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Modelling silicosis: existence, uniqueness and basic properties of solutions

We present a model for the silicosis disease mechanism following the original proposal by Tran, Jones, and Donaldson (1995) [9], as modified recently by da Costa, Drmota, and Grinfeld (2020) [4]. The model consists in an infinite ordinary differential equation system of coagulation-fragmentation-death type. Results of existence, uniqueness, continuous dependence on the initial data and differentiability of solutions are proved for the initial value problem.


Introduction
Silicosis is an occupational lung disease characterized by inflammation and fibrosis of the lungs caused by the inhalation of crystalline silica dust (also known as quartz or silicon dioxide dust, with chemical composition SiO 2 ) and can progress to respiratory failure and death [8]. Although it is a preventable disease, it is presently incurable and affects hundreds of thousands of persons, killing an estimated eleven thousand annually (see figures for 2017 in [5]).
In this paper we consider a mathematical model for the process used by the immune system to clear inhalated silica particles from the lungs' alveoli. The model is a slight modification proposed in [4] of one introduced in [9] that, to the best of our knowledge, has not received the attention we think it deserves.
We start by giving a simplified description of the physiological processes, essentially using [9], followed by the presentation of the mathematical model to be studied.
When a silica particle enters the respiratory system down to the pulmonary alveoli (the innermost part of the lungs) its presence can be detected by cells of the immune system called alveoli macrophages, either by random encounters or by chemotaxis generated by the silica particles. Macrophages can then engulf the silica particle in a process known as phagocytosis. Typically, the phagocytosis of organic materials (pathogens, dying or dead cells and cellular debris) result in its destruction by the macrophage with the production of basic molecular fragments followed by their expulsion or assimilation [1]. In the case of silica particles, however, there is essentially nothing to be destroyed from a molecular point of view and the role of the macrophages is just to expel their load of silica by travelling up the mucociliary escalator to get eliminated out of the respiratory system [10]. However, due to a mechanism not yet completely understood, silica particles are toxic to the macrophages [6] and lead to their death, which can occur while they are still in the lungs, thus liberating the silica particles back into the respiratory system.
A macrophage can ingest more than one silica particle. The hability of a given macrophage already containing i particles to ingest an additional particle typically decreases with i and in the model in [9] a maximum load capacity of n max < ∞ is assumed a priori. Given the toxicity of the silica particles, macrophages with a higher load of silica particles die at an higher rate. Additionally, the capacity of the macrophages to migrate up the mucocilliary escalator is also impaired by an increase in their load of silica particles. It is the balance of these three types of processes that leads to the mathematical model in [9] and that we also consider here with the modifications introduced in [4].
Let us denote by M i = M i (t) the concentration of macrophages containing i silica particles (which we will call the i-th cohort) at time t, by x = x(t) the concentration of silica particles, and by r (which, in principle, can be a function of x) the rate of supply of new (empty of silica particles) macrophages. Following [9], we obtain the following equations for the mechanism described above: where k i is the rate of phagocytosis of a silica particle by a macrophage already containing i particles, p i , is the transfer rate of macrophages in the i-th cohort to the mucocilliary escalator, i.e. the rate of their removal from the pulmonary alveoli together with their quartz load, and q i is the rate of death of the macrophages in the i-th cohort which results in the release of the quartz burden back into the lungs. Unlike in [9] we do not impose an upper limit on the number of particles n max < ∞ a macrophage can contain, the existence of such a load capacity will be a consequence of the assumptions on the rate coefficients k i and q i . In [4] the following rate equation for the evolution of the concentration of silica particles in the system under the assumption of an inhalation rate α was postulated, valid under the same assumption about the validity of the mass action law used to obtain the equations for the M i : A scheme of the processes modelled by the above rate equations is presented in Figure 1.
In [4] the equilibria of this system (with a countable infinite number) of equations was studied. An important modelling goal is, of course, the study of the dynamical stability properties of those equilibria and for this we need to consider solutions which are not equilibria and to study their behaviour as t → +∞. Thus, the existence of solutions of the initial value problem for this system with a countable infinite number of ordinary differential equations is a problem that has to be addressed first. This is the goal of the present paper. Figure 1. Reaction scheme of the mechanism studied in this paper. The input rates of quartz and of macrophages with no quartz particles are α and r, respectively. The concentration of free quartz particles and of macrophages containing j quartz particles are represented by x and M j , respectively. Macrophages M j can be destroyed, releasing j quartz particles (dashed lines), or they can be removed by rising in the mucocilliary escalator (vertical downward lines), or they can ingest an additional quartz particle becoming an M j+1 macrophage (horizontal rightward arrows).
In section 2 we state, in precise mathematical terms, the problem to be studied and the basic approach we shall use, and relate this system to others studied in the literature of cluster growth modelling. In section 3 we consider a finite dimension truncated system that shall be used, in section 4, to prove the existence of solutions to the Cauchy problem, which can be considered the main result of the paper. To prove differentiability of the solutions, in section 6, as well as uniqueness, in section 7, and the semigroup property, in section 8, a number of a priori estimates and evolution equations for the solution moments are needed and will be stated and proved in section 5.

Preliminaries
The Cauchy problem we will consider in this paper is the following system of a countable number of ordinary differential equations with nonnegative initial condi- This is an infinite dimensional system of ordinary differential equations of the type coagulation-fragmentation-death type and its analysis requires the definition of an Banach space to work in, as well as an adequate definition of solution. From the biological interpretation of the phase variables x and M i (i = 0, 1, . . .) given in the Introduction we can define the following quantities: Amount of quartz particles: It is easy to conclude from (1) that, at a formal level, Remark 2.1. These formal computations are justified if, for all i = 0, 1, 2, . . . , and all t 0, we have, Therefore, at least at a formal level, we have the following a priori bounds:Ṁ r, X α, andU r + α .
These formal computations and a comparison with the case of Becker-Döring equations [2] and with other more general coagulation-fragmentation systems [3] suggest that considering a space of sequences (x, M 0 , M 1 , . . .) where, for nonnegative sequences, the quantity U is finite can be expected to be a mathematically reasonable space to work in and, furthermore, will also have biological meaning.
Thus, let us denote an element of R N by y = (y n ) = (x, M 0 , M 1 , . . . ). Define and X = {y = (y n ) : y < ∞} . We say that y 0 if and only if y ∈ (R + 0 ) N and will denote the nonnegative cone of X by X + := {y ∈ X : y 0}.
Similarly, comparison with other types of coagulation-fragmentation equations suggests the adoption of a definition of solution that is very close to that used in [2] for the Becker-Döring system. This is not surprising, since an inspection of the model to be considered shows that, as in the Becker-Döring system, there is a phase variable that has a crucially distinct role in the dynamics: in the Becker-Döring system was the concentration of monomers [2], in system (1) it is the concentration of crystaline silica, x. This does not mean that (1) can be automatically transformed into the Becker-Döring system, but it suggests analogies in the definition of solution and in the general approch in the proofs we shall present.
A standard approach used in many studies of coagulation-fragmentation type equations is to consider a finite n-dimensional truncation of the infinite dimensional system and prove a priori bounds for the evolution of quantities associated with the truncated system that are independent of the truncation dimension n. This allows us to take limits as n → ∞ in the truncated system and, in this way, to prove results for the infinite system. In order to apply the same strategy here, we will present, in the next section, a truncated system and a useful auxiliary time evolution result.

The truncated silicosis system
In (1) consider rate coefficients satisfying k i = 0, for i n, and p i = q i = 0, for i n + 1 . It is clear from the biological meaning of these constants (and also from the kinetic scheme in Figure 1) that, with these conditions, if we take initial data satisfying M i (0) = 0 if i n + 1 then, for these i, M i (t) will remain zero for all t > 0. It is due to this invariance that the above conditions give rise to a finite (n+2)-dimensional approximation of the infinite dimensional system (1). The corresponding Cauchy problem, which we shall use extensively in the next section, can be explicitly written as follows: x n (0) = x 0 , M n i (0) = M 0i , i = 0, 1, . . . , n .
For the truncated versions of M and X , M n and X n say, we have: Therefore,U For now, let's only assume that the coefficients k i , p i and q i in (5), as well as the constants r and α, are nonnegative.
Since (5) is an initial value problem for a finite dimensional ODE with a smooth vector field, the standard Picard-Lindelöff theorem allows us to conclude that, for each initial condition y n 0 = (x n 0 , M n 0 , M n 1 , . . . , M n 0 ) ∈ R n+2 , there is a unique local solution in the classical sense, y n i (·) := (x n (·), M n 0 (·), M n 1 (·), . . . , M n n (·)), such that y n (0) = y n 0 . Moreover, we have: Lemma 3.1. Let n ∈ N 2 . For each y n 0 ∈ (R + 0 ) n+2 , the unique local solution of (5) satisfying y n (0) = y n 0 , y n (·), is extendable to the interval [0, +∞), and for each t 0, y n (t) ∈ (R + 0 ) n+2 . This solution satisfies, for all t 0, Furthermore, assume that (g i ) ∈ R n+2 . Then, for each positive integer m < n, Proof. Equations (5) have the property that allows us to apply the argument in the proof of Theorem III-4-5 in [7] to prove the nonnegativity property, that is, by adding any ε > 0 to all equations, and being y n,ε the corresponding solution, for any t 0 0, such that y n,ε (t 0 ) ∈ (R + 0 ) n+2 and, for some j ∈ N, y n,ε j (t 0 ) = 0, then, y n,ε j (t 0 ) > 0. Since (6) has been proved before, then we can say that y n is bounded in any bounded subset of its maximal domain of existence. This, together with the smoothness of the vector field in R n+2 , implies that this solution is extendable to [0, +∞). Equation (7) is easily obtained by multiplying the equation for eachṀ n i in (5) by g i and subsequently adding them together.

Existence
We consider the existence theorem for the Cauchy problem (1) for a relevant class of rate coefficients and initial conditions. In the proof we start by obtaining a candidate to a local solution as the limit of a subsequence of solutions of the truncated systems. This is done by establishing estimates that allow us to apply Ascoli-Arzel theorem. Then, we pass to the limit in the integral version of the truncated system by using some a priori estimates obtained via Gronwall's inequality. These ideas were originally used in the seminal paper [2] on the Becker-Döring system.
We shall regard y n as an element of X + , by considering y n r (t) ≡ 0, if r > n + 2, or equivalently M n r (t) ≡ 0 if r > n. With this identification we have y n = U n , and therefore, for all t 0, y n (t) y n (0) + (α + r)t y 0 + (α + r)t.
Since, for each i = 0, 1, 2, . . . , n, we obviously have (i + 1)M n i (t) y n (t) , we conclude that, for all t ∈ [0, T ], 0 M n i (t) Now, in order to apply Ascoli-Arzelà theorem, we need estimates on |Ṁ n i |, i = 0, 1, . . . , n. From (5) we have, for all t > 0, By the estimates (9), (10) and (11) with t ∈ [0, T ], and taking into account that x n y n , we conclude An important fact to be retained from estimates (12) is that, for each i = 0, 1, 2, . . . , |Ṁ n i | is bounded uniformly in n, and therefore the set of functions M n i (·), n = 1, 2, . . . , forms an equibounded and equicontinuous set of functions in [0, T ]. Therefore, by applying Ascoli-Arzelà theorem, and by a standard diagonalization procedure, we can establish the existence, for each i = 0, 1, 2, . . . , of a continuous function M i : [0, T ] → R, and a strictly increasing sequence of natural numbers (n j ), such that, Considering now the sequence (x n (·)), observe that the equation forẋ n in system (5) contains sums that are not clear how can controlled, as n → ∞, by estimates similar to the above ones. Therefore, we adopt the following different approach already used in the proof of Theorem 2.2 in [2]: we can extract a subsequence of (n j ), which we still designate by (n j ), such that, for some non-negative x ∈ L ∞ (0, T ), which means that Now, from (5), we have that, for all t 0, Since, for all t ∈ [0, T ], using the convergence of M nj 0 → M 0 , uniformly on [0, T ], and the weak * convergence (15) combined with estimate (14), we can pass to the limit in (16) to obtain, thus proving (2) in Definition 2.2. Now, fix i ∈ N. For j sufficiently large, by (5), Using the same arguments, now with the uniform convergence M nj i → M i , on [0, T ], i = 0, 1, 2, . . . , we can pass to the limit in (17) to obtain In order to prove the other assertions in Definition 2.2 and, in particular, to pass to the limit in the integral version of the last equation of (5) we need some further estimates. Using estimates (13) and (14) in the moments' equation (7) with m = 1, and using the hypothesis of the theorem, we obtain 1 , after integration, for t ∈ [0, T ], Cg i . Then, due to the nonnegativity of (g i ), (p i ), (q i ) and (M nj i ), we can apply Gronwall's lemma to the above inequality to deduce that, for Now, consider any integer N 2. Again, by the nonnegativity referred to above, the bound (18) implies that, for all j, so that, by making j → ∞, we obtain, due to the uniform convergence M Then, taking N → ∞, we conclude that where we have used the monotone convergence theorem for the limit of the integral. This clearly proves assertions (8). Using the assumptions on (g i ), namely that for for all t ∈ [0, T ). 1 We recall the reader that for the sequences (M This proves the second assertion in (iii) in Definition 2.2. The first assertion of (iii) is a direct consequence of (15) and the uniform boundedness of x nj , j ∈ N. Since properties (i), (ii) are easily obtained for the coordinates y i , i = 2, 3, . . . , that is, for M i , i = 0, 1, 2, . . . , by the uniform convergence properties, what is still lacking are the corresponding ones for y 1 , that is, for x, and (4) in Definition 2.2.
Our next main goal is to prove that (x n j ) is a Cauchy sequence in the uniform convergence norm. From the integrated version of (5) and letting χ i,p be the indicator function of {0, 1, . . . , n p − 1}, we can write, for j > l ∈ N and t ∈ [0, T ], where, We will deal first with I 2 and I 3 . Given N ∈ N, and for j > l sufficiently large, we use the splitting i 0 = We start by proving that the parts of I 2 and I 3 that correspond to the sums with i N can be made arbitrarily small by choosing j and l sufficiently large. In order to achieve this we will now deduce a number of adequate estimates.
Using again (7) with 2 m < n j and with g i = 1, for all i ∈ N, we can write after integration (remembering that M Next, we pass to the limit n j → ∞ in (22). Write the integrand of the first integral in the form We can then conclude that For the second integral in the right-hand side of (22) we observe that, for every m > 0 and due to (18), the following holds uniformly in n j and t For every ε > 0 fix m such that the last right-hand side in (24) is less than 1 2 ε. By the uniform in t convergence of M nj i to M i we conclude that there exists a ν = ν( m) such that, for all t ∈ [0, T ] and n j > ν, Thus, (24) and (25) imply that for every ε > 0 there exists m and ν such that, for all t ∈ [0, T ] and n j > ν, we have and, due to the arbitrariness of m, we conclude that, as n j → ∞, Next we will deal with the convergence of the left-hand side of (22). Again let us consider a fixed m > m. Then, for large j so that n j > m, write Estimates (9) and (13) imply that so that, for each t ∈ [0, T ], the series ∞ i= m in the right-hand side of (27) converges, uniformly with respect to j ∈ N, which allows us to conclude that, Plugging (23), (26) and (28) We shall now use (29) to obtain an estimate that will be crucial to handle the limit of I 2 and I 3 in (21) as n j → ∞. For each m ∈ N and t 0 define where (g m ) is a positive sequence satisfying the assumptions of the theorem. We can write the product of (29) by g m in the following form: Now, using the assumption g i+1 − g i δ > 0, we have, for t ∈ [0, T ], and, by (19), we conclude that, for each t ∈  Using this in (31), we obtain, for all t ∈ [0, T ], Using (23) and (26) we conclude that and consequently, by the dominated convergence theorem, Hence, for every ε > 0 there exists m 0 , j 0 such that, for all m > m 0 and n j > j 0 the following hold Now, we consider the integrated version of the moments' equation (7) with m = N. Taking into consideration the hypothesis of the theorem, namely that k i (g i+1 −g i ) = O(g i ) K g g i for some constant K g independent of i, and recalling (9), we can write, where, we recall the reader, C 2 := y 0 + (α + r)T . Let The nonnegativity of M nj i , the inequality just obtained, and (32) imply that, for all t ∈ [0, T ], Applying Gronwall's inequality we get, with K 1 = K g C 2 and for all N sufficiently large, we finally conclude that, for sufficiently large fixed N and for all j j 0 , for all t ∈ [0, T ].
We have now enough estimates to deal with the expressions in (21). We consider the splitting i 0 = for some K 2 > 0, independent of l, j, t. Furthermore, since i = O(g i ), by (33) we have, for some K 3 > 0 independent of l and j, uniformly for t ∈ [0, T /2]. Combining the above statements, we conclude that there is j 1 j 0 , such that, for j l j 1 , and all t ∈ [0, T /2], But, by the same reasoning, there is K 4 > 0 independent of l, j, t such that, for j l j 1 , and all t ∈ [0, T ], for all t ∈ [0, T /2], so that, by Gronwall's inequality, there is C > 0, such that sup Due to the arbitrariness of ε we conclude that (x nj ) is a Cauchy sequence in C([0, T /2]), and therefore x nj → x uniformly for t ∈ [0, T /2]. This also proves the continuity of x in [0, T /2].
We are now able to pass to the limit in the integrated version of the last equation of (5), that is, in where χ i,j is the indicator function of the set {0, 1, . . . , n j − 1}. Splitting again i 0 = N −1 i=0 + i N , using (33), and the bound on x nj , we observe that there is a constant K 5 > 0 such that, for j j 1 , for all t ∈ [0, T /2]. By a procedure already used above, writing in this estimate, i N = l−1 i=N + i l and making first j → ∞ and then l → ∞, using the uniform convergence of (M nj i ) and (x nj ), we conclude that for all t ∈ [0, T /2]. Thus, from (34) and (35), for j j 1 , for all t ∈ [0, T /2]. Now letting j → ∞ and using the same uniform convergence property we obtain, from the last inequality, and hence, by (36), for all t ∈ [0, T /2]. Since ε was chosen arbitrarily small we conclude that (4) in Definition 2.2 holds for all t ∈ [0, T /2].
We have showed that for any given T > 0 and for each initial condition y 0 ∈ X + there exists a solution y defined in [0, T /2]. Since the equations in system (2)-(4) are autonomous this allows us to extend y to [0, ∞).
Setting g i = i, for i = 1, 2, . . . , in the above theorem we obtain the following result:

The moments' equation and some a priori results
The next theorem states an a priori equality which is an integrated version for any solution of (1), of a result stated in Lemma 3.1 for the solutions of the truncated version of the silicosis system. Take any pair (t 1 , t 2 ), such that 0 t 1 < t 2 < T . Suppose that, and, furthermore, one of the following two sets of conditions, (A) or (B), holds: Then, for m 1, the following integrated version of (7) holds for the solution y: Proof. Take any n > m 1. From (3) we easily obtain We now pass to the limit n → ∞ in both sides. By the boundedness of x and hypothesis (37), we readily conclude that, as n → ∞, Next we claim that the second integral in the right-hand side of (39) vanishes in this limit: from (ii) and (iii) of Definition 2.2, we readily obtain t2 t1 x(s)k n M n (s) ds → 0, as n → ∞ ; (41) on the other hand, by setting g i = 1 for all i = 0, 1, 2, . . . in (39), we obtain, x(s)k n M n (s) ds , so that, from (41) and again the properties (ii) and (iii) of Definition 2.2, by letting n → ∞, we obtain, Let us now consider hypothesis (A). This implies that, for p = 1, 2, for some constant C > 0, and thus Replacing m by n + 1 in (42), multiplying both members by |g n+1 |, letting n → ∞, and using (iii) of Definition 2.2 together with (43), we conclude that, t2 t1 g n+1 x(s)k n M n (s) ds → 0 , as n → ∞ , thus proving our claim under the first condition in (A). Using the second condition in (A) together with Definition 2.2, (40), (44), and the bounded convergence theorem, we can pass to the limit n → ∞ in (39) and prove (38).
Now we prove the claim under the hypothesis (B). This implies that, for some constant C > 0, and thus, under these conditions, also (43) and (44) hold true. Therefore, using hypothesis (B), (40), and (44) we conclude that all terms in (39) but the integral in the left-hand side converge as n → ∞. But then, using the monotone convergence theorem we conclude that in (38) we also have, as n → ∞, thus completing the proof.
Then, for all t ∈ [0, T ), the following integrated version of (6) holds for that solution: Also, for every m 1, Proof. Equations (46) and (47) are obtained by setting g i = 1 and g i = i, resp., in the previous theorem. The first choice trivially satisfies all conditions of the theorem. The second, if we recall Definition 2.2, satisfies all statements corresponding to the hypothesis (B). Adding the equation for x(t), (4), to the equation (47) with m = 1, we obtain, On the other hand, adding the equation for M 0 (t), (2), to equation (46) with m = 1, we obtain, Finally, adding together (48) and (49) we obtain (45) thus completing the proof.
The previous results allow us to directly deduce the following conclusion on the solution regularity:  (i + 1)M i (t), for all t ∈ [0, T ). Each φ n is a continuous function on [0, T ). Take now any T ∈ (0, T ). By (45) we know that, for all n ∈ N and t ∈ [0, T ], φ n (t) U(y(t)) U(y(0)) + (r + α)T .
But, for each t ∈ [0, T ), the real sequence (φ n (t)) is increasing, and thus, by Dini's theorem, the series To prove the continuity of [0, T ) t → y(t) ∈ X, we fix any t 0 ∈ [0, T ). Let T ∈ (t 0 , T ) so that the series But then, by continuity of the real valued functions x and M i , i = 0, 1, 2, . . . , From this and the arbitrariness of ε we conclude that lim t→t0 y(t)−y(t 0 ) = 0 .
Only from the definition of solution and without further assumptions we know, by conditions (iii) in Definition 2.2, that . t in (0, T ). This means, in particular, that if y(0) ∈ X + but the previous series is divergent in t = 0 then, generically, for t > 0, it is convergent and thus the sequences (y n (t)) n∈N will decay faster than y(0) as n → ∞. However, the next two propositions show that with additional hypothesis we can be more specific about this behaviour. Proof. Let the first set of conditions be fulfilled. Then, the sequence g i = iq i satisfies (37) and hypothesis (A) of Theorem 5.1, so that we consider the corresponding version of (38) with m = 1. Take any T ∈ (0, T ). Then, that expression shows that, for each t ∈ [0, T ), Q(t) − Q(0) can be written as a sum of integrals of Lebesgue integrable functions on [0, T ). Therefore, it is an absolutely continuous function of t on [0, T ). The same applies to P (·) if we consider the more strict hypothesis on the p i coefficients.
The previous Proposition imposes too stringent restrictions to the sequences (p i ) and (q i ) that might not be satisfied in some applications. Having this in mind we can still obtain the following result without those restrictive hypothesis at the expense of not guaranteeing the absolute continuity of Q(·) down to t = 0.
Consider a solution y on some interval [0, T ). Then, the function Q(·) is absolutely continuous on each compact subinterval of (0, T ).
Proof. Take any pair (t 1 , t 2 ) such that, 0 < t 1 < t 2 < T , for some fixed T ∈ (0, T ), and ∞ i=1 iq i M i (t p ) is finite for p = 1, 2. We claim that g i = iq i satisfies (37) and hypothesis (B) of Theorem 5.1. Since ((i + 1)q i+1 − iq i )k i = O(iq i ) and by (ii) and (iii) of Definition 2.2, we can conclude that (37) is satisfied. On the other hand, the hypothesis also states that, for our choice of the sequence (g i ) i∈N0 , we have g i+1 g i , for sufficiently large i. This, together with the way t 1 and t 2 were chosen, proves our claim. This establishes the validity of (38) with m = 1 for this choice of t 1 , t 2 and g i = iq i , i = 0, 1, 2, . . . . But the first integral on the right-hand side of (38) is bounded by a constant depending only on T , and, by the hypothesis, for some C > 0 independent of the particular choice of t 1 , t 2 , we have Therefore, by (38), there is another positive constant C depending only of T such that, so that, by fixing t 0 for which Q(t 0 ) is finite, we have, by Gronwall inequality, for t a.e. in (0, T ). Thus, Q ∈ L ∞ (0, T ). Using this fact again in (38) together with our hypothesis, we conclude that Q is, in fact, in C((0, T )) and it is absolutely continuous on this interval. This completes the proof.

Differentiability
We state the first order differentiability properties of solutions for the silicosis system under two different sets of assumptions. In the first one, we can draw a conclusion slightly stronger than in the second one. Proposition 6.1. Let (k i ) i∈N0 be a nonnegative sequence such that k i = O(i). Let (q i ) i∈N0 be bounded. Let y be a solution of (1) in some interval [0, T ), 0 < T ∞.
Then, x ≡ y 1 and M i ≡ y i+1 , i ∈ N 0 , are C 1 functions on [0, T ) and satisfy the equations (1) in the classical sense that is, in their differential form, for all t ∈ [0, T ). Proposition 6.2. Let (k i ) i∈N0 and (q i ) i∈N0 satisfy the hypothesis of Proposition 5.5 and k i = O(i). Let y be a solution of (1) in some interval [0, T ), 0 < T ∞. Then, x ≡ y 1 and M i ≡ y i+1 , i ∈ N 0 , are C 1 on (0, T ) and they satisfy equations (1) in the classical sense that is, in their differential form, for all t ∈ (0, T ).
Proof. This is just a consequence of the same arguments used in the proof of the previous proposition together with Proposition 5.5. Corollary 6.3. Let (k i ) i∈N0 , (p i ) i∈N0 , and (q i ) i∈N0 be nonnegative sequences such that, k i = O(i), and, for sufficiently large i, q i = µi η , for some µ, η > 0. Then, the same conclusions in Proposition 6.2 hold.
Proof. From the fact that, as i → ∞, and from Proposition 6.2, the result immediately follows.

Uniqueness
Theorem 7.1. Let (k i ) i∈N0 and (g i ) i∈N0 be as in the hypothesis of the existence Proof. Let y be a solution as stated above, proved to exist in Theorem 4.1, and letỹ be another solution withỹ(0) = y 0 . Define z(t) = y(t) −ỹ(t). Fix any finite T > 0. By Definition 2.2 all the coordinates of y,ỹ and, therefore, of z are absolutely continuous functions on [0, T ]. Hence, t → |z(t)| has the same property and furthermore, for t ∈ [0, T ] a.e., d dt |z(t)| = (sgn z(t)) dz dt (t).
We then know that the differential versions of equations (2) and (3)  Having in mind that y 1 = x and y i = M i−2 , i = 2, 3, . . . , and the same forỹ, rewriting each one of those equations using y i ,ỹ i notation, we can obtain, for a.e. t ∈ [0, T ], d dt Now for the first sum in the right-hand side of the last equation we observe that, for i = 2, 3, . . . , and using this in (50) we obtain, after integrating in [0, t], for every t ∈ [0, T ], Since, by hypothesis, ik i = O(g i ), and y satisfies (8), we know that By (45), and using the hypothesis on (p i ) and (q i ), we obtain, for all t ∈ [0, T ], for some positive constant C 0 . Therefore, for t ∈ [0, T ], for some positive (T -dependent) constant, C 1 .
For the last term in (51), we use (47) to prove that it converges to zero as N → ∞. In fact, from the definition of solution and the bounded convergence theorem we have that Proceeding similarly with respect toỹ, and since Using in (51) the boundedness ofỹ and the hypothesis on k i , together with (52), (54), and (55), we deduce that there exits some constant C > 0 such that, for each and thus, since by the definition of z, z i (0) = 0, we conclude that for all t ∈ [0, T ]. Hence, for i = 2, 3, . . . , z i (t) = 0, which means that y i (t) =ỹ i (t). But then, by (53) we also have z 1 (t) = 0, which means that y 1 (t) =ỹ 1 (t). Since T > 0 can be chosen arbitrarily large, the proof is complete.
Observe that the uniqueness theorem in [2] (Theorem 3.6) for the Becker-Döring equations, which corresponds to the uniqueness result, Theorem 7.1, in our present setting, does not require any type of restrictions on the fragmentation coefficients b r , in contrast with our restrictions on the coefficients p i and q i . The reason for this is that, to estimate y 1 in terms of y i , i 2, that is, to obtain (53), we used the mass balance equation (45) thus requiring some hypothesis on the coefficients p i , q i , while in [2] to estimate c 1 in terms of the other coordinates c r , the authors were able to use the conservation of mass property which do not required restrictions on the growth rate of the b r coefficients. Corollary 7.3. Let (k i ), (p i ), (q i ) be nonnegative sequences such that k i = O(i), q i = O(i), and (p i ) is bounded. Then, for each y 0 = (y 0i ) i∈N ∈ X + such that there is a unique local solution y of the silicosis system (1) satisfying y(0) = y 0 and, moreover, it is extendable to [0, ∞). If (k i ) is also bounded, then, the same is true for solutions with initial conditions with finite mass, that is, if The following result is also a consequence of the uniqueness theorem: Corollary 7.4. Let y 0 ∈ X + and suppose that the hypothesis of Theorem 7.1 are satisfied. Let y be the unique solution of the silicosis system (1) such that y(0) = y 0 . If (y n ) n∈N is the sequence of solutions of the truncated systems (5) considered in the proof of Theorem 4.1, then, as n → ∞, y n (t) → y(t) uniformly on compact intervals of [0, ∞).
Proof. In order to reach a contradiction, suppose that there is a subsequence of (y n ), (y nj ) say, for which there is T > 0 and η > 0, such that, for all j ∈ N, sup t∈[0,T ] |y nj (t) − y(t)| η. We can take this subsequence in place of the whole approximating sequence in the proof of Theorem 4.1 and conclude by that proof and the uniqueness theorem, that there is another subsequence (y n j ) such that, for each i ∈ N, as j → ∞, y n j i (t) → y i (t), on [0, T ].

Semigroup property
The last result we shall consider is a property that, together with the existence and the uniqueness results, is crucial for the study of the dynamics of solutions: the semigroup property of the set of solutions. Denote by T (·)y 0 the unique solution y(· ; y 0 ) of (1) satisfying the initial condition y(0; y 0 ) = y 0 . Then, {T (t) : X + → X + | t 0} is a C 0 -semigroup, i.e., (i) T (0) = id, the identity operator; (ii) T (t + s) = T (t)T (s), for all t, s 0; (iii) (t, y 0 ) → T (t)y 0 is a continuous mapping from [0, ∞) × X + to X + .
Proof. Conditions (i) and (ii) are obvious from the fact that equations (2)-(4) are autonomous. The continuity of each solution T (·)y 0 : [0, ∞) → X + was concluded in Corollary 5.3. It remains to prove the continuity with respect to y 0 in (iii). Consider a sequence (y n 0 ) n∈N0 and assume that y n 0 → y 0 in X + as n → ∞, for some y 0 ∈ X + . By our present assumptions, for each y n 0 there is a unique globally defined solution y n (·) := T (·)y n 0 , and also a unique global solution y(·) := T (·)y 0 . We now need to prove that y n → y strongly in X + . To this end we can start by repeating the proof of the existence result, Theorem 4.1, with y n taking the place of the solutions of the n-truncated systems. Actually, the only difference between what we do now and what was done in the proof of Theorem 4.1 is that now one must take χ i,j ≡ 1 and, instead of using Lemma 3.1 for estimations of the moment of y n , we need to use Theorem 5.1 or Corollary 5.2. This allows us to conclude that, for the above functions y n = (x n , M n 0 , M n 1 , . . .), etc., we have, when n → ∞, y n * y in X + , and, for all t ∈ [0, ∞), the following is valid: Thus, since by assumption y n 0 → y 0 strongly in X + , and in X + we have U(·) = · , we can let n → ∞ in the equations (45) written for the functions y n , namely U(y n (t)) = U(y n 0 ) + (r + α)t − Being y(·) the solution of (1) with initial condition y 0 , (45) in Corollary 5.2 allows us to conclude that the right-hand side of the above equality is equal to U(y(t)), that is: lim n→∞ U(y n (t)) = U(y(t)), ∀t 0. Now, by Lemma 3.3 in [2], this, together with the weak * convergence above, implies that T (·)y n 0 → T (·)y 0 in X + , as we wanted to conclude.