 𝑅₀=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)=𝑅₀=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 (viii) hold. Then 𝑅₀=1 implies (4.6).

Proof. First let us assume 𝛼𝑦(0)≤𝛽. (4.8) From Lemma 4.4 we have 𝑦=𝛼/𝛽, so there exists a sequence {𝑛_𝑘}^∞_𝑘=1 such that lim_𝑘→∞𝑦(𝑛_𝑘)=𝑦. Let us define the two sets: 𝐸_𝑔𝑡={𝑛_𝑘∶𝑦(𝑛_𝑘−1)≥𝑦(𝑛_𝑘𝐸)},_𝑙𝑡={𝑛_𝑘∶𝑦(𝑛_𝑘−1)≤𝑦(𝑛_𝑘)}. (4.9) Let us consider two cases with respect to the cardinality of the set 𝐸_𝑔𝑡.
Case 1 (𝐸_𝑔𝑡=∞). Let us consider the subsequence {𝑛_𝑘𝑗}^∞_𝑗=1 corresponding to indexes belonging to 𝐸_𝑔𝑡.
In this case we easily see (from Lemma 3.3  𝑦≤𝛼/𝛽) that lim_𝑗→∞𝑦(𝑛_𝑘𝑗−1)=𝑦. From the first of (1.1) computed in 𝑛_𝑘𝑗,𝑗=1,2,… we have 𝑦(𝑛_𝑘𝑗)=𝛼+(1−𝛽)𝑦(𝑛_𝑘𝑗−1)−^𝑚_𝑖=1𝜓_𝑖(𝑦(𝑛_𝑘𝑗))𝑥_𝑖(𝑛_𝑘𝑗), (4.10) and as 𝑗 goes to infinity: ^𝑚_𝑖=1𝜓_𝑖(𝑦)lim_𝑗→∞𝑥_𝑖(𝑛_𝑘𝑗)=0. (4.11) We know that there exists î such that 𝜓_î is strictly increasing, so from positivity of 𝜓_î(𝑦), we obtain: lim_𝑗→∞𝑥_î(𝑛_𝑘𝑗)=0. (4.12)
Case 2 (𝐸_𝑔𝑡<∞). In this case we have |𝐸_𝑙𝑡|=∞. Let us consider the subsequence corresponding to indexes belonging to 𝐸_𝑙𝑡, name it {𝑛_𝑘𝑗}^∞_𝑗=1 again, so we have 𝑦(𝑛_𝑘𝑗)−𝑦(𝑛_𝑘𝑗−1)≥0∀𝑘∈ℕ. (4.13) From the first of (1.1) we have 𝛽𝑦(𝑛_𝑘𝑗)=𝛼+(1−𝛽)𝑦(𝑛_𝑘𝑗−1)−(1−𝛽)𝑦(𝑛_𝑘𝑗)−^𝑚_𝑖=1𝜓_𝑖(𝑦(𝑛_𝑘𝑗))𝑥_𝑖(𝑛_𝑘𝑗) (4.14) and then 𝑦(𝑛_𝑘𝑗)=𝛼−(1−𝛽)[𝑦(𝑛_𝑘𝑗)−𝑦(𝑛_𝑘𝑗−1)]−∑^𝑚_𝑖=1𝜓_𝑖(𝑦(𝑛_𝑘𝑗))𝑥_𝑖(𝑛_𝑘𝑗)𝛽≤𝛼𝛽. (4.15) As 𝑗 goes to infinity, we obtain lim_𝑗→∞𝛼−(1−𝛽)[𝑦(𝑛_𝑘𝑗)−𝑦(𝑛_𝑘𝑗−1)]−∑^𝑚_𝑖=1𝜓_𝑖(𝑦(𝑛_𝑘𝑗))𝑥_𝑖(𝑛_𝑘𝑗)𝛽=𝛼𝛽 (4.16) and then lim_𝑗→∞((1−𝛽)[𝑦(𝑛_𝑘𝑗)−𝑦(𝑛_𝑘𝑗−1)]+^𝑚_𝑖=1𝜓_𝑖(𝑦(𝑛_𝑘𝑗))𝑥_𝑖(𝑛_𝑘𝑗))=0. (4.17) This leads (from (4.13) and positivity of ∑^𝑚_𝑖=1𝜓_𝑖(𝑦(𝑛_𝑘𝑗))𝑥_𝑖(𝑛_𝑘𝑗)) to lim_𝑗→∞𝑦(𝑛_𝑘𝑗)=lim_𝑗→∞𝑦(𝑛_𝑘𝑗−1)=𝛼𝑦=𝛽,lim^𝑚_{𝑗→∞𝑖=1}𝜓_𝑖(𝑦(𝑛_𝑘𝑗))𝑥_𝑖(𝑛_𝑘𝑗)=0. (4.18) Once again from this last statement, from the strict monotonousness of 𝜓_î and Theorem 3.1 we obtain lim_𝑗→∞𝑥_î(𝑛_𝑘𝑗)=0. (4.19) So we proved in both cases that exists a sequence {𝑛_𝑘𝑗}_𝑗 such that lim_𝑗→∞𝑦(𝑛_𝑘𝑗)=lim_𝑗→∞𝑦(𝑛_𝑘𝑗−1)=𝛼𝑦=𝛽,lim_𝑗→∞𝑥_î(𝑛_𝑘𝑗)=0. (4.20) In the same way, we can prove that there exists a subsequence of {𝑛_𝑘𝑗}_𝑗, for the sake of simplicity we name it {𝑛_𝑙}_𝑙, such that lim_𝑙→∞𝑦(𝑛_𝑙−𝐿)=lim_𝑙→∞𝑦(𝑛_𝑙−𝐿−1)=𝛼𝑦=𝛽,lim_𝑙→∞𝑥_𝑗(𝑛_𝑙−𝐿)=0,𝑗=1,…,𝐿. (4.21)(4.22) Let us consider the positive and bounded sequence 𝐴(𝑥(𝑛))=𝑥_𝐿(𝑛)+^𝐿−1_𝑖=1^𝐿−1_𝑗=𝑖𝜙_𝑗+1(𝛼/𝛽)𝑎_𝑗𝑥_𝑖(𝑛). (4.23) Assume 𝐿≠1 and compute its first difference, there results 𝐴(𝑥(𝑛+1))−𝐴(𝑥(𝑛))=𝑥_𝐿(𝑛)[−𝑎_𝐿+^𝐿−1_𝑗=1𝜙_𝑗+1(𝛼/𝛽)𝑎_𝑗𝜙₁+(𝑦(𝑛))]_{𝐿−1𝑖=2𝐿−1𝑗=𝑖}𝜙_𝑗+1(𝛼/𝛽)𝑎_𝑗[−𝑎_𝑖𝑥_𝑖(𝑛)+𝜙_𝑖(𝑦(𝑛))𝑥_𝑖−1(𝑛)]+𝜙_𝐿(𝑦(𝑛))𝑥_𝐿−1−(𝑛)_{𝐿−1𝑗=1}𝜙_𝑗+1(𝛼/𝛽)𝑎_𝑗𝑎₁𝑥₁(𝑛)=𝑥_𝐿(𝑛)[−𝑎_𝐿+^𝐿−1_𝑗=1𝜙_𝑗+1(𝛼/𝛽)𝑎_𝑗𝜙₁+(𝑦(𝑛))]_{𝐿−1𝑖=1𝐿−1𝑗=𝑖}𝜙_𝑗+1(𝛼/𝛽)𝑎_𝑗[−𝑎_𝑖𝑥_𝑖(𝑛)]+^𝐿_𝑖=2^𝐿−1_𝑗=𝑖𝜙_𝑗+1(𝛼/𝛽)𝑎_𝑗𝜙_𝑖(𝑦(𝑛))𝑥_𝑖−1(𝑛), (4.24) where ∏^𝑖−1_𝑗=𝑖=1. Hence 𝐴(𝑥(𝑛+1))−𝐴(𝑥(𝑛))=𝑥_𝐿(𝑛)[−𝑎_𝐿+^𝐿−1_𝑗=1𝜙_𝑗+1(𝛼/𝛽)𝑎_𝑗𝜙₁−(𝑦(𝑛))]_{𝐿−1𝑖=1𝐿−1𝑗=𝑖+1}𝜙_𝑗+1(𝛼/𝛽)𝑎_𝑗𝜙_𝑖+1(𝛼𝛽)𝑥_𝑖+(𝑛)_{𝐿−1𝑖=1𝐿−1𝑗=𝑖+1}𝜙_𝑗+1(𝛼/𝛽)𝑎_𝑗𝜙_𝑖+1(𝑦(𝑛))𝑥_𝑖(𝑛). (4.25) As it can be easily seen this equality also holds for 𝐿=1. From the first of (3.3) and (4.8) it is 𝛼𝑦(𝑛)≤𝛽, (4.26) then, by taking into account the fact that 𝜙_𝑖 is nondecreasing, we have 𝐴(𝑥(𝑛+1))−𝐴(𝑥(𝑛))≤𝑎_𝐿𝑥_𝐿𝜙(𝑛)[−1+¹(𝑦(𝑛))𝜙₁(𝛼/𝛽)^𝐿_𝑗=1𝜙_𝑗(𝛼/𝛽)𝑎_𝑗],1≤𝐿≤𝑚 (4.27) and from definition of 𝑅₀𝐴(𝑥(𝑛+1))−𝐴(𝑥(𝑛))≤𝑎_𝐿𝑥_𝐿𝜙(𝑛)[−1+¹(𝑦(𝑛))𝜙₁𝑅(𝛼/𝛽)⁰],1≤𝐿≤𝑚. (4.28) Once again by recalling that 𝜙₁ is nondecreasing and 𝑅₀=1 we have 𝐴(𝑥(𝑛+1))−𝐴(𝑥(𝑛))≤0,1≤𝐿≤𝑚. (4.29) This implies that the sequence 𝐴(𝑥(𝑛)) is convergent. Since, from (4.22), lim_𝑙→∞𝐴(𝑥(𝑛_𝑙−𝐿))=0, we obtain lim_𝑛→∞𝐴(𝑥(𝑛))=0, and considering that 𝑥_𝐿(𝑛)≤𝐴(𝑥(𝑛)), we obtain lim_𝑛→∞𝑥_𝐿(𝑛)=0. (4.30) Equation (4.6) can be easily obtained from this last statement, from Lemma 3.3 and (1.1).
Now let us consider the case 𝛼𝑦(𝑖)>𝛽∀𝑖∈ℕ. (4.31) From the first of (1.1), we obtain 𝑦(𝑛+1)−𝑦(𝑛)=𝛼−𝛽𝑦(𝑛)−^𝑚_𝑖=1𝜓_𝑖(𝑦(𝑛+1))𝑥_𝑖(𝑛+1)≤0, (4.32) so the bounded sequence 𝑦(𝑛) is monotonic, then it converges. From this and Lemma 3.3 we obtain: lim_𝑛→∞𝛼𝑦(𝑛)=𝛽. (4.33) This implies that lim^𝑚_{𝑛→∞𝑖=1}𝜓_𝑖(𝑦(𝑛))𝑥_𝑖(𝑛)=0. (4.34) We know that exists î such that 𝜓_î is strictly increasing, and so we have lim_𝑛→∞𝑥_î(𝑛)=0. (4.35) From (1.1) we obtain lim_𝑛→∞𝑥_𝑗(𝑛)=0 for 𝑗<î and lim_𝑛→∞𝑥_𝐿(𝑛)=0. Moreover, from Lemma 3.3 we obtain lim_𝑛→∞𝑥_𝑘(𝑛)=0 for 𝑘>î.
Hence, we have: lim_𝑛→∞𝑥_𝑖(𝑛)=0,𝑖=1,…,𝑚. (4.36) Otherwise, if (4.31) does not hold, there exists 𝑟∈ℕ such that 𝑦(𝑟)≤𝛼/𝛽 then we can use 𝑦(𝑟) as starting value instead of 𝑦(0).

Lemma 4.6. Let one assume that hypotheses of Theorem 3.2 and (viii) hold. Then, if 𝑅₀>1, then there exist a unique solution 0<𝜆=𝜆^∗<𝛼/𝛽 of 𝑃(𝜆)=1 and positive constant solutions of (1.1) such that for any 𝑛=0,1,2,…𝑦(𝑛)=𝜆^∗<𝛼𝛽,𝑥₀(𝑛)∶=𝑥_𝐿(𝑛)=𝛼−𝛽𝜆^∗∑^𝑚_𝑙=1𝜓_𝑙(𝜆^∗)∏^𝑙_𝑗=1(𝜙_𝑗(𝜆^∗)𝑎_𝑗)𝑥>0,_𝑖(𝑛)=^𝑖_𝑗=1𝜙_𝑗(𝜆^∗)𝑎_𝑗𝑥₀(𝑛),𝑖=1,2,…,𝑚. (4.37)

Proof. If 𝑅₀=𝑃(𝛼/𝛽)>1, then by the fact that 𝑃(𝜆) is a strictly increasing positive function of 𝜆 on (0,+∞) and 0≤𝑃(0)<1, we have that 𝑃(𝜆)=1 has a unique solutions 0<𝜆=𝜆^∗<𝛼/𝛽. Then, by Lemma 3.3 and (3.19), we can easily see that there are positive constant solutions of (1.1) defined by (4.37).

From Lemmas 4.4–4.6, we obtain the following.

Lemma 4.7. Let one assume that hypotheses of Theorem 3.2 and (viii) hold. Then (4.6) implies 𝑅₀≤1.

Proof. If 𝑅₀=𝑃(𝛼/𝛽)>1, then by Lemma 4.6, we can see that there are positive constant solutions of (1.1). Hence the thesis holds.