A Discrete Time Population Genetic Model for X-Linked Recessive Diseases

INTERNATIONAL JOURNAL OF BIOLOGY AND BIOMEDICAL ENGINEERING Volume 11, 2017

A discrete time population genetic model for X-linked recessive diseases

Carmen Del Vecchio1, Francesca Verrilli1, Luigi Glielmo1 and Martin Corless2

Abstract—The epidemiology of X-linked recessive dis- this number can not be inferred from genotypes’ frequency eases, a class of genetic disorders, is modeled with a discrete- distribution when assuming infinite population size. time, structured, mathematical model. The model accounts The results in this paper are an extension of some pre- for both de novo mutations and different reproduction rates vious preliminary works on the same topic ([8], [9]). The of procreating couples depending on their health conditions. major original result of our work is to model the peculiar Relying on Lyapunov theory, asymptotic stability properties inheritance mechanism of X-linked diseases while taking of equilibrium points of the model are demonstrated. The into account the reduced reproduction capacity of affected model describes the spread over time in the population of individuals as well as the occurrence of the diseases in any recessive genetic disorder transmitted through the X- healthy couple progeny due to de novo genetic mutations. chromosome. This represents an advancement over current epidemiological Keywords—Population genetic dynamics, nonlinear dy- models, and could be exploited to better understand the namic analysis, stability, epidemiology. epidemiology of X-linked genetic diseases; moreover, it could be generalized to allow application to other classes of I.MOTIVATION genetic disorders. Although the mathematical model developed to describe the epidemiology of these diseases within We study a specific class of genetic disorders named a population is nonlinear, it is suitable for analyses using X-linked recessive diseases; these conditions include the classical nonlinear instruments to gain information about serious diseases hemophilia A, Duchenne-Becker muscular system behavior and equilibrium properties. dystrophy, and Lesch-Nyhan syndrome as well as common The paper is structured as follows: a brief description of X- and less serious conditions such as male pattern baldness linked genetic diseases and their peculiar inheritance pattern and red-green color blindness. A major reason for devoting is provided in Section II-A. We pose our model and we attention to this topic is the inadequacy of the currently used derive general system properties and solutions characteristic mathematical instruments to describe the transmission of a in Sections II-B and III.Finally, we discuss the physiological genetic disease within a predefined population. implications corresponding to the mathematical properties Related studies analyzed the inheritance mechanism of any derived from our model and some special model cases. gene —not necessarily responsible for a genetic disease— II.MATHEMATICAL MODEL OF X-LINKEDRECESSIVE placed on the X-chromosome ([1], [2], [3]); they belong to DISEASES the field of population genetics. In these works genotypes frequencies —i.e., the frequency or proportion of genotypes A. The transmission mechanism of X-linked recessive dis- in a population— are frequently chosen as model’s variables. eases Under the hypothesis of infinite population and starting from An X-linked recessive disease may be inherited as per the a genotypes’ distribution, the genotypes’ proportions in the following rules (see [9] for details): next generation are evaluated according to the inheritance • Affected males never spread the disease to their sons, mechanism and to the effects of selection or mutation. The as no male-to-male transmission of the X chromosome average fitness (see [4] pag. 385-387) is frequently studied as occurs. a suitable Lyapunov function candidate to analyze stability • Affected males pass the abnormal X chromosome to properties of model’s equilibrium points. all of their daughters, who are described as obligate Even in this generic scenario seldom contributions ex- carriers. amine the combined effects of selection and mutation on • On average, female carriers pass a defective X chro- population’s dynamics and equilibrium (see [5], [6], [7] mosome to half of their sons (who will born affected) and reference therein). Moreover, results of these researches and half of their daughters (who become carriers). The cannot be applied to epidemiological studies. In fact the remaining half of their siblings inherit a normal copy ultimate goal of genetic epidemiology is to predict the of the chromosome. number of individuals carrying the disease responsible gene, • Affected females are the rare result of an affected male and a carrier female mating. 1 Dipartimento di Ingegneria, Universita` Figures 1 and 2 depict the inheritance patterns described degli Studi del Sannio, Benevento (Italy) above. {fverrilli,glielmo,c.delvecchio}@unisannio.it 2 School of Aeronautics and Astronautics, Purdue University, West X-linked recessive conditions include the serious diseases Lafayette, Indiana (USA) [email protected]. Duchenne/Becker muscular dystrophy, hemophilia A, and

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healthy carrier father mother affected healthy father mother due to premature, spontaneous intrauterine death. In modern population genetics, prenatal diagnosis and birth control measures can play a major role and signiﬁcantly inﬂuence the rate of affected cases in a given generation. Couples with a family history of X-linked genetic disorders often access prenatal or pre-implantation embryonic genetic screening, with the consequent negative selection of affected ones or selective therapeutic abortion.

healthy son carrier carrier healthy son affected son healthy carrier healthy son daughter daughter daughter daughter B. Model formulation Fig. 1. Inheritance pattern Fig. 2. Inheritance pattern In this section we present a mathematical model we of affected father and healthy of healthy father and carrier have developed to describe the epidemiology of genetic mother. mother. diseases linked to the X chromosome. Our model fits in the category of structured models. In these models, a population Lesch-Nyhan syndrome as well as common and less serious is divided into homogeneous groups according to some major conditions such as male pattern baldness and red-green color parameters, such as subject’s age, sex or health conditions blindness. X-linked dominant diseases are very uncommon, with respect to a disease. The model dynamics describes although some inherited forms of rickets are transmitted in the distribution of the population over time according to the this manner. Unlike the recessive diseases discussed above, chosen parameters ([11], [12]). the prevalence of X-linked dominant diseases is similar in Population dynamic models differ depending on assumptions males and females, even though the absence of male-to-male regarding population size and mating rules among groups. transmission distinguishes them from autosomal dominant The population is often assumed to be isolated (i.e. migration diseases. and selection are not modeled) and of constant finite size. Other than the result of the described, well characterized This allows mathematical tractability. However, models with patterns of inheritance, the spread of genetic disorders within variable population size or including selection factors (such a given population is influenced by additional factors, includ- as the early death of affected individuals, or selection due to ing sporadic mutations and prenatal diagnosis. prenatal diagnosis) are definitely more realistic. A genetic disease that occurs when neither parent is The most frequently adopted rule for mating is that individ- affected or a carrier of any genetic defect is called sporadic uals in the population mix randomly, i.e. individuals mate mutation or de novo gene mutation. These cases arise via according to the product rule of probability; this is more random genetic mutations within the DNA sequence; the realistic in large populations and assumes that the studied mutation can occur in the the germ-line cell population — trait does not influence reproduction ([13]). i.e. in eggs and sperm cells— in subjects without any prior We have developed a discrete-time dynamic model to genetic defect and can be transmitted down to one of the describe the inheritance mechanism of X-linked recessive offspring. The genetic mutation can also occur in the zygote diseases in a finite size population grouped by sex and health cell, i.e. the initial cell formed when two gametes cell are condition with respect to the disease. joined. A sporadic mutation can be the cause of an X-linked We divide the population into four classes, namely healthy recessive disease (whereas it is unlikely for an autosomal and affected males, healthy and carrier females. We do not recessive disorder) as a single mutation is enough in males consider affected females as they very rarely occur in nature. to cause the disease. Males can be born affected due to a Thus our model has four variables: spontaneous gene mutation as a single abnormal gene copy • x1(k): the number of healthy males at time k is enough for the disease to become symptomatic; females • x2(k): the number of affected males at time k can also be born carriers owing to random mutations. • x3(k): the number of healthy females at time k The rate of de novo mutations varies widely among • x4(k): the number of carrier females at time k. different genetic regions, and depends on a number of factors, We make the following assumptions. including environmental exposure to mutagenic agents, the • In each generation there is an equal number of males length of the gene sequence and ability of the cell machinery and females; thus there is the the same number of males to actually repair or correct the mutations. For instance, and females in newborn children. the dystrophin gene, whose mutations may give rise to • Each person breeds with a person of the opposite sex Duchenne/Becker muscular dystrophy, is particularly prone from his/her own generation. to de novo mutations due to its massive length ([10]). It is • The number of sons (which is equal the number of estimated that up to a third of all cases of this disorder are daughters) of each couple varies according to the par- due to de novo mutations, a rate considerably higher than any ents’ health conditions and is modeled through the other X-linked disorder. Not all mutations in X-chromosome fertility factors wij. genes confer sufficient survival fitness to give rise to a viable • Spontaneous genetic mutations are modeled; they are embryo. Therefore, an individual born affected due to de assumed to occur in the zygote cells; a child of a healthy novo mutations may not pass on the affected gene to progeny couple can be born affected or a carrier due to mutation.

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The number of males (and females in view of the ﬁrst rate of the disease in males and females respectively. Their assumption above) born to a person of class i ∈ {1, 2} values strictly depend on the genetic disease and they range breeding with a person of class j ∈ {3, 4} is between 10−3 and 10−8 ([14]). We will consider α and β to be strictly less then 1 ; otherwise spontaneous genetic 1 xi xj 2 wij (x1 + x2 + x3 + x4), (1) mutation would be more relevant than the ordinary disease 2 (x + x ) (x + x ) 1 2 3 4 transmission mechanism; in contrast setting β = 0 and α = 0 where implies that gene mutations do not apply to the disease. Thus w ≥ 0 (2) 1 ij in what follows α and β will range in [0, 2 ). Finally we explicitly note that once system (3)-(4) is is the fertility factor, procreation rate or reproduction rate initialized with all state variables non-negative (that is, non- of couples of type (i, j). The parameters w are bounded ij negative initial populations) —the state variables remain by clinical considerations; the more severe the disease the nonnegative for all times k ≥ 0: smaller its value in couples formed by affected males and/or carrier females. A son (daughter) is a healthy or affected xi(k) ≥ 0 for i = 1, 2, 3, 4 when xi0 := xi(0) ≥ 0 male (healthy or carrier female) depending on the health (6) conditions of his (her) parents. As an example consider sons for i = 1,..., 4. Hence we are dealing with a positive system. born from couples formed by a healthy father (an individual Due to the model hypotheses discussed above —each of class 1) and a carrier mother (an individual of class 4). couple procreates an equal number of males and females— According to the inheritance patterns of X-linked recessive one can easily see that diseases (see Figure 2), half of these sons (on average) will x (k) + x (k) = x (k) + x (k) (7) be affected and half will be healthy. Hence the number of 1 2 3 4 affected males in the next generation due to such couples is for all k > 0. Hence P (x) in equation (5) simpliﬁes to

1 x1 x4 1 1 w14 (x1 + x2 + x3 + x4) P (x) = = 4 (x1 + x2) (x3 + x4) x1 + x2 x3 + x4 which is the same as the number of healthy males in the next that is, P (x) is the inverse of half the total population. Since generation due to these couples. wij ≥ 0 for i = 1, 2 and j = 3, 4 and assuming wij > 0 for For ease in describing our model we introduce the state at least one couple (i, j), it is not difficult to show that vector T x1(k) + x2(k) > 0 for all k when x10 + x20 > 0 . x := [x1 x2 x3 x4] . (8) According to the previous assumptions and to the inheritance This guarantees that P (x(k)) is always well defined. Let pattern of X-linked recessive disease described in Section X := {x : xi ≥ 0 for i = 1,..., 4 and x1+x2 = x3+x4 > 0}. II-A, the population dynamics is described by the following non-linear discrete-time system: Then this set is invariant for system (3)-(4), that is, if the state starts in this set, it never leaves it. x(k + 1) = f(x(k)) (3) The model presented in this paper is a generalization T of previous models in ([8], [9]). It significantly improves where the vector function f = [f1 f2 f3 f4] is given by the modeling of de novo mutations (i.e. affected or carrier h f1(x) = P (x) (1 − α)w13x1x3 + w23x2x3 + children born to healthy parents) and reproduction rates consistent with the health conditions of reproducing couples. 1 1 i w x x + w x x (4a) These model features, enabling the modeling of any X-linked 2 14 1 4 2 24 2 4 h 1 disease, were not included in the first version ([8]). Moreover f (x) = P (x) αw x x + w x x + the results we gain on system equilibrium, stability and 2 13 1 3 2 14 1 4 1 i convergence properties are more general than those in [9] w24x2x4 (4b) where only the special case of negligible de novo mutations 2 h 1 i and few combinations of reproduction rates values could be f (x) = P (x) (1 − β)w x x + w x x (4c) 3 13 1 3 2 14 1 4 analyzed. Finally in this paper the sporadic genetic mutation h 1 can affect sons and daughters born from healthy couples with f (x) = P (x) βw x x + w x x + w x x + 4 13 1 3 23 2 3 2 14 1 4 different rates α and β; in [9] it was assumed that only males i could be born affected due to a spontaneous mutation. w24x2x4 (4d) III.SOME SYSTEM PROPERTIES with x + x + x + x A. A Lyapunov function P (x) := 1 2 3 4 . (5) 2(x1 + x2)(x3 + x4) Noting that

The term αw13 (βw13) is the fraction of affected sons x1 = P (x)(x1x3 + x1x4), x2 = P (x)(x2x3 + x2x4) (carrier daughters) born from healthy parents due to de novo x3 = P (x)(x1x3 + x2x3), x4 = P (x)(x1x4 + x2x4) gene mutation; thus α and β model the spontaneous mutation (9)

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system (3)-(4) can be described by Proposition 3.1: Consider system (3)-(4) with 0 ≤ wij ≤ 1 for i = 1, 2 and j = 3, 4. If the initial state x0 lies in X , x(k + 1) = x(k) + g(x(k)) (10) then, for all k ≥ 0, x(k) ∈ X and T where g = [g1 g2 g3 g4] and x1(k) + x2(k) 6 x10 + x20 (18) h x3(k) + x4(k) 6 x30 + x40 (19) g1(x) = P (x) ((1 − α)w13 − 1)x1x3 + x (k) x + x for i = 1,..., 4. (20) w w i i 6 10 20 14 − 1 x x + w x x + 24 x x 2 1 4 23 2 3 2 2 4 h 1 a) Special case: wij = 1 for all i and j: In this case, g2(x) = P (x) αw13x1x3 + w14x1x4 − x2x3 + ∆V (x) = 0 for all x; hence V (x(k + 1) = V (x(k)) for all 2 1 w i k which implies that V1(x(k)) = V1(x(0)), that is, 24 − 1 x x 2 2 4 h x1(k) + x2(k) = x10 + x20 (21) g3(x) = P (x) ((1 − β)w13 − 1)x1x3 + for all k. This means that the total male (hence female) 1 i w14x1x4 − x2x3 population remains constant. We examine this special case in 2 further detail in Section IV Now we consider what happens h w14 g4(x) = P (x) βw13x1x3 + − 1 x1x4 + when w < 1 for at least one ij. 2 ij i w23x2x3 + (w24 − 1)x2x4 . B. Some convergence properties

Our ﬁrst result tells us that if one of the fertility rates wij is In investigating the behavior of system (3)-(4) the function strictly less than one, then the state converges to a limit with the number of the affected males and carrier males equal to V1 deﬁned by zero. If in addition w < 1 or there is a non-zero mutation V (x) = x + x (12) 13 1 1 2 rate then the whole population goes to zero. is very useful. This is simply the total male population which Proposition 3.2: Consider system (3)-(4) with initial stat is the same as the total female population, that is, x0 in X and 0 < wij ≤ 1 for i = 1, 2 and j = 3, 4. If wij < 1 for at least one ij, then V1(x) = x3 + x4. (13) lim x2(k) = lim x4(k) = 0 (22) k→∞ k→∞ This function will be called a Lyapunov function. The change in this population from one stage k to the next stage k + 1 and is given by lim x1(k) = lim x3(k) = x1 for some x1 ≥ 0. (23) k→∞ k→∞ V1(x(k + 1)) − V1(x(k)) = ∆V1(x(k)) (14) If either w13 < 1 or α > 0 or β > 0 then x1 = 0, that is, where lim x(k) = 0. k→∞ Proof: Since w ≤ 1 for all i and j, it follows that ∆V1(x) := V1(f(x)) − V1(x) ij (16) holds, that is, {V (x(k))} is a non-increasing sequence. = g (x) + g (x) 1 1 2 Since this sequence is bounded below by zero, it converges = −P (x)[(1−w13)x1x3 + (1−w14)x1x4 + that is + (1−w23)x2x3 + (1−w24)x2x4]. lim V1(k) = V1 (24) k→∞ (15) for some V1 ≥ 0. Hence

If wij ≤ 1 for all i, j then, for all k ≥ 0, we have lim ∆V1(x(k)) = lim V1(x(k + 1)) − lim V1(x(k)) ∆V1(x(k)) ≤ 0 and k→∞ k→∞ k→∞ = V1 − V1 V1(x(k + 1)) ≤ V1(x(k)) . (16) = 0 . (25) Therefore Suppose that wij < 1 for some ij. Since wij ≤ 1 for all V1(x(k)) ≤ V1(0) (17) ij it follows from (15) that for all k ≥ 0. That is the total population is bounded by the ∆V1(x(k)) 6 −P (x(k))(1 − wij)xi(k)xj(k) ≤ 0 initial population. We can now readily obtain the following Since ∆V (x(k)) → 0, it now follows that boundedness result. Noting that 1 P (x(k))xi(k)xj(k) tends to 0. Noting that P (x(k)) ≥ P (x(0)) we obtain that x (k)x (k) goes to zero. This xi(k) 6 V1(k) for i = 1,..., 4. i j implies that fi(x(k))fj(x(k)) = xi(k + 1)xj(k + 1) also we obtain our ﬁrst result. converges to zero.

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First suppose that w14 < 1. Then and from this we can deduce that

P (x(k))x1(k)x4(k) → 0 . (26) f2(x(k))f4(x(k)) → 0. (32)

Also f1(x(k))f4(x(k)) → 0. Since all terms in (4a) and (4d) are non-negative we must have It follows from (32) that P (x(k))x (k)x (k) → 0 P (x(k))x (k)x (k) → 0 . 2 3 and 2 4 P (x(k))x1(k)x4(k) → 0. (33) (27) Using the second relationship in (9) it follows from (27) Using the fourth relationship in (9) it follows from (31) and that x (k) → 0 and, recalling (24), we also have x (k) → 2 1 (33) that x4(k) → 0. It now follows that f4(x(k)) = x4(k + x1 := V1. The fourth relationship in (9) along with (26) 1) goes to zero and, hence and the second condition in (27) imply that x4(k) → 0 and, recalling (24), we also have x (k) → x . 3 1 P (x(k))x2(k)x3(k) → 0 (34) Now suppose that β > 0. Since f1(x(k))f4(x(k)) → 0 equations (4a) and (4d) imply that Using the second relationship in (9) it follows from (31) and (34) that x2(k) → 0. The proof of (23) is the same as that P (x(k))x1(k)x3(k) → 0. (28) in Proposition 3.2

When α > 0, the fact that f2(x(k)) = x2(k + 1) → 0 and b) The special case: w13 = 1 and α, β = 0: In this (4b) also implies (28). Using (28) and (26) along with the special x(k) does not always converge to zero; only x2(k) ﬁrst relationship in (9) we obtain that x1(k) → 0, that is, and x4(k) always converge to zero. This can be seen by T x1 = 0. observing that, in this case, any state of the form [x1 0 x3 0] Now suppose that w24 < 1. Then f2(x(k))f4(x(k)) → 0 is an equilibrium state of system (3)-(4). If the system starts and it follows from (4b) and (4d) that (26) holds and as we in one of these equilibrium states it remains there. Such a have just shown this results in (22) and (23) with x1 = 0 state corresponds to all the population being healthy. when either α > 0 or β > 0. In a similar fashion one can show that if w23 < 1 then, When all wij are strictly less than one, the next result (22) and (23) hold with x1 = 0 when either α > 0 or β > 0. claims that the population exponentially decays to zero. Finally suppose w13 < 1. Then f1(x(k))f3(x(k)) → 0 Proposition 3.4: Consider system (3)-(4) with wij < 1 and it follows from (4a) and (4c) that (26) and (28) hold. for i = 1, 2 and j = 3, 4 and let From this we can conclude as before that (22) and (23) hold with x1 = 0 w¯ := max{wij} < 1. The next result tells us that even if wij = 1 for all ij, properties (22) and (23) still hold. To prove this we need to If the initial state x0 lies in X then, for all k ≥ 0, introduce a new function V2. Proposition 3.3: Consider system (3)-(4) with initial state k x1(k) + x2(k) 6 w¯ (x10 + x20) (35) x0 in X and 0 < wij ≤ 1 for i = 1, 2 and j = 3, 4. If k x3(k) + x4(k) 6 w¯ (x30 + x40) (36) α = β = 0 then (22) and (23) hold. k Proof: To prove this result we consider the behavior of xi(k) 6 w¯ (x10 + x20) for i = 1,..., 4.(37) the following function: Proof: It follows from (15) that

V2(x) := x2 + x4 (29) 1 ∆V1(x) 6 − (1 − w¯)(x1x3 + x1x4 + x2x3 + x2x4) V1 Along any solution x(·) we have (1 − w¯) = − [(x1 + x2)(x3 + x4)] V2(x(k + 1)) = V2(x(k)) + ∆V2(x(k)) (30) V1 = (−1 +w ¯)V1(x). and with α = β = 0 we have ∆V (x) = g (x) + g (x) It now follows from (14) that V1(x(k + 1)) 6 wV¯ 1(x(k)) 2 2 4 k h and, consequently, V1(x(k)) 6 w¯ V1(x(0)) which yields the = −P (x) (1 − w14)x1x4 + desired results. 3w i Remark 1: Note that the results in propositions 3.1 and (1 − w )x x + (2 − 24 )x x 23 2 3 2 2 4 3.4 are independent of the mutation rates α and β. Hence, 1 3w24 these results hold for any α and β in [0, ). −P (x)(2 − )x x 0. 2 6 2 2 4 6 Proceeding as in the proof of Propositions 3.2, we can show Now we derive system’ solutions upper and lower bounds that we must have and provide the exact solutions in some speciﬁc cases. Comments on the medical implications of the mathematical P (x(k))x2(k)x4(k) → 0, (31) results can be found in the Discussion Section.

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C. Lower bound on convergence values of x1 and x3 E. A special case

To obtain estimates of the values to which x1 and x3 Consider the case in which w13 = w14=w23=1 that converge, consider correspond to diseases where the fertility is altered only in couples formed by affected males and carrier females, that V3(x) = x1 + x3 − x2 − x4. is only w24 ranges in [0, 1]. In this case from (38) we can deduce that V4 is constant; hence Along any solution x(·) we have V3(x(k+1)) = V3(x(k))+ ∆V3(x(k)) where 1h w24 i x1 = x3 = x10 + x30 + (x20 + x40) , 2 4 − 3w24 ∆V3 = g1(x) + g3(x) − g2(x) − g4(x) which is positive for non-zero initial conditions. = P (x)(2 − w24)x2x4 Another interesting case is when w13 = 1, w14 in > 0. [0,1] and w23 = w24 = 0. This models the reproduction scenario for severe X-linked recessive diseases such as This guarantees that V does not decrease. If x and x are 3 1 3 the aforementioned hemophilia A and ectodermal dysplasia the values to which x and x converge, then 1 3 where healthy males and females always contribute to next generation, while healthy males and carrier women have 1 1 a variable reproduction capacity (w ) depending on the x = x V (x ) = [x + x − x − x ]. 14 1 3 > 2 3 0 2 10 30 20 40 disease gravity. Affected males do not contribute to next generation, independently from the females health condition For the above bound to be positive, it must hold as they rarely reach the reproduction age. For this combination of reproduction rates the exact solu- x + x x + x , 10 30 > 20 40 tion of system (3)-(4) is: that is in the initial population distribution the number of Qk i Pi (i−j) j healthy people must be greater or equal to the number i=1 2 x30 + j=1 2 w14x40 of affected and carrier one. This hypothesis is reasonable x1(k) = A1 Qk−1 2i−1x + Pi Qi−p−1 2 wp x and consistent with the epidemiological observation of the i=1 30 p=1 j=1 14 40 diseases. Qk−1 i Pi i−j j i=1 2 x30 + j=1 2 w14x40 x2(k) = A2 Qk−1 i−1 Pi Qi−p−1 p 2 x30 + 2 w x40 D. Upper bound on convergence values of x1 and x3 i=1 p=1 j=1 14 Qk 2ix + Pi 2(i−j)wj x To obtain an upper bound on x1 and x3 consider i=1 30 j=1 14 40 x3(k) = A1 Qk−1 i−1 Pi Qi−p−1 p 4 − 2w 2 x30 + 2 w x40 V (x) = V (x) + 24 V (x) i=1 p=1 j=1 14 4 3 4 − 3w 2 24 Qk−1 2ix + Pi 2i−jwj x w24 i=1 30 j=1 14 40 = x1 + x3 + (x2 + x4). x4(k) = A2 4 − 3w24 Qk−1 i−1 Pi Qi−p−1 p i=1 2 x30 + p=1 j=1 2 w14x40

Along any solution x(·) we have V4(x(k+1)) = V4(x(k))+ ∆V4(x(k)) where where

4 − 2w24 x10(x10 + x20 + x30 + x40) ∆V4 = ∆V3(x) + ∆V2(x) A1 = 2k 4 − 3w24 2 (x10 + x20)(x30 + x40) = P (x)(2 − w24)x2x4 − k w14x10x40(x10 + x20 + x30 + x40) 4 − 2w24 A2 = 2k . P (x)[(1 − w14)x1x4 + (1 − w23)x2x3] − 2 (x10 + x20)(x30 + x40) 4 − 3w24 4 − 2w24 3w24 P (x) 2 − x2x4 4 − 3w24 2 The demonstration is straightforward and can be easily 4 − 2w24 obtained substituting the previous solution in system (3)-(4). = − P (x)[(1 − w14)x1x4 + 4 − 3w24 (1 − w23)x2x3] (38) IV. SPECIALCASE: UNITARY FERTILITY FACTORS 6 0 In this section we consider a special case of model (3)-(4). This implies that V4 does not increase and yields the upper bounds: This case is obtained setting all reproduction rates equal to one (wij ≡ 1), thus modeling non-disabling diseases which 1h w24 i do not affect the fertility of affected/carrier couples. x1 = x3 6 x10 + x30 + (x20 + x40) 2 4 − 3w24 When wij = 1 for i = 1, 2 and j = 3, 4, the functions fi

ISSN: 1998-4510 12 INTERNATIONAL JOURNAL OF BIOLOGY AND BIOMEDICAL ENGINEERING Volume 11, 2017 in (4) simplify to Proof: The first part of the proposition can be straight- h 1 forwardly derived from the previous discussions on system f (x) = P (x) (1 − α)x x + x x + x x + 1 1 3 2 3 2 1 4 properties. 1 i Consider now any two pairs (z2, z4) and (¯z2, z¯4) with x x (40a) 2 2 4 z2, z4, z¯2, z¯4 in [0, 1] and let h 1 1 i f2(x) = P (x) αx1x3 + x1x4 + x2x4 (40b) z˜2 , z2 − z¯2 , z˜4 , z4 − z¯4 2 2 h 1 i and f (x) = P (x) (1 − β)x x + x x (40c) 3 1 3 2 1 4 h 1 i q˜2 , q2(z2, z4)−q2(¯z2, z¯4) , q˜4 , q4(z2, z4)−q4(¯z2, z¯4) f4(x) = P (x) βx1x3 + x2x3 + x1x4 + x2x4 (40d). 2 Using equation (44) one can easily show that It follows from Section III-A that x1(k + 1) + x2(k + 1) = 1 x (k) + x (k) for all k, that is, that the number of males, q˜ = − α z˜ − αz˜ + α(z z − z¯ z¯ ) (45a) 1 2 2 2 4 2 2 4 2 4 equal to the number of females at each generation is constant. 2N 1 Consider a population of individuals; the following q˜4 = − β z˜4 + (1 − β)˜z2 − constraints apply: 2 1 x (k) + x (k) = N for all k, (41a) − β (z z − z¯ z¯ ) . (45b) 1 2 2 2 4 2 4 x3(k) + x4(k) = N for all k; (41b) The term (z2z4 − z¯2z¯4) can be factorized as z˜2z4 +z ¯2z˜4; thus system dynamics can be rewritten using one state substitution in (45) gives variable of the male population (i.e. x or x ) and one 1 2 1 variable of the female population (i.e. x3 or x4); moreover, q˜ = −α(1 − z )˜z + − α(1 − z¯ ) z˜ 1 2 4 2 2 2 4 P (x) as given (5) simplifies to P = . N 1 1 We normalize the system states by introducing z = q˜4 = 1 − β − − β z4 z˜2 + − β (1 − z¯2)˜z4. T 2 2 [z1 z2 z3 z4] where x Recalling that 0 6 z2 6 1, z := i for i = 1,..., 4 i N 1 |q˜ | α(1 − z )|z˜ | + − α |z˜ | The evolution of z is governed by 2 6 4 2 2 4 1 z(k + 1) = h(z(k)) (42) α|z˜ | + − α |z˜ |. 6 2 2 4 T where h = [h1 h2 h3 h4] with Last inequality holds because of z2 and α bounds. Similarly 1 1 h1(z) = (1 − α)z1z3 + z2z3 + z1z4 + z2z4(43a) we have that 2 2 1 1 1 h (z) = αz z + z z + z z (43b) |q˜4| 6 (1 − β)|z˜2| + − β |z˜4| 2 1 3 2 1 2 2 2 4 2 1 β < 1 (z z − h3(z) = (1 − β) z1z3 + z1z4 (43c) when 2 . Note that different factorizations of 2 4 2 z¯ z¯ ) would give the same bounds for |q˜ | and |q˜ |. Using 1 2 4 2 4 h4(z) = βz1z3 + z2z3 + z1z4 + z2z4 . (43d) Lemma 7.1 in the appendix one can deduce there exists a 2 κ < 1 such that Exploiting constraints (41), formulas (43) can be rewritten as λ|q˜2| + |q˜4| 6 κ(λ|z˜2| + |z˜4|) (46) 1 1 − β 1 + 2β h (z) = q (z , z ) = −αz + − α z + αz z + α for λ ∈ , and for any α and β in [0, 1 ]. 2 2 2 4 2 2 4 2 4 1 − α 1 − 2α 2 Hence q = (q , q ) is a contraction in [0, 1] × [0, 1] with h4(z) = q4(z2, z4) 2 4 1 1 respect to the norm = (1 − β) z2 + − β z4 − − β z2z4 + β 2 2 V5(z2, z4) = λ|z2| + |z4|. h (z) = 1 − q (z , z ) 1 2 2 4 From this we conclude that q possesses an attractive fixed h (z) = 1 − q (z , z ) 3 4 2 4 point (¯z2, z¯4),that is The stability analysis of system (3)-(40) can be achieved by q2(¯z2, z¯4) =z ¯2, q4(¯z2, z¯4) =z ¯4. studying the stability properties of system (42)-(44). It follows from (46) that Proposition 4.1: The set {z|z +z = z +z } is invariant 1 2 3 4 V (˜z (k), z˜ (k)) ≤ κkV (˜z (0), z˜ (0)) for system z(k + 1) = h(z(k)) and contains a unique 5 2 4 2 4 exponentially globally stable equilibrium point. for all k ≥ 0. This yields the desired result.

ISSN: 1998-4510 13 INTERNATIONAL JOURNAL OF BIOLOGY AND BIOMEDICAL ENGINEERING Volume 11, 2017

V. DISCUSSION There are about 1,098 genes on the human X-chromosome. Most of them code for characteristics other than female anatomical traits. Many of the non-sex determining X- linked genes are responsible for abnormal conditions such as hemophilia A, Duchenne muscular dystrophy, fragile- X syndrome, some high blood pressure dysfunctionalities, congenital night blindness, G6PD deﬁciency, and the most common human genetic disorder, red-green color blindness. X-linked genes are also responsible for a common form of baldness referred to as “male pattern baldness”. Some of these diseases are severely disabling and affected (a) Trend of x1 and x3. people usually do not reach the reproduction age. Model (3)–(4) reproduces the severity of the X-linked recessive diseases through an appropriate choice of reproduction rates wij. Disabling diseases such as hemophilia A and Duchenne muscular dystrophy can be modeled by assigning small values or zero to reproduction rates w23 and w24. Less serious conditions (such as the red and green color blindness or male pattern baldness) do not cause premature death of affected males and these males usually reach reproduction age. System (3)–(4) could also be exploited to model these diseases by setting all wij to the same value.

However in these cases the number of affected women in (b) Trend of x2 and x4. the population should also be considered; in fact affected daughters can be born from affected fathers and carrier or Fig. 3. An example of state evolutions for hemophilia A. affected mothers; they can reach the reproduction age and contribute to the next generation. Model (3)–(4) does not consider affected woman thus it is more suitable to model the VI.CONCLUSIONS epidemiology of severe X-linked diseases. The development We have presented a discrete-time nonlinear model for of a model with a ﬁve vector state comprising the class of X-linked recessive diseases aiming at describing the spread affected woman is the object of current research. of such diseases in a population; the model takes into We present some numerical results obtained simulating the account the role of sporadic or de novo mutations on the distribution of hemophilia A disease, a hereditary bleeding inheritance pattern and distinct reproduction rates according disorder caused by a lack of the blood clotting factor VIII, to the health conditions of breeding couples. We analyzed a protein encoded by gene F8 on the X-chromosome. It is system properties and performed stability analysis of the largely an inherited disorder; affected males show a reduced equilibrium point using Lyapunov functions. Future studies reproduction capacity related to the severity of the disease will also need to take into account carrier females who could symptoms; carrier females do not usually show any sign contribute to disease spread in less severe diseases, as well of the disease ([15]). The spontaneous mutation rate of the as the effects on the population of control actions such as disease is very small; it has been approximately evaluated to prenatal diagnosis. be 2.67 · 10−5 for both males and females ([16]). Relying on clinical observations we assigned a reproduc- ACKNOWLEDGEMENTS tion rate of w13 = w14 = 1 to couples formed by a healthy male and a carrier woman, while we choose w23 = w24 = Authors are grateful to Emanuele Durante Mangoni, MD 0.62. The following initial values have been assigned PhD, from the Internal Medicine Unit, Second University of x(0) = [29323162 3275 29320437 6000]T Naples, Italy, for helpful discussions and literature sugges- tions on X-linked diseases and their inheritance pattern and The initial distribution of males and females (i.e. x1(0) and for his critical review of the manuscript. x3(0)) has been chosen according to the Italian population (58652874 in 2008 according to [17]). According to the VII.APPENDIX census in [18], 3275 males were affected by hemophilia A.

Simulation results are reported in Figure 3; note that, accord- Lemma 7.1: Consider two real-valued functions g1 and g2 ing to the theoretical results we demonstrated (Proposition of two real variables y1 and y2, and suppose that there are 3.3), the simulation depicts the extinction of the population; scalars aij > 0, i, j = 1, 2 with particularly this should happen in a very long period (105 generations). aii < 1 and a12a21 < (1 − a11)(1 − a22) ISSN: 1998-4510 14 INTERNATIONAL JOURNAL OF BIOLOGY AND BIOMEDICAL ENGINEERING Volume 11, 2017

such that for any (y1, y2), (y¯1, y¯2), [14] J. W. Drake, B. Charlesworth, D. Charlesworth, and J. F. Crow, “Rates of spontaneous mutation,” Genetics, vol. 148, pp. 1667–1686, 1998. |g˜1| 6 a11|y˜1| + a12|y˜2| (47a) [15] D. J. Bowen, “Haemophilia a and haemophilia b: molecular insights,” Molecular Pathology, vol. 55, pp. 1–18, 2002. |g˜2| 6 a21|y˜1| + a22|y˜2|. (47b) [16] J. Haldane, “The mutation rate of the gene for hemophilia, and its segregation ratios in males and females,” Annuals of Eugenics, vol. 13, where pp. 262–71, 1947. [17] http://www.istat.it/it/charts/popolazioneresidente, 2014, authors’ last y˜1 , y1 − y¯1 y˜2 , y2 − y¯2 visit on February 2014. g˜1 , g1(y1, y2) − g1(¯y1, y¯2)g ˜2 , g2(y1, y2) − g2(¯y1, y¯2) [18] M. R. Gualan, A. Sferrazza, C. Cadeddu, C. de Waure, F. D. Nardo, G. L. Torre, and W. Ricciardi, “Epidemiologia dell’emoﬁlia A nel Then for any mondo e in Italia,” Italian Journal of public health, vol. 8, no. 2, 2011, in italian. a 1 − a λ ∈ 21 , 22 1 − a11 a12 there exists a κ < 1 such that

λ|g˜1| + |g˜2| 6 κ(λ|y˜1| + |y˜2|). Proof: If λ > 0, combining inequalities (47) yields: a λ|g˜ | + |g˜ | a + 21 λ|y˜ | + (λa + a )|y˜ | 1 2 6 11 λ 1 12 22 2 a Deﬁne κ = max(κ , κ ) with κ = a + 21 and κ = 1 2 1 11 λ 2 λa12 +a22; clearly κ > 0 and the following inequality holds:

λ|g˜1| + |g˜2| 6 κ(λ|y˜1| + |y˜2|). (48) It is straightforward to verify that if

a21 1 − a22 a12a21 < (1−a11)(1−a22) and < λ < 1 − a11 a12 then κ1, κ2 < 1 ; hence κ < 1 and the desired result follows.

T Notice that the function g = [g1 g2] satisfying the hypotheses of the above lemma is a contraction under a suitable norm.

REFERENCES [1] J. H. Bennett and C. R. Oertel, “The approach to a random associations of genotypes with random mating,” Journal of Theoretical Biology, vol. 9, pp. 67–76, 1965. [2] C. Cannings, “Equilibrium, convergence and stability at a sex-linked locus under natural selection,” Genetics, vol. 56, pp. 613–618, 1967. [3] G. Palm, “On the selection model for a sex-linked locus,” Journal of Mathematical Biology, vol. 1, pp. 47–50, 1974. [4] D. Luenberger, Introduction to Dynamic Systems. John Wiley and Sons, 1979. [5] T. Nagylaki, “Selection and mutation at an x-linked locus,” Annals of Human Genetics, vol. 41, no. 2, pp. 241–248, 1977. [6] J. M. Szucs, “Selection and mutation at a diallelic X-linked locus,” Journal of Mathematical Biology, vol. 29, pp. 587–627, 1991. [7] I. Heuch, “Partial and complete sex linkage in inﬁnite populations,” Journal of Mathematical Biology, vol. 1, no. 4, pp. 331–343, 1975. [8] C. Del Vecchio, L. Glielmo, and M. Corless, “Equilibrium and stability analysis of x-chromosome linked recessive diseases model,” in Proc. IEEE 51st Annual Conference on Decision and Control (CDC), 2012, pp. 4936 – 4941. [9] ——, “Non linear discrete time epidemiological model for x-linked recessive diseases,” in 22nd Mediterranean Conference on Control and Automation, June, Palermo, Italy, 2014. [10] V. Dubowitz, Muscle Disorders in Childhood, Second Edition. Saun- ders, 1995. [11] A. W. F. Edwards, Foundations of Mathematical Genetics. Cambridge University Press, 2000. [12] W. J. Ewens, Mathematical Population Genetics I: Theoretical Intro- duction. Springer, 2004. [13] M. Lachowicz and J. Miekisz, From Genetics to Mathematics, ser. Series on Advances in Mathematics for Applied Sciences. World Scientiﬁc Publications, 2009, vol. 79. ISSN: 1998-4510 15