Inmate population models with nonhomogeneous sentence lengths and their effects in an epidemiological model
aa r X i v : . [ q - b i o . P E ] J a n Inmate population models with nonhomogeneous sentencelengths and their effects in an epidemiological model
P. Gajardo and V. Riquelme Departamento de Matemática, Universidad Técnica Federico Santa María,Avenida España 1680, Valparaíso, Chile [email protected], [email protected]
February 1, 2021
Abstract
In this work, we develop an inmate population model with a sentencing length structure.The sentence length structure of new inmates represents the problem data and can usually beestimated from the histograms corresponding to the conviction times that are sentenced in agiven population. We obtain a transport equation, typically known as the McKendrick equa-tion, the homogenous version of which is included in population models with age structures.Using this equation, we compute the inmate population and entry/exit rates in equilibrium,which are the values to consider in the design of a penitentiary system. With data from theChilean penitentiary system, we illustrate how to perform these computations. In classifyingthe inmate population into two groups of sentence lengths (short and long), we incorporatethe SIS (susceptible-infected-susceptible) epidemiological model, which considers the entry ofinfective individuals. We show that a failure to consider the structure of the sentence lengths—as is common in epidemiological models developed for inmate populations—for prevalences ofnew inmates below a certain threshold induces an underestimation of the prevalence in theprison population at steady state. The threshold depends on the basic reproduction numberassociated with the nonstructured SIS model with no entry of new inmates. We illustrateour findings with analytical and numerical examples for different distributions of sentencinglengths.
Keywords: Prison population dynamics, sentencing length structure, McKendrick equation,SIS epidemiological model, health in prisons
Currently, several communicable diseases, such as sexually transmitted infections (STIs), remain apublic health problem that is far from being controlled [20]. Generally, prisons present far higherprevalences of certain diseases than the general population. This is due, among many factors, tothe existence of crowded environments, high-risk behaviors such as unprotected sexual relations[17], and the increase in the probability of the appearance of disease risk factors, such as depressionand drug use, during imprisonment. Another relevant factor in penitentiary systems is related tothe deficiencies of prison health systems, which imply the existence of barriers to access to care,delayed diagnoses and prolonged contagion times [4, 15, 16, 17]. This health problem is not onlya penitentiary concern but also a general social issue because prisons act as reservoirs for diseases,which are later transmitted to the community when inmates are released or come into contact withthe outside population, such as visitors and prison workers. For the aforementioned reasons, theWorld Health Organization (WHO) includes prisoners among the key populations that should bethe focus of interventions designed to reduce the burden of diseases [14, 20].Mathematical models of communicable diseases in penitentiary systems can be found in therecent literature [3, 7, 9, 19, 21]. The main objective of these models is to develop and assesscontrol strategies regarding the spread of these diseases. The main characteristics of these modelsand two of the difficulties in conducting a theoretical analysis are the entry or immigration ofinfected individuals and a variable population size. These features are also present in age-structuredepidemiological models with vertical transmission, but as indicated in [7], vertical transmission1odels generally include a flow of new infectives proportional to the number of infectives alreadyin the population and thus may have a disease-free equilibrium [8]. To the best of our knowledge,the first theoretical analysis for this class of models was conducted in [7], where the SIS (susceptible-infected-susceptible) and SIR (susceptible-infected-recovered) models are studied.These previous studies ignore the sentence length structure of inmates and assume the entry(and exit) rate of new inmates to be the inverse of the average sentence length (or residence timein prisons). However, it seems appropriate to consider the sentence length structure of inmates inepidemiological models, since individuals with a longer sentence length are exposed to the diseaseunder study for a longer time.Under the framework of discrete-time models, we find epidemiological models developed forprison populations that consider the sentence length structure. In [2], an age-structured operationalresearch model is developed, where the sentence length structure is included for studying hepatitisC treatment strategies in U.S. prisons. In [13], a model (also in discrete time) is introduced for thespread and control of HIV in prison populations. In this work, the authors consider two classes ofsentencing lengths (short and long) and conclude, from numerical simulations, that the epidemicis not very sensitive to the length of sentences in the range of parameters used in their simulations.In this paper, we first introduce an inmate population model with a sentencing length structure.We obtain a transport equation in the form of a partial differential equation, typically known asthe McKendrick equation, the homogenous version of which is included in population modelswith an age structure [6]. This equation allows us to compute the inmate population and theentry/exit rates at steady state, which are values to consider in the design of a penitentiarysystem. Our main objective is not to solve or analyze the obtained McKendrick equation, whichcan be achieved using the method of characteristics. Instead of coupling the McKendrick equationwith an epidemiological model (which for homogeneous equations —obtained from age-structuredpopulation models— is analyzed in [1, 6, 18]), we divide the inmate population into two classesdepending on their sentencing lengths (short and long), and in the obtained model, we incorporatethe SIS epidemiological model. This epidemiological model is compared, in equilibrium, with themodel obtained when ignoring the sentencing length structure. We prove that failing to consider thestructure of the sentence lengths for prevalences of new inmates below a certain threshold inducesan underestimation of the prevalence in the prison population at steady state. The thresholddepends on the basic reproduction number associated with the nonstructured SIS model with noentry of new inmates.The remainder of the paper is organized as follows: In Section 2, we develop a model for aninmate population that considers an nonhomogeneous distribution of the sentence lengths of newinmates and the removal rate from the prison. We perform the analysis at steady state for themodel, and we obtain expressions for the population distribution (with respect to the remainingsentence time) and the population size with respect to the data of the problem. Using real datafrom a prison in Chile, in Section 2.4 we illustrate how to obtain these key values. In Section 3, westudy the effect of considering one or two classes of initial sentencing lengths in the estimation of theproportion of infected people in an epidemiological population model in equilibrium. We provideconditions under which the homogeneous model (single class without a sentencing length structure)underestimates the number of infective individuals relative to the two-class model. Finally, inSection 3.2, we illustrate our findings with two numerical examples for different distributions ofinitial sentencing lengths and then conclude with some final remarks in Section 4. In this section, we develop a population model for a prison population consisting of a continuous-state, continuous-time model for the number of inmates inside the prison. Suppose that at time t ,prisoners enter the prison at a rate λ t ≥ (the number of people per unit of time) correspondingto inmates newly sentenced, or transferred from other institutions, with a distribution of initialsentence length q t ( dr ) that, for each t ≥ , is a nonnegative measure on B ( C ) , the Borel σ − algebraon the set of sentence times C = (0 , ∞ ) , with total mass 1 (that is, a probability measure).Consider the existence of a removal rate of prisoners d t ≥ corresponding to transfer of inmates We assume that the prevalence in new inmates is the steady-state prevalence of the population outside theprison.
2o other institutions, pardon or commutation of sentences, death, etc., in units of time − , whichis independent of their respective remaining sentence time in prison.Throughout this work, we make the following assumption regarding the distribution of initialsentence lengths q t ( · ) : Assumption 1.
For all t ≥ , the distribution q t ( · ) is a mixture measure of a continuous distri-bution and a discrete distribution with finite support. That is, it can be decomposed as q t ( dr ) = q a,t ( dr ) + X r jt ∈D t p jt δ r jt ( dr ) , where, for all t ≥ , one has the following: • q a,t ( · ) is a nonnegative measure, absolutely continuous with respect to the Lebesgue measureon R ; that is, there exists a density function q ′ a,t ( · ) that is Lebesgue measurable, such that q a,t ( I ) = R I q ′ a,t ( r ) dr for all I ∈ B ( C ) . • D t ⊆ C is a finite (possibly empty) set. • For all r jt ∈ D t , δ r jt ( dr ) denotes the Dirac measure concentrated on { r jt } . • For all r jt ∈ D t , p jt > .Moreover, we assume that for all t ≥ , the mean value of the distribution q t ( · ) , denoted by T q t = R (0 , ∞ ) rq t ( dr ) , is finite. Under Assumption 1, q t ( · ) has a generalized density function q ′ t ( r ) , in the sense that we canwrite q ′ t ( r ) = q ′ a,t ( r ) + X r jt ∈D t p jt δ ′ r jt ( r ) , where the Dirac generalized function δ ′ x ( r ) can be regarded as a density for the Dirac measure δ x ( dr ) , in the sense that δ x ( I ) = Z I δ ′ x ( r ) dr = ( , x ∈ I, , x / ∈ I. In what follows, we do not make distinctions between the absolutely continuous and discreteparts of q t ( · ) under the sign of the integral, meaning the following: for every Borel-measurablefunction f : C → R , Z C f ( r ) q t ( dr ) = Z C f ( r ) q ′ a,t ( r ) dr + X r jt ∈D t f ( r jt ) p jt . Remark 1. If ν ( · ) : B ( R + ) → R + is a nonnegative Borel measure concentrated on the set R + and f : R + → R is a Borel-measurable function, we have Z R + f ( x ) ν ( dx ) = Z ∞ ν ( { x ∈ R + | f ( x ) > α } ) dα, according to [5, Theorem 2.9.3]. Since for all t ≥ , q t ( · ) is concentrated on the positive real line,its mean value T q t := R ∞ rq t ( dr ) can be computed as T q t = Z ∞ q t (( r, ∞ )) dr. (1)Given a set I ⊆ C , let us define N It as the number of prisoners at time t , whose remaining timesin prison belong to the set I . Let us suppose that, for each t , there exists a population density ρ rt (the number of people per unit of time), with respect to the remaining time in prison r , such that N It = Z I ρ rt dr, ∀ I ∈ B ( C ) . Define E It and D It as the rates of prisoners (number of prisoners per unit of time) that enter theprison and are removed from prison, respectively, at time t , whose remaining sentence times belongto I ∈ B ( C ) .The following proposition characterizes the population density ρ rt as a function of the rates λ t and d t and the distribution q t ( · ) . 3 roposition 1. Suppose that the functions t λ t and t d t are continuous and that the familyof measures ( q t ( · )) t ≥ satisfies the following continuity hypothesis: ∀ t ≥ , ∀ A ∈ B ( R + ) , ∀ ε > , ∃ η ∈ (0 ,
1) : | z | , | z ′ | < η ⇒ | q t + z ( A + z ′ ) − q t ( A + z ′ ) | < ε. (2) Let us consider ≤ s < r . Then, the number of prisoners whose remaining sentence times belongto the interval ( s, r ] satisfies the equation ddt N ( s,r ] t = ρ rt − ρ st + λ t q t (( s, r ]) − d t N ( s,r ] t , a.e. t ≥ . (3) Therefore, under Assumption 1, the density of the prison population ρ rt satisfies the McKendricktransport equation ∂ρ rt ∂t + d t ρ rt = ∂ρ rt ∂r + λ t q ′ t ( r ) , a.e. t ≥ , r ≥ , (4) from an initial distribution ( ρ r ) r ≥ .Proof. We know that E It = Z I λ t q t ( dr ) = λ t q t ( I ) , D It = d t Z I ρ rt dr = d t N It . Consider an interval I = ( s, r ] of remaining sentence times and a time step ∆ t . Mass balanceanalysis yields N ( s,r ] t +∆ t = N ( s +∆ t,r +∆ t ] t + Z ∆ t E ( s +∆ t − τ,r +∆ t − τ ] t + τ dτ − Z ∆ t D ( s +∆ t − τ,r +∆ t − τ ] t + τ dτ. (5)We note that for I , I ∈ B ( C ) disjoint sets, N I t + N I t = N I ∪ I t . Thus, N ( s +∆ t,r +∆ t ] t = N ( s,r ] t + N ( r,r +∆ t ] t − N ( s,s +∆ t ] t . Then, (5) becomes the mass balance equation on the interval ( s, r ] : N ( s,r ] t +∆ t − N ( s,r ] t = N ( r,r +∆ t ] t − N ( s,s +∆ t ] t + Z ∆ t λ t + τ q t + τ (( s + ∆ t − τ, r + ∆ t − τ ]) dτ − Z ∆ t d t + τ N ( s +∆ t − τ,r +∆ t − τ ] t + τ dτ. (6)Note that for all t ≥ fixed, ϕ t ( · ) = λ t q t ( · ) and φ t ( · ) = d t N · t define positive measures that, underthe hypotheses of the proposition, satisfy hypothesis (2). Thus, by Proposition 8 in the Appendix,dividing (6) by ∆ t and taking limits as ∆ t ց , we obtain balance equation (3).Now, under Assumption 1, q t ( dr ) has a density q ′ t ( r ) , that is, for all I ∈ B ( C ) , q ( I ) = R I q ′ ( r ) dr .Since N ( s,r ] t = R ( s,r ] ρ αt dα , (3) can be rewritten as Z ( s,r ] (cid:20) ∂ρ αt ∂t + d t ρ αt (cid:21) dα = ρ rt − ρ st + λ t Z ( s,r ] q ′ t ( α ) dα, ≤ s < r. (7)Dividing (7) by r − s and taking limits as s ր r , by [5, Theorem 5.4.2], we obtain the transportequation (4). (cid:4) Remark 2.
The McKendrick equation appears naturally in population models with an age structure[6]. Our prison population model is in some sense an age-structured model where, instead ofaccounting for age, we consider the remaining life of an individual, given a distribution of lifespan. One of the differences between equation (4) and that arising in age-structured populationmodels is the nonhomogeneous term λ t q ′ t ( r ) , which appears because the entry of new inmates is notproportional to the current population, contrary to typical age-structured models. .1.1 Steady-state analysis In this section, we study the behavior of the system in the long term, assuming that λ, d, q ( · ) does not depend on t . Regarding the solution of (4) under this assumption, we have the followingproposition: Proposition 2.
Suppose that λ, d, q ( · ) does not depend on t and that Assumption 1 holds with T q < ∞ being the mean value of q ( · ) and D being the support of the discrete part of q ( · ) . If theinitial density ( ρ r ) r ≥ defines a finite measure consisting of a mixture of a continuous measure anda discrete measure with finite support, then:1. For all t , the solution ( ρ rt ) r ≥ of equation (4) has finite mass. Moreover, for all t ≥ , ρ rt converges to 0 as r tends to infinity in a set whose complement has a null Lebesgue measure.2. The population density satisfies lim t →∞ ρ rt = λ Z ( r, ∞ ) e − d ( α − r ) q ( dα ) , a.e. r ≥ . (8) Moreover, the previous formula holds for all but finitely many r ≥ .3. For all I ∈ B ( R + ) , we have lim t →∞ N It = λ Z I Z ( r, ∞ ) e − d ( α − r ) q ( dα ) dr. (9) Proof.
Equation (4) can be explicitly solved via the method of characteristics. Integration alongthe characteristics leads to the possible solutions ρ r − t := ρ r + t e − dt + λ Z [ r,r + t ] e − d ( α − r ) q ( dα ) ,ρ r + t := ρ r + t e − dt + λ Z ( r,r + t ] e − d ( α − r ) q ( dα ) . (10)Note that the set of points r ∈ C for which ρ r − t = ρ r + t is at most finite. Indeed, it coincides with D . Thus, the solution ρ rt of (4) satisfies, for all t ≥ , ρ rt = ρ r − t = ρ r + t , a.e. r > (indeed, for all r / ∈ D ). Thus, ρ rt ≤ ρ r − t ≤ ρ r + t e − dt + λ q ( { r } ) + Z ( r, ∞ ) e − d ( α − r ) q ( dα ) ! ≤ ρ r + t e − dt + λq ( { r } ) + λq (( r, ∞ )) , (11)where q ( { r } ) > only for r ∈ D , which is a set with a null Lebesgue measure. The right-hand-sideexpression in (11) is Lebesgue-integrable with respect to r .1. Integrating (11) with respect to r , on the set (0 , ∞ ) , we obtain N t ≤ N ( t, ∞ )0 e − dt + λ Z ∞ q (( r, ∞ )) dr. According to Remark 1, T q = R ∞ q (( r, ∞ )) dr , which is finite by hypothesis. This, combinedwith the fact that N ( t, ∞ )0 ≤ N < ∞ , proves the finiteness of the total mass N t , t ≥ .Since N t = R ∞ ρ rt dr is finite, then r ρ rt is an integrable function. Thus, ρ rt converges to 0as r ր ∞ in a set whose complement has a null Lebesgue measure.2. From (10), we note that for t large enough, the points with an initial positive measure, withrespect to ( ρ α ) α ≥ , are no longer involved in the expressions for ρ r − t and ρ r + t . This, alongwith the fact that the support of the discrete part of q ( · ) is finite, implies that for t largeenough, ρ r + t = ρ r − t except at a finite set, which implies that r ρ rt is a function continuousby parts, with at most finitely many discontinuities.Formula (8) is directly obtained from (10).5. From (8), we have the r − a.e. pointwise convergence lim t →∞ ρ rt I ( r ) = λ Z ( r, ∞ ) e − d ( α − r ) q ( dα ) I ( r ) , (12)where I ( r ) denotes the indicator function of the set I , which means that I ( r ) = 1 if r ∈ I ,and I ( r ) = 0 otherwise. Moreover, from (11), we obtain for t large enough (fixed), thebound ρ rt I ( r ) ≤ (cid:0) ρ r + t + λq (( r, ∞ )) (cid:1) I ( r ) , which is an integrable function. Integrating (12), by the Lebesgue dominated convergencetheorem [5, Theorem 2.8.1], we conclude that (9). (cid:4) Remark 3.
For the time-dependent case, a similar result to point 1. of Proposition 2 can beproved, if the initial density ( ρ r ) r ≥ satisfies the hypotheses of Proposition 2, ( λ t ) t ≥ is bounded,and the family ( q t ( · )) t ≥ is uniformly tight [10] (or does not depend on t ), with every q t ( · ) satisfyingAssumption 1, under the hypotheses of Proposition 1. Then, if in (3) we replace the particular value s = 0 and take limits as r ր ∞ , we obtain the ODE associated with the total prison populationwith positive remaining time N t := N (0 , ∞ ) t : ddt N t = λ t − ρ t − d t N t , a.e. t ≥ , (13) with ρ rt being the solution of (4) . The quantity ρ t corresponds to the exit rate of the prison (thenumber of people per unit of time), which is interpreted as the rate of inmates that have effectivelycompleted their sentences at instant t . Proposition 2 shows that the density ρ rt converges as t → ∞ for all but finitely many r ≥ . Assume now that the parameters λ, d, q ( · ) are time independent and that the system is in astationary regime. Denote the stationary density ρ r and the stationary population with remainingsentence time in the interval I ∈ B ( C ) by N I . The mass-balance equation (3) in equilibrium statesthat ρ r is the solution of the stationary equation ρ r − ρ s + λq (( s, r ]) = dN ( s,r ] , for all ≤ s < r. (14)Let us define N := N (0 , ∞ ) = R ∞ ρ α dα as the total number of prisoners in equilibrium. Inwhat follows, for a function f : [0 , ∞ ) → R , we denote its Laplace transform by L ( f )( ξ ) = R ∞ e − ξx f ( x ) dx , while for a signed measure ν : B ([0 , ∞ )) → R ∪ { + ∞} , we denote its Laplacetransform by ˆ ν ( ξ ) = R [0 , ∞ ) e − ξx ν ( dx ) .In the following proposition, we show that the limiting density (8) obtained in point 2 ofProposition 2 coincides with the unique integrable solution of (14). Proposition 3.
Suppose that Assumption 1 holds. Let ρ r be a solution of (14) with finite mass,that is, such that N = R ∞ ρ r dr < ∞ . Then, ρ r = λ Z ( r, ∞ ) e − d ( α − r ) q ( dα ) , a.e. r ≥ . (15) That is, ρ r corresponds to the limit of the solution ρ rt of (4) starting from an initial condition ( ρ r ) r ≥ with finite mass.Proof. Suppose that ρ r is a density with finite mass, solution of (14). Thus, r ρ r is an integrablefunction, and ρ r converges to 0 in a set whose complement has a null Lebesgue measure.Define Q ( r ) = q ((0 , r ]) , r ≥ . Evaluating s = 0 and applying the Laplace transform withrespect to the variable r in (14), with N (0 ,r ] = R (0 ,r ] ρ α dα , we obtain L ( ρ )( ξ ) = ρ ξ − d − λ L ( Q )( ξ ) − λd L ( Q )( ξ ) ξ − d = ρ L ( e dr )( ξ ) − λ L ( Q )( ξ ) − λd L ( Q )( ξ ) · L ( e dr )( ξ ) , ρ r = ρ e dr − λQ ( r ) − λdQ ∗ e dr ( r ) , (16)where the operator ∗ stands for convolution. By definition, ( Q ∗ e dr )( r ) = Z r Q ( α ) e d ( r − α ) dα = e dr Z r Q ( α ) e − dα dα. (17)Note that Q ( · ) is a right-continuous function with left limits at every point of the interval [0 , ∞ ) with at most a finite number of discontinuity points (under Assumption 1). Thus, Q ( · ) definesa Lebesgue-Stieltjes measure µ Q ( · ) , which coincides with q ( · ) on B ([0 , ∞ )) . Thus, integrating byparts [10, Proposition 5.3.3], for f continuously differentiable, Z [0 ,r ] f ( α ) µ Q ( dα ) = Q ( r ) f ( r ) − Q (0) f (0) − Z r Q ( α ) f ′ ( α ) dα. (18)Taking f ′ ( α ) = e − dα in (18) and noting that Q (0) = q ((0 , , we obtain Z r Q ( α ) e − dα dα = 1 d − Q ( r ) e − dr + Z [0 ,r ] e − dα q ( dα ) ! . (19)Thus, combining (17), (19) and (16), we obtain ρ r = ρ e dr − λe dr Z [0 ,r ] e − dα q ( dα ) . (20)Dividing by e dr and taking limits as r ր ∞ in (20), we obtain ρ = λ Z ∞ e − dα q ( dα ) = λ ˆ q ( d ) . (21)Replacing (21) in (20), we obtain (15). (cid:4) Let us define the relative entry and exit rates µ in and µ out as the entry and exit rates relativeto the total prison population in equilibrium N = R ∞ ρ α dα : µ in := λN , µ out := ρ N , (22)with units of time − . Recall that, under Assumption 1, the mean value of the probability measure q ( · ) , denoted by T q , is finite. We refer to T q as the mean initial sentence length.In the remainder of this section, we obtain explicit formulas for the quantities of interest atsteady state. Proposition 3 allows us to characterize these quantities of interest in terms of thestationary density ρ r from the stationary equation (14). Proposition 4.
1. The relative entry and exit rates µ in , µ out , defined in (22) , are linked by µ out = µ in − d. (23)
2. Suppose that d > . The total number of prisoners in equilibrium is N = λ − ˆ q ( d ) d . (24)
3. Suppose that d > . The relative rates µ in , µ out have the explicit formulas µ in = d − ˆ q ( d ) , µ out = d ˆ q ( d )1 − ˆ q ( d ) = µ in ˆ q ( d ) . (25)7 roof. In (14), evaluating s = 0 and taking limits as r ր ∞ , since lim r →∞ ρ r = 0 , we obtain ρ = λq ((0 , ∞ )) − dN (0 , ∞ ) = λ − dN. (26)Dividing (26) by N , for the relative rates µ in , µ out defined in (22), we obtain (23). Now,combining (26) and (21), we obtain (24). Replacing (24) and (21) in the definitions of µ in and µ out given in (22), we obtain (25). (cid:4) Remark 4.
The total population at equilibrium given by (24) is a key value for the design of apenitentiary system, which involves all the data of the problem: the entry rate, removal rate fromthe prison and the initial sentence length distribution.
Let us define R ρ as the mean remaining sentence time inside the prison. This term can becomputed from ρ r as R ρ = 1 N Z ∞ rρ r dr. Proposition 5. If d > , the mean remaining sentence time inside the prison R ρ has the expression R ρ = T q − ˆ q ( d ) − d . (27) Proof.
Swapping r and s and taking limits as s ր ∞ in (14), we obtain ρ r = λq (( r, ∞ )) − dN ( r, ∞ ) . (28)If we integrate (28), Z ∞ ρ r dr = λ Z ∞ q (( r, ∞ )) dr − d Z ∞ N ( r, ∞ ) dr. (29)Note that ˜ ρ r = ρ r /N is a probability density function on the set C corresponding to the densityof the remaining time in prison. Thus, following Remark 1, the mean sentence time inside prison R ρ can be computed as R ρ = Z ∞ Z ( r, ∞ ) ˜ ρ α dα ! dr = 1 N Z ∞ Z ( r, ∞ ) ρ α dα ! dr = 1 N Z ∞ N ( r, ∞ ) dr. (30)Thus, from (1) and (30), (29) translates into N = λT q − dN R ρ . (31)Since d > , replacing N from (24) and isolating R ρ in (31), we obtain (27). (cid:4) To obtain the values deduced in Proposition 4 when there is no removal from the prison (i.e., d = 0 ), let us define the following functions: d N = ( λ − ˆ q ( d ) d , d > ,λT q , d = 0 , d µ in = ( d − ˆ q ( d ) , d > , T q , d = 0 ,d µ out = ( d ˆ q ( d )1 − ˆ q ( d ) , d > , T q , d = 0 . (32) Proposition 6.
The functions d N , d µ in , and d µ out defined in (32) are right-continuousat d = 0 .Proof. When taking d = 0 , from (31), we directly obtain N = λT q . From this equation and (26),we directly obtain ρ = λ = N/T q , and dividing by N , we obtain µ in = µ out = 1 /T q .To prove the continuity of N , µ in and µ out at d = 0 , we note that ˆ q (0) = R ∞ q ( dr ) = q ((0 , ∞ )) =1 . Thus, lim d ց − ˆ q ( d ) d = − lim d ց ˆ q ( d ) − ˆ q (0) d − − ˆ q ′ (0) , ˆ q ′ ( d ) = Z ∞ ∂∂d e − dr q ( dr ) = Z ∞ − re − dr q ( dr ) , which implies ˆ q ′ (0) = R ∞ − rq ( dr ) = − T q . Then, lim d ց − ˆ q ( d ) d = − ˆ q ′ (0) = T q , Note that, taking limits as d ց in N, µ in , µ out in (32), N = λ − ˆ q ( d ) d → λT q , µ in = d − ˆ q ( d ) → T q , µ out = µ in ˆ q ( d ) → T q , which proves the result. (cid:4) Remark 5.
As a consequence of Proposition 6, we henceforth develop the results using the obtainedformulas for the case of d > , taking into consideration that the case of d = 0 is obtained as thelimiting result when d converges to 0. Remark 6.
Typically, in mathematical models of communicable diseases for penitentiary systems,the sentence length structure of inmates is ignored [3, 7, 9, 19, 21], and the entry (and exit) rateof new inmates is assumed to be the inverse of the average sentencing length. In other words, therelative rate of entry µ in , with respect to the total population, is taken as the inverse of the meantime of residence, which in this case corresponds to T q . In Propositions 4 and 6, we show that thisis true when the removal rate is null. If the removal rate is not null, µ in is no longer a functionof the mean time of residence| T q but depends on the complete distribution of sentence length q ( · ) via its Laplace transform. A high removal rate can be interpreted as a bad situation, for instance,if this rate consists mainly of deaths and transfers due to problems in the prison (overcrowding,riots), therefore, to assume that this rate is close to zero is almost to assume an ideal situation. Suppose that we want to classify the inmates according to the length of their initial sentencewhile not affecting the homogeneous mixing among all prisoners (which is important from theepidemiological perspective to be developed later). This means the existence of a threshold r ⋆ > ,such that every sentence in the set C := (0 , r ⋆ ] (resp. C := ( r ⋆ , ∞ ) ) is considered a short (resp.long) sentence length. Note that C = C ∪ C is a disjoint union. We say that a prisoner belongsto the class i , or his/her type is i , if his/her initial sentence length belongs to C i ( i = 1 , ).To apply the results of the previous section, we note that each class has its own entry rate andits own distribution of initial sentence length. Indeed, let us define p it := q t ( C i ) , i = 1 , , (33)the proportions of individuals entering the prison with a short ( i = 1 ) or long ( i = 2 ) initialsentence length. Thus, the entry rate of prisoners to class i is λ it := λ t q t ( C i ) = λ t p it , and thedistribution of initial sentence lengths of class i corresponds to the distribution of initial sentencelengths, conditional on the inmate entering said class: q it ( I ) := q t ( I ∩ C i ) q t ( C i ) = q t ( I ∩ C i ) p it , I ∈ B ( C ) . (34)Suppose that the removal rate for each class is d it ≥ for all t ≥ . In this setting, from (13),the total number of prisoners in each class N t , N t follows the equations ddt N t = λ t − ρ , t − d t N t , ddt N t = λ t − ρ , t − d t N t , a.e. t ≥ , (35)where ρ , t and ρ , t are the exit rates of classes 1 and 2, respectively, and solve the McKendrickequation (4), that is, ∂ρ r, t ∂t + d t ρ r, t = ∂ρ r, t ∂r + λ t q ′ t ( r ) , ∂ρ r, t ∂t + d t ρ r, t = ∂ρ r, t ∂r + λ t q ′ t ( r ) , (36)where q i ′ t ( · ) denotes the density of the distribution q it ( · ) at time t , i = 1 , .9 emma 1. Suppose that d t = d t =: d t ≥ for all t ≥ . Then, the density of the total population ρ rt satisfies ∂ρ rt ∂t + d t ρ rt = ∂ρ rt ∂r + λ t q ′ t ( r ) . (37) Consequently, the total population N t inside the prison satisfies ddt N t = λ t − ρ t − d t N t . (38) Proof.
Adding the two equations in (36), we obtain ∂∂t ( ρ r, t + ρ r, t ) + d t ( ρ r, t + ρ r, t ) = ∂∂r ( ρ r, t + ρ r, t ) + λ t ( p t q ′ t ( r ) + p t q ′ t ( r )) . Note that, from (33) and (34), p t q ′ t ( r ) + p t q ′ t ( r ) = p t q ′ t ( r ) p t C + p t q ′ t ( r ) p t C = q ′ t ( r ) . (39)Since ρ rt = ρ r, t + ρ r, t is the density of the total prison population, we conclude (37).Now, adding the two equations in (35), we obtain ddt ( N t + N t ) = λ t ( p t + p t ) − ( ρ , t + ρ , t ) − d t ( N t + N t ) . Since N = N + N is the total prison population and in the previous computations we obtainedthat ρ = ρ , + ρ , , we conclude (38). (cid:4) Let us define π it := N it /N t as the proportions of inmates of each class i = 1 , , relative tothe total prison population at each time t . Define also the entry/exit rates of each class and theentry/exit rates of the whole prison relative to the class size: µ in t := λ t N t , µ out t := ρ t N t , µ in ,it := λ it N it , µ out ,it := ρ ,it N it , i = 1 , . (40) Remark 7.
Suppose that d t = d t for all t ≥ . Then, we have the following relations between theentry/exit rates of each class and the entry/exit rates of the whole prison: µ in , t π t + µ in , t π t = λ t N t N t N t + λ t N t N t N t = λ t + λ t N t = λ t N t = µ in t ,µ out , t π t + µ out , t π t = ρ , t N t N t N t + ρ , t N t N t N t = ρ , t + ρ , t N t = ρ t N t = µ out t . (41) We are now interested in obtaining explicit expressions and relations between the sizes of thedifferent classes in equilibrium. Suppose that the entry rate λ , the distribution q ( · ) , and theremoval rate d do not depend on t and that the system operates at steady state. In this setting,(24) applies to each class separately. From (24), the total number of prisoners in class i is N i = λ i d i (1 − ˆ q i ( d i )) , i = 1 , , (42)where ˆ q i ( · ) denotes the Laplace transform of the distribution of initial sentence lengths with respectto each class q i ( · ) . If we suppose that the removal rates are equal for the two classes, namely, d = d =: d > , from Lemma 1, we have N = N + N , where, according to (24), N = λd (1 − ˆ q ( d )) . (43)10hen, the proportion of prisoners of each class in equilibrium is π i = N i /N , where, from (42)and (43), we have π i = p i − ˆ q i ( d )1 − ˆ q ( d ) , i = 1 , , (44)where p i = q ( C i ) , as in (33). Additionally, the entry rates µ in ,i = λ i /N i , µ in = λ/N , relative tothe stationary population sizes N i , N , (as defined in (40)) in equilibrium, are (from (25)) µ in = d − ˆ q ( d ) , µ in , = d − ˆ q ( d ) , µ in , = d − ˆ q ( d ) . (45)In a similar way, the exit rates µ out ,i = ρ ,i /N i , µ out = ρ /N , relative to the stationarypopulation sizes N i , N in equilibrium, are (from (25)) µ out = d ˆ q ( d )1 − ˆ q ( d ) , µ out , = d ˆ q ( d )1 − ˆ q ( d ) , µ out , = d ˆ q ( d )1 − ˆ q ( d ) . (46) Lemma 2.
Suppose that d = d = d > . Then, µ in = p µ in , + p µ in , . Proof.
From (45), following a similar calculation as in (39), p µ in , + p µ in , = p − ˆ q ( d ) d + p − ˆ q ( d ) d = 1 d (cid:0) ( p + p ) − ( p ˆ q ( d ) + p ˆ q ( d ) (cid:1) = 1 − ˆ q ( d ) d = 1 µ in . (cid:4) Corollary 1.
We have π ≤ p , π ≥ p , and µ in , ≤ µ in ≤ µ in , .Proof. These results are obtained from the relation ˆ q ( d ) ≤ ˆ q ( d ) ≤ ˆ q ( d ) . Indeed, ˆ q ( d ) − ˆ q ( d ) = Z ∞ e − dr q ( dr ) − p Z ∞ r ⋆ e − dr q ( dr )= (cid:18) − p (cid:19) Z ∞ r ⋆ e − dr q ( dr ) + Z r ⋆ e − dr q ( dr ) ≥ (cid:18) − p (cid:19) e − dr ⋆ p + e − dr ⋆ p ≥ e − dr ⋆ ( p − p ) = 0 . The proof of the other inequality is analogous. Thus, the results follow. (cid:4)
Remark 8.
From Proposition 6, taking limits as d ց in (42) , (43) , the proportions of prisonersthat belong to each class, relative to the total number of prisoners, converge to π i := N i N → λ i T q i λT q = p i T q i T q , i = 1 , , with T q i being the mean initial sentence length relative to class i (that is, the mean of q i ( · ) ), whichcorresponds to the case without removal (see Proposition 6 and Remark 5). Additionally, the entryand exit rates relative to the stationary population sizes from (45) , (46) , converge to µ in , µ out → T q , µ in , , µ out , → T q , µ in , , µ out , → T q . Remark 9.
The previous procedure can be performed for a finite arbitrary number of classes,obtaining analogous results. Additionally, the classification in a prison population can be performedusing other criteria. For instance, one can classify individuals into dangerous and nondangerousinmates or, related to a communicable disease, a possible classification can be a high-risk population(superspreader individuals) and a low-risk population. .3 Examples Consider the constant entry rate λ t = λ > and death rate d t = d ≥ , and suppose that thedistribution q ( · ) is discrete, with only two possible initial sentence lengths < T < T , withprobabilities p and p , respectively, such that p + p = 1 , that is, q ′ ( r ) = p δ ′ T ( r ) + p δ ′ T ( r ) , r ≥ , with δ ′ x ( · ) , the Dirac function concentrated on x associated with the Dirac measure δ x ( · ) concen-trated on { x } . Then, T q = p T + p T , ˆ q ( d ) = p e − dT + p e − dT . The total population, entryand exit rates at steady state are shown in Table 1.Table 1: Total population, entry and exit rates at steady state for a discrete initial sentence lengthdistribution. Quantity d > d = 0 N λd (1 − ( p e − dT + p e − dT )) λ ( p T + p T ) µ in d − ( p e − dT + p e − dT ) 1 p T + p T µ out d ( p e − dT + p e − dT )1 − ( p e − dT + p e − dT ) 1 p T + p T In this example, the population density with respect to the remaining sentence time is givenby ρ = λ ˆ q ( d ) = λ ( p e − dT + p e − dT ) and ρ r = λ Z ∞ r e − d ( α − r ) q ′ ( α ) dα = λ ( p e − d ( T − r ) + p e − d ( T − r ) ) , r < T ,λp e − d ( T − r ) , T ≤ r < T , T ≤ r, for both cases d > and d = 0 .If C = (0 , r ⋆ ] , C = ( r ⋆ , ∞ ) , with T < r ⋆ < T , then T q = T , T q = T , and we obtain theexpressions in Table 2.Table 2: Total population, entry and exit rates at steady state for a discrete initial sentence lengthdistribution considering two classes. Quantity d > d = 0 Class 1 Class 2 Class 1 Class 2 ˆ q i ( d ) e − dT e − dT N i λp d (1 − e − dT ) λp d (1 − e − dT ) λp T λp T µ in ,i d − e − dT d − e − dT T T µ out ,i de − dT − e − dT de − dT − e − dT T T Consider a constant entry rate λ t = λ > and a death rate d t = d ≥ , and suppose that q ( · ) follows an exponential distribution with rate c > , that is, its probability density function12orresponds to q ′ ( r ) = ce − cr , r ≥ . Then, T q = 1 c and ˆ q ( d ) = cd + c , obtaining the expressions in Table 3. The population densityfunction with respect to the remaining sentence time is ρ r = λcd + c e − cr , r ≥ . Table 3: Total population, entry and exit rates at steady state for an exponential sentence lengthdistribution.
Quantity d > d = 0 N λd + c λcµ in d + c cµ out c c If C = (0 , r ⋆ ] , C = ( r ⋆ , ∞ ) , then p = 1 − e − cr ⋆ , p = e − cr ⋆ , T q = c − r ⋆ e − cr⋆ − e − cr⋆ , and T q = c + r ⋆ . In Table 4, we report the corresponding expressions when d > and we do so inTable 5 when d = 0 .Table 4: Total population, entry and exit rates at steady state for an exponential sentence lengthdistribution considering two classes when d > . Quantity Class 1 Class 2 ˆ q i ( d ) cd + c − e − ( d + c ) r ⋆ − e − cr ⋆ cd + c e − ( d + c ) r ⋆ e − cr ⋆ N i λd (cid:18) (1 − e − cr ⋆ ) − cd + c (1 − e − ( d + c ) r ⋆ ) (cid:19) λd (cid:18) e − cr ⋆ − cd + c e − ( d + c ) r ⋆ (cid:19) µ in ,i d (cid:16) − cd + c − e − ( d + c ) r⋆ − e − cr⋆ (cid:17) − d (cid:16) − cd + c e − dr ⋆ (cid:17) − µ out ,i cdd + c − e − ( d + c ) r ⋆ − e − cr ⋆ (cid:16) − cd + c − e − ( d + c ) r⋆ − e − cr⋆ (cid:17) − cdd + c e − ( d + c ) r ⋆ e − cr ⋆ (cid:16) − cd + c e − dr ⋆ (cid:17) − Table 5: Total population, entry and exit rates at steady state for an exponential sentence lengthdistribution considering two classes when d = 0 . Quantity Class 1 Class 2 N i λ (cid:18) − e − cr ⋆ c − r ⋆ e − cr ⋆ (cid:19) λe − cr ⋆ (cid:18) r ⋆ + 1 c (cid:19) µ in ,i (cid:18) c − r ⋆ e − cr ⋆ − e − cr ⋆ (cid:19) − (cid:18) c + r ⋆ (cid:19) − µ out ,i (cid:18) c − r ⋆ e − cr ⋆ − e − cr ⋆ (cid:19) − (cid:18) c + r ⋆ (cid:19) − .4 Example of application: estimation of initial sentence distribution It is often difficult to have access to the distribution q ( · ) of initial sentence lengths. Nevertheless,it is possible that the information of the operation of the prison is stored or reported in theform of periodic snapshots or averages of its status, consisting of the entry rate (new inmatesor transferred), exit rate (by finishing the sentence), removal rate (by transfer, death, pardon orcommutation of the sentence of prisoners), and histograms of the initial sentence lengths relativeto the existing population.Suppose that our source of information about the prison contains the aggregated entry rate λ data , the removal rate d data , the exit rate ρ , and a histogram of the current state of the prison,consisting of the frequencies N i data of inmates with respect to their initial sentence length, splitin n classes corresponding to the intervals C i = ( T i − , T i ] , with T < · · · < T n = T max < ∞ .We wish to estimate the initial sentence distribution q ( · ) from the known data. For this, weconsider the same interval classification of sentence lengths, and write q ( · ) = P ni =1 p i q i ( · ) , wherethe probabilities by class ( p i ) ni =1 and the distributions conditional to the classes ( q i ( · )) ni =1 areunknown.Notice that the information given in the histogram corresponds to the number of inmateswhose initial sentence length belongs to the interval C i . Thus, defining N data := P ni =1 N i data , theproportion π i data := N i data /N data is an estimator of the proportion π i of prisoners at each class.Based on (44) and Remark 9, we can obtain an estimator of ( p i ) ni =1 . Indeed, if ( π i ) ni =1 , ( q ( · ) i ) ni =1 ,and d were known, from (44) we would obtain − ˆ q ( d ) = p i − ˆ q i ( d ) π i = p j − ˆ q j ( d ) π j , ∀ i, j = 1 , . . . , n, which implies, choosing a particular (fixed) index j ⋆ ∈ { , . . . , n } , that p i = p j ⋆ π i π j ⋆ − ˆ q j ⋆ ( d )1 − ˆ q i ( d ) , ∀ i = 1 , . . . , n. Imposing P ni =1 p i = 1 , we obtain p j ⋆ = π j⋆ − ˆ q j⋆ ( d ) (cid:16)P nj =1 π j − ˆ q j ( d ) (cid:17) − , and then, p i = π i / (1 − ˆ q i ( d )) P nj =1 π j / (1 − ˆ q j ( d )) , i = 1 , . . . , n. (47)Given the discrete nature of a histogram, we can suppose that for each i ∈ { , . . . , n } thedistribution q i ( · ) is concentrated in a point S i ∈ ( T i − , T i ] which operates as a representative ofthe interval C i = ( T i − , T i ] , that is, q i ( · ) = δ S i ( · ) , whose Laplace transform is ˆ q i ( d ) = e − dS i . Then,our method to estimate ( p i ) ni =1 is:1. Compute π i data := N i data /N data
2. Compute ˜ p i = π i data / (1 − e − d data S i ) P nj =1 π j data / (1 − e − d data S j ) = N i data / (1 − e − d data S i ) P nj =1 N j data / (1 − e − d data S j ) .It is possible to compare the theoretical quantities (given by the model) with those computedfrom the real data. For instance, using ˆ q ( d ) = P ni =1 ˜ p i e − dS i the Laplace transform of the estimateddistribution q ( · ) , we can:1. compare N data with ˜ N model = λ data 1 − ˆ q ( d data ) d data .2. compare µ indata = λ data /N data with ˜ µ inmodel = d data − ˆ q ( d data ) .3. compare µ outdata = ρ /N data with ˜ µ outmodel = d data ˆ q ( d data )1 − ˆ q ( d data ) .4. compare ρ with ˜ ρ = ˜ µ outmodel ˜ N model .14 .4.1 Application to a real prison In this part, we apply the procedure described in the previous section to estimate the total num-ber of inmates, entry, removal and exit rates, using real data from the prison
Colina 1 in theMetropolitan Region (Santiago), Chile. This prison is a penitentiary center in closed regime whereconvicted inmates, eventually transferred from other prisons, serve their sentences.To estimate the distribution ( π i ) i we consider the average of histograms of the initial sentencinglengths of the existing inmate population in the whole Metropolitan Region. Figure 1 shows theproportions by class associated to these histograms for the years 2016-2019, obtained from [11],which are quite similar among different years. In these histograms, the frequencies are distributedin the following intervals of initial sentencing lengths: (0 , days, (15 , days, (600 days , years ] , (3 , years, (5 , years, (10 , years, (15 , years, and (20 , years. The averagevector obtained is π data = (0 . , . , . , . , . , . , . , . .Figure 1: Histograms of the initial sentencing lengths of the existing inmate population in thewhole Metropolitan Region for the years 2016-2019.Regarding the estimation of the other parameters, we had access to the data from prison Colina 1 for years 2010-2014 and 2019-2020 in [12]. The data we use is presented in Table 6. Thisinformation includes:• monthly averages of the number of total inmates N data by year;• yearly entry rate λ data of new inmates or transferred from other prisons;• yearly number D data of removed inmates due to deaths, transfers to other prisons, etc.;• yearly exit rate ρ of inmates who are finishing their sentences.We estimate each of the parameters listed above by the average of the corresponding variablesin Table 6. The removal rate d data is estimated as the average of the ratios D data /N data .Once obtained the proportions π i data , to estimate the probabilities ˜ p i we consider as represen-tative S i of the sentencing length interval ( T i − , T i ] the value S i = T i . The obtained distributionof π i data and ˜ p i (and then q ( · ) ) are depicted in Figure 2.The comparison between the total number of inmates ( N ), the entry and exit rates relativeto the populations sizes ( µ in and µ out respectively), obtained directly from the data and obtainedfrom the model (using λ data , d data and the histograms), is shown in Table 7.The differences observed in the Table 7 are not very large and they can be explained by: (i) Theconsideration of the histograms of the initial sentencing lengths of the existing inmate populationin all the Metropolitan Region and not just those of the prison studied (not available); (ii) Theassumption that all the admitted inmates start serving their sentence in the prison under study,which is not totally true for inmates transferred from other prisons where they have served partof their sentences; (iii) The consideration of the value T i as representative of the sentence length15able 6: Data obtained from [12] corresponding to prison Colina 1 (Santiago, Metropolitan Region,Chile): Number of total inmates ( N data ), total entry rate λ data of new inmates or transferred fromother prisons; the removal rate d data of inmates due to deaths, transfers to other prisons, etc., andthe exit rate ρ of inmates who are finishing their sentences. Year N data λ data ρ D data d data Averages π i data obtained from histograms of the initial sentencing lengths of theexisting inmate population in the all Metropolitan Region for the years 2016-2019 (see Fig. 1) andestimated distribution q ( · ) given by estimated probabilities ˜ p i .Table 7: Comparison between the total number of inmates ( N ), the entry and exit rates relativeto the populations sizes ( µ in and µ out respectively) obtained directly from the data and obtainedfrom the model (using N data , λ data , d data and the histograms). Data Value Model Value % Relative difference N data ˜ N model = λ data 1 − ˆ q ( d data ) d data µ indata = λ data /N data ˜ µ inmodel = d data − ˆ q ( d data ) µ outdata = ρ /N data ˜ µ outmodel = d data ˆ q ( d data )1 − ˆ q ( d data ) ρ ˜ ρ = ˜ µ outmodel ˜ N model
299 12.83% interval ( T i − , T i ] ; (iv) The assumption that the removal rate d is independent of the remainingsentence time in prison, which, depending on the reason of the removal (for instance, by pardon orcommutation of the sentence) can be a too strong hypothesis. Nevertheless, the results obtained16ith the model suggest a reliable approximation of the analyzed prison situation. In this section, we consider the spread of a communicable disease in a prison population, modelingthe disease dynamics by the SIS (susceptible-infected-susceptible) model (i.e., the disease confersno immunity). In general, an SIS model is appropriate for a bacterial disease. Suppose that diseasetransmission occurs at the per capita contact rate (sufficient to transmit the disease) β t > , andinfective individuals recover from the disease at a rate γ t > with no immunity. Assume furtherthat removals from the prison occur at a rate d t ≥ , the entry rate of inmates to the prison is λ t ≥ , and the distribution of initial sentence lengths is q t ( · ) (as considered in Section 2). Weassume that a proportion α I > of the new inmates is infective and that a proportion α S = 1 − α I is susceptible to the infection. The proportion α I > is assumed to be the steady-state prevalenceof the population outside the prison.First, we consider the whole prison as a single class. Define S t ≥ and I t ≥ as the quantitiesof susceptible and infective individuals, respectively, and N t as the population size at time t ≥ .Then, the disease transmission dynamics in the population can be described by the following systemof differential equations: ˙ N t = λ t − ρ t − d t N t , ˙ S t = α S λ t − β t S t I t N t + γ t I t − ρ t S t N t − d t S t , ˙ I t = α I λ t + β t S t I t N t − γ t I t − ρ t I t N t − d t I t , (48)where the first equation comes from (13), with ρ t being the instantaneous exit rate of prisoners,which is a solution of (4). We note that the set { ( N, S, I ) | N − ( S + I ) = 0 } is invariant under(48). Indeed, ddt ( N t − ( S t + I t )) = − (cid:20) ρ t N t + d t (cid:21) ( N t − ( S t + I t )) . Since we suppose that, at the beginning of the process, N = S + I , we can replace S t = N t − I t in the equation for I , and defining x t = I t /N t , we obtain the equation ˙ x t = α I µ in t + β t x t (1 − x t ) − γ t x t − µ in t x t , µ in t := λ t N t . (49)Now, suppose that the population is divided into two classes of initial sentence lengths, asdescribed in Section 2.2, where i = 1 (resp. i = 2 ) stands for the class of short (resp. long)sentences, without affecting the homogeneous mixing of inmates. Define S it ≥ and I it ≥ as thequantities of susceptible and infective individuals that belong to class i , respectively, with N it beingthe population size of class i at time t ≥ ( i = 1 , ). Each susceptible individual of a class mayhave contact with an infective individual of his/her own class or of the other class. We supposethat both classes share the same removal rate d t ≥ . Then, the disease transmission dynamicscan be described by the following system of coupled differential equations for i = 1 , : ˙ N it = λ it − ρ ,it − d t N it , ˙ S it = α S λ it − β t S it I t + I t N t + γ t I it − ρ ,it S it N it − d t S it , ˙ I it = α I λ it + β t S it I t + I t N t − γ t I it − ρ ,it I it N it − d t I it , (50)where the equations for N i come from (35) and ρ ,it are solutions of (36). In this case, we alsohave the invariance of the set { ( N i , S i , I i ) | N i − ( S i + I i ) = 0 , i = 1 , } , and then we can replace S it = N it − I it in the equation for I i in (50). Thus, I i solves ˙ I it = α I λ it + β t I t + I t N t + N t ( N it − I it ) − γ t I it − ρ ,it I it N it − d t I it . N = N + N solves the same equation as in (48). Defining x it := I it /N t asthe proportion of infective people in class i with respect to the total prison population, x i satisfies ˙ x it = α I µ in ,it π it + β t ( x t + x t )( π it − x it ) − γ t x it − µ out ,it x it − ( µ in t − µ out t ) x it , i = 1 , , (51)where π it = N it /N t is the proportion of inmates of each class relative to the total prison populationat each time t and µ in t , µ out t , µ in ,it , and µ out ,it as in (40). In this section, we analyze and compare the equilibria of equations (49) (single-class model) and(51) (two-class model) and provide conditions under which the single-class model underestimatesthe proportion of infected inmates with respect to the two-class model.
Remark 10.
Note that the equations associated with the populations
N, N , N in models (48) and (50) are independent of the epidemiological partition S, I , since we do not consider specificremovals due to illness. Thus, if λ , d and q ( · ) do not depend on t , we can perform a partial analysisconsidering the populations N, N , N in equilibrium. Then, the total populations N , N , N areconstant (given by (42) and (43) ), as are the rates µ in , µ out , µ in ,i , µ out ,i and the proportions π i , i = 1 , which, from (40) , become µ in = d − ˆ q ( d ) , µ out = ˆ q ( d ) µ in ,µ in ,i = d − ˆ q i ( d ) , µ out ,i = ˆ q i ( d ) µ in ,i , π i = p i − ˆ q i ( d )1 − ˆ q ( d ) , (52) As we suppose the same (constant) removal rate for both classes, (23) states that µ in − µ out = d = µ in ,i − µ out ,i , i = 1 , . Then, supposing that β, γ does not depend on t , the equations corresponding to the epidemio-logical parts of the models (49) and (51) , under the assumption of population at equilibrium, takethe simpler form ˙ x t = α I µ in + βx t (1 − x t ) − ( γ + µ in ) x t , (53) ˙ x it = α I π i µ in ,i + β ( x t + x t )( π i − x it ) − ( γ + µ in ,i ) x it , i = 1 , . (54) Since in this section we study the behavior in equilibrium, it suffices to study the equilibria of (53) and (54) . Let us denote the positive equilibrium of the single-class model (53) by x ⋆ , and the positive equi-librium of the two-class model (54) by ( x ⋆ , x ⋆ ) . Define the total proportion of infective individualsin equilibrium in the two-class model by ω ⋆ := x ⋆ + x ⋆ .We present the main result of this section in Proposition 7: Proposition 7.
Suppose that α I , µ in ,i > , i = 1 , . Then, x ⋆ , ω ⋆ > . Moreover, x ⋆ < ω ⋆ ( >, = resp.) if and only if α I < − γβ ( >, = resp. ).Proof. For the single-class model, from (53), we have the equilibrium equation α I µ in + βx ⋆ (1 − x ⋆ ) − γx ⋆ − µ in x ⋆ = 0 , µ in = d − ˆ q ( d ) , with µ in given in (52). Then, x ⋆ satisfies β ( x ⋆ ) − ( β − ( γ + µ in )) x ⋆ − α I µ in = 0 , from which we obtain the alternative equation ( x ⋆ ) = 1 β (cid:2) ( β − ( γ + µ in )) x ⋆ − α I µ in (cid:3) , (55)18nd the explicit expression for the nonnegative solution x ⋆ = 12 β h ( β − ( γ + µ in )) + p ( β − ( γ + µ in )) + 4 α I µ in β i , (56)which is strictly positive if α I µ in > .For the two-class model, from (54), we have the equilibrium equations ( α I π µ in , + β ( x ⋆ + x ⋆ )( π − x ⋆ ) − γx ⋆ − µ in , x ⋆ = 0 ,α I π µ in , + β ( x ⋆ + x ⋆ )( π − x ⋆ ) − γx ⋆ − µ in , x ⋆ = 0 , (57)with π i and µ in ,i given by (52). From (57), we obtain β ( x ⋆ + x ⋆ ) = ( γ + µ in , ) x ⋆ − α I π µ in , π − x ⋆ = ( γ + µ in , ) x ⋆ − α I π µ in , π − x ⋆ . (58)From (58), we can write x ⋆ and x ⋆ as functions of ω ⋆ = x ⋆ + x ⋆ as x ⋆ = g ( ω ⋆ ) := π βω ⋆ + α I µ in , βω ⋆ + γ + µ in , , x ⋆ = g ( ω ⋆ ) := π βω ⋆ + α I µ in , βω ⋆ + γ + µ in , . (59)Thus, summing both expressions in (59), ω ⋆ is a nonnegative solution of ω ⋆ = g ( ω ⋆ ) + g ( ω ⋆ ) . (60)We contend that, under the hypotheses α I , µ in ,i > , i = 1 , , there exists a unique strictlypositive solution of (60). Indeed, each of the functions g i ( · ) has the form g i ( ω ) = a i ω + b i c i ω + d i , with a i = βπ i > , b i = α I π i µ in ,i > , c i = β > , d i = γ + µ in ,i > . Then, g i ( · ) has a uniquezero at ω i = − b i /a i = − α I µ in ,i /β < , it is undefined at ω i ∞ = − d i /c i = − ( γ + µ in ,i ) /β < ,and it holds that ω ∞ < ω . Indeed, this is equivalent to a i d i − b i c i = βπ i ( γ + (1 − α I ) µ in ,i ) > .Moreover, g ′ i ( ω ) = a i d i − b i c i ( c i ω + d i ) which is strictly positive for ω > ω i ∞ and decreases to 0 as ω → ∞ . This shows that g i ( · ) is strictlyincreasing and concave on the interval ( ω i ∞ , ∞ ) and positive on the interval ( ω i , ∞ ) ⊆ ( ω i ∞ , ∞ ) ,with g i (0) = b i /d i > and lim ω →∞ g i ( ω ) = a i /c i = π i > .Now, consider the function g , ( ω ) = g ( ω ) + g ( ω ) . This function is strictly increasing in theinterval (max { ω ∞ , ω ∞ } , ∞ ) , with g , (0) > , and it has a horizontal asymptote as ω converges toinfinity. Thus, there exists a unique strictly positive solution of equation g , ( ω ) = ω , that is, of(60).We refer to the unique positive solution of (60) as ω ⋆ , which can be equivalently written (undera rearrangement of the terms of (60)) as the unique positive root of the function g ( ω ) := β ω + β ( µ in , + µ in , + 2 γ − β ) ω + (( γ + µ in , )( γ + µ in , ) − π β ( α I µ in , + γ + µ in , ) − π β ( α I µ in , + γ + µ in , )) ω − α I ( π µ in , ( γ + µ in , ) + π µ in , ( γ + µ in , )) . The function g ( · ) is a third-degree polynomial, with lim ω →∞ g ( ω ) = ∞ , g (0) < , and g ( ω ⋆ ) =0 . Thus, on the interval [0 , ∞ ) , g ( ω ) < if and only if ω < ω ⋆ , and g ( ω ) > if and only if ω > ω ⋆ .To compare x ⋆ and ω ⋆ , it suffices to compute the sign of g ( x ⋆ ) , provided that x ⋆ ≥ . Using (55)and (41), after a lengthy computation, we arrive at g ( x ⋆ ) = ( µ in − µ in , )( µ in − µ in , )( x ⋆ − α I ) , µ in , ≤ µ in ≤ µ in , . Then, x ⋆ < ω ⋆ ( resp. >, =) ⇔ g ( x ⋆ ) < resp. >, =) ⇔ x ⋆ > α I ( resp. <, =) . Using the formula for x ⋆ from (56), we obtain the condition for the equilibrium x ⋆ to be lessthan (resp. greater than, equal to) ω ⋆ : x ⋆ > α I ( resp. <, =) ⇔ − γβ > α I ( resp. <, =) , which concludes the proof. (cid:4) Remark 11.
From Proposition 7, the condition − γβ > α I for obtaining x ⋆ < ω ⋆ does not dependon the removal rate d or on the parameters of the class separation. On the other hand, the threshold − γβ is exactly the herd immunity threshold (i.e., − / R with R = β/γ ) associated with thesingle-class model (53) when µ in = 0 . For the numerical simulations, we consider the epidemiological parameters β = 0 . and γ = 0 . andfour cases of removal rates: null removal rate ( d = 0 ) and positive removal rates d = 0 . , . , . .We consider r ⋆ = 5 [years] as the maximum time for a sentence to be considered short . Underthese parameters, Proposition 7 states that x ⋆ < ω ⋆ if α I < − γβ = 0 . .We study the cases of initial sentence lengths given by an exponential function q ′ ( r ) = 110 e − r , r ≥ , the mean sentence time of which is T q = 10 [years], and by a bimodal function q ′ ( r ) = 0 . p , φ ( r ; 5 , ) + 0 . p , . φ ( r ; 10 , . ) , r ≥ where φ ( · ; µ, σ ) denotes the probability density function of a normal distribution with mean µ and variance σ , and p µ,σ = R ∞ φ ( t ; µ, σ ) dt is a normalizing constant, having the mean sentencetime T q = 7 [years].In the first example (exponential distribution), we obtain the proportions p = 0 . and p =0 . and the mean initial sentence lengths by class T q = 2 . [years] and T q = 15 [years]. In thesecond example, we have the proportions p = 0 . and p = 0 . and the mean initial sentencelengths by class T q = 4 . [years] and T q = 8 . [years].In Figures 3 (exponential) and 4 (bimodal), we compute the underestimation of the infectedproportion of the population incurred by the single-class model with respect to the two-class modelrelative to the disease prevalence of new inmates α I , that is, ( ω ⋆ − x ⋆ ) /α I , and relative to thesingle-class prevalence x ⋆ , that is, ( ω ⋆ − x ⋆ ) /x ⋆ .We show the plots for prevalences in the interval α I ∈ [0 , . , which is a range for α I in which ω ⋆ > x ⋆ , that is, the single-class model underestimates the number of infected people in the prison,according to Proposition 7.For the second example (bimodal distribution), we depict in Figure 5 a comparison betweenthe initial sentence length distribution and the remaining sentence time distribution. Note thatthe remaining sentence time distribution inside the prison approaches the initial sentence lengthas d increases. In this paper, we introduce an inmate population model with a sentencing length structure and findthat the density of the number of inmates (with respect to the remaining time in prison) followsa transport equation, typically known as the McKendrick equation. We compute the inmatepopulation and the entry/exit rates at steady state, showing that the sentencing length structureand the removal rate have a strong influence on these values. We illustrate how to obtain thesevalues with real data from a prison in Chile. Since a typical assumption in prison models is a20igure 3: Underestimation of total infected proportion by the single-class model with respect tothe two-class model, relative to outside prevalence (left) and relative to the single-class modelprevalence (right) when considering an exponential distribution of initial sentencing lengths.Figure 4: Underestimation of total infected proportion by the one-class model with respect tothe two-class model, relative to outside prevalence (left) and relative to the single-class modelprevalence (right) when considering a bimodal distribution of initial sentencing lengths.constant prison population, the obtained values can be used by decision-makers in the design oroptimization of a penitentiary system.To study the effect of considering or not considering sentence length structure in the estimationof the infected population, we divide the inmate population into two classes depending on theirsentencing lengths (short and long), and we couple the SIS epidemiological model to the obtainedmodel. This epidemiological model is compared, in equilibrium, with the model obtained whenignoring the sentencing length structure. We prove that not accounting for the structure of thesentence lengths for disease prevalences of new inmates below a certain threshold induces an un-derestimation of the prevalence in the prison population at steady state. The involved thresholddepends on the basic reproduction number associated with the nonstructured SIS model with noentry of new inmates.In epidemiological models for inmate populations, assuming that the prevalence of new inmatesis low represents a situation where the disease under study is almost eradicated from the generalpopulation (outside the prison), but since the prison can act as a reservoir, the prevalence of thisdisease in the inmate population can eventually be much higher (see, for instance, [9, 19, 21]).Therefore, the message from our results is that, in these situations, a recommendation from themodeling perspective is to include the sentencing length structure of the prison population when21igure 5: Comparison of initial sentence length distribution (bimodal) and remaining sentencetime distribution inside the prison for different values of d .analyzing epidemiological models. This work was funded by FONDECYT grants N 1200355 (first author) and N 3180367 (secondauthor) from ANID-Chile. The authors are very grateful to professors Heliana Arias (Universidaddel Valle, Colombia) and Carla Castillo-Laborde (Universidad del Desarrollo, Chile) for fruitfuldiscussions.
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A Convergence of nonnegative measures
Proposition 8.
Consider ( ϕ t ) t ≥ a family of nonnegative measures on B ( R + ) . Suppose that thisfamily satisfies the following hypothesis: ∀ t ≥ , ∀ A ∈ B ( R + ) , ∀ ε > , ∃ η ∈ (0 ,
1) : | z | , | z ′ | < η ⇒ | ϕ t + z ( A + z ′ ) − ϕ t ( A + z ′ ) | < ε. (61) Then, . For almost every t ≥ , ≤ s < r , lim ∆ t ց t Z ∆ t | ϕ t +∆ t − u (( s + u, r + u ]) − ϕ t (( s + u, r + u ]) | du = 0 .
2. For almost every t ≥ , ≤ s < r , lim ∆ t ց t Z ∆ t ϕ t +∆ t − u (( s + u, t + u ]) du = ϕ t (( s, r ]) . Proof.
1. Let ε > . Take, for t ≥ and A = ( s, r ] ∈ B ( R + ) , η ∈ (0 , from (61). Then, thereexists ∆ t < η such that if z ′ = u ≤ ∆ t < η , z = ∆ t − u < η , and then, t Z ∆ t | ϕ t +∆ t − u (( s + u, r + u ]) − ϕ t (( s + u, r + u ]) | du ≤ t ǫ ∆ t = ǫ, which proves the result.2. We have Z ∆ t ϕ t +∆ t − u (( s + u, r + u ]) du ≤ Z ∆ t | ϕ t +∆ t − u (( s + u, r + u ]) − ϕ t (( s + u, r + u ]) | du + Z ∆ t ϕ t (( s + u, r + u ]) du. The first term on the right-hand side can be bounded by the result in point 1. For the secondterm, Z ∆ t ϕ t (( s + u, r + u ]) du = Z ∆ t ϕ t (( s, r ]) du + Z ∆ t ϕ t (( r, r + u ]) du − Z ∆ t ϕ t (( s, s + u ]) du The first term on the right-hand side of the previous expression is equal to ∆ t · ϕ t (( s, r ]) . De-fine, for t, r fixed, the function f r ( u ) = ϕ t (( r, r + u ]) . This function is measurable, increasing,and bounded. Then, by [5, Theorem 5.4.2], noting that f r (0) = ϕ t (( r, r ]) = 0 , lim ∆ t ց t Z ∆ t ϕ t (( r, r + u ]) du = lim ∆ t ց t Z ∆ t f r ( u ) du = f r (0) = 0 , and similarly replacing r by s . Thus, lim ∆ t ց Z ∆ t ϕ t (( s + u, r + u ]) du = ϕ t (( s, r ]) . (cid:4)(cid:4)