A General Discrete Time Model of Population Dynamics in the Presence of an Infection
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.
Consider the following set of difference equations:
where , . Moreover , , and at least one of the () is strictly monotone increasing functions and
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 , while the latter is represented by one of the sequences ; we name it where is an integer between 1 and .
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 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 ). This continuous model represents the spread of HIV-1 infection inside the human organism. Here represents the number of susceptible cells which are present at time in a unit of plasma. The process of infection of a cell is divided into several sequential stages; therefore, is the number of infected cells at time at stage . The variable is the number of viruses at time . The meaning of the rest of symbols can be found in 
Rewrite (1.1) at a general time step the length of which is and put
This can be easily seen (see, e.g., ) to be the discrete analogue of (2.1) by dividing each equation by .
In conclusion, by assuming , we have that (1.1) is the discrete counterpart of (2.1) provided that The role of is related to that of the variable of (2.1) according to the following scheme:
Observe that and play the role of and , respectively, and therefore they correspond to the susceptible and infective populations as we mentioned in the introduction.
Example 2.2 (see ). This represents the spread of HTLV-I infection in a human organism we refer to  for the meaning of the symbols
As in Example 2.1 we can see that (1.1) is the discrete counterpart of (2.5) provided that The correspondence between the variables of (1.1) and (2.5) is summarized by
Let us note that this model is mathematically equivalent to the classical SIR model [4, model (2.5)].
Example 2.3 (see ). This represents the spread of HIV-I infection in a human organism Here we have
It is worth to note that all the continuous models just proposed can be discretized by means of the discrete model proposed in . 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 ). This represents the spread of HIV-I infection in a human organism, too This continuous model cannot be discretized by means of the discrete model proposed in  because of the presence of the two nonlinear terms ( and ) in the first equation. Instead, that can be done by means of (1.1) and we have .
3. Basic Properties
Since functions and () 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);(ii)(iii)(iv) is not decreasing and (v)(vi) s.t. and is strictly increasing. Then
Proof. From , and the positivity of we have Now assume From (v), (vi), we get and from (i), (ii) and the first of (1.1) we obtain which contradicts (3.1). The rest of the theorem can be proved in the same way (by induction).
Theorem 3.2. Assume that (i);(ii);(iii);(iv) is not decreasing and ;(v)(vi) s.t. and is strictly increasing;(vii) s.t. and
Then, the sequences 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]):
Since the hypotheses of the previous theorem hold, positivity of the sequences is assured. From the first of (3.3) we obtain
and so the boundedness of . From the first of (1.1) and (3.4) we also obtain for all
Assume that there exists such that
From the second of (3.3), , ( and (3.5) we have
Let us consider the third of (3.3) for . From the boundedness of and we have The boundedness of the remaining sequences can be proved in the same way.
If (3.6) does not hold then, from and , there exists and such that and Thus, by the third of (3.3), and (3.5), the boundedness of (and then the sequences, ) can be proved with the same argumentation used before for and . Since by , is bounded, we obtain the boundedness of the remaining sequences .
In order to simplify the theorems' proofs of the remaining section, let us set
and introduce the following basic lemma.
Lemma 3.3. Let one assume that hypotheses of Theorem 3.2 hold. Then
Proof. From (1.1) and Theorems 3.1 and 3.2, we easily have that
In the same way it can be proved that
Also, we easily have that and we have that and then Similarly, we have that The remaining parts are obtained similarly.
4. Asymptotic Properties
We assume that hypotheses of Theorem 3.2 hold and(viii) on and
and put We have the following theorems.
Theorem 4.1. Let one assume that hypotheses of Theorem 3.2 and hold. Then if and only if the desease free equilibrium point is global asymptotically stable. In this case
Theorem 4.3. Let one assume that hypotheses of Theorem 3.2 and hold and let one suppose that there exists a globally asymptotically stable endemic equilibrium point , then if and only if there exists a unique solution such that , and
Lemma 4.4. Let one assume that hypotheses of Theorem 3.2 and hold. Then, if , then If , then
Proof. If , then by (3.17) and Lemma 3.3, we have that and which implies (4.6). Therefore, if , then and hence, (4.6) holds.
If , then , because if then by the fact that is a strictly increasing positive function of on , we have that and by the above discussion, we obtain (4.6), which is a contradiction.
Proof. First let us assume
From Lemma 4.4 we have , so there exists a sequence such that . Let us define the two sets:
Let us consider two cases with respect to the cardinality of the set .
Case 1 (). Let us consider the subsequence corresponding to indexes belonging to .
In this case we easily see (from Lemma 3.3 ) that . From the first of (1.1) computed in we have and as goes to infinity: We know that there exists such that is strictly increasing, so from positivity of , we obtain:
Case 2 (). In this case we have . Let us consider the subsequence corresponding to indexes belonging to , name it again, so we have From the first of (1.1) we have and then As goes to infinity, we obtain and then This leads (from (4.13) and positivity of ) to Once again from this last statement, from the strict monotonousness of and Theorem 3.1 we obtain So we proved in both cases that exists a sequence such that In the same way, we can prove that there exists a subsequence of , for the sake of simplicity we name it , such that Let us consider the positive and bounded sequence Assume and compute its first difference, there results where Hence As it can be easily seen this equality also holds for . From the first of (3.3) and (4.8) it is then, by taking into account the fact that is nondecreasing, we have and from definition of Once again by recalling that is nondecreasing and we have This implies that the sequence is convergent. Since, from (4.22), , we obtain , and considering that , we obtain Equation (4.6) can be easily obtained from this last statement, from Lemma 3.3 and (1.1).
Now let us consider the case From the first of (1.1), we obtain so the bounded sequence is monotonic, then it converges. From this and Lemma 3.3 we obtain: This implies that We know that exists such that is strictly increasing, and so we have From (1.1) we obtain for and . Moreover, from Lemma 3.3 we obtain for .
Hence, we have: Otherwise, if (4.31) does not hold, there exists such that then we can use as starting value instead of .
Proof. If , then by the fact that is a strictly increasing positive function of on and , we have that has a unique solutions . 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).