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June  2017, 14(3): 755-775. doi: 10.3934/mbe.2017042

## Effects of selection and mutation on epidemiology of X-linked genetic diseases

 1 Dipartimento di Ingegneria, Università degli Studi del Sannio, Benevento (Italy), Piazza Roma, 21, Benevento 82022, Italy 2 School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran 3 School of Aeronautics and Astronautics, Purdue University, West Lafayette, Indiana, USA

* Corresponding author: fverrilli@unisannio.it

Received  March 02, 2016 Accepted  November 11, 2016 Published  December 2016

The epidemiology of X-linked recessive diseases, a class of genetic disorders, is modeled with a discrete-time, structured, non linear mathematical system. The model accounts for both de novo mutations (i.e., affected sibling born to unaffected parents) and selection (i.e., distinct fitness rates depending on individual's health conditions). Assuming that the population is constant over generations and relying on Lyapunov theory we found the domain of attraction of model's equilibrium point and studied the convergence properties of the degenerate equilibrium where only affected individuals survive. Examples of applications of the proposed model to two among the most common X-linked recessive diseases (namely the red and green color blindness and the Hemophilia A) are described.

Citation: Francesca Verrilli, Hamed Kebriaei, Luigi Glielmo, Martin Corless, Carmen Del Vecchio. Effects of selection and mutation on epidemiology of X-linked genetic diseases. Mathematical Biosciences & Engineering, 2017, 14 (3) : 755-775. doi: 10.3934/mbe.2017042
##### References:

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##### References:
Inheritance pattern of X-linked recessive disease
">Figure 2.  Region of attraction corresponding to wi parameters in Table 3
using the two Lyapunov functions.">Figure 3.  Region of attraction corresponding to wi parameters in Table 3 using the two Lyapunov functions.
B = (72.5, 26.5, 68.9) and system's parameters as in Section 5.2.">Figure 4.  Region of attraction of xB = (72.5, 26.5, 68.9) and system's parameters as in Section 5.2.
Trajectories with different initial conditions and wi in (27)
B">Figure 6.  Trajectories converging to xB
Static sensitivity analysis with respect to wi
X-linked recessive inheritance probabilities for sons
 PARENTS SONS father mother healthy$X^A Y$ affected$X^a Y$ $X^A Y$ $X^A X^A$ $1-\gamma$ $\gamma$ $X^A Y$ $X^A X^a$ $\frac{1}{2}(1-\gamma)$ $\frac{1}{2}(1+\gamma)$ $X^A Y$ $X^a X^a$ $0$ $1$ $X^a Y$ $X^A X^A$ $1-\gamma$ $\gamma$ $X^a Y$ $X^A X^a$ $\frac{1}{2}(1-\gamma)$ $\frac{1}{2}(1+\gamma)$ $X^a Y$ $X^a X^a$ $0$ $1$
 PARENTS SONS father mother healthy$X^A Y$ affected$X^a Y$ $X^A Y$ $X^A X^A$ $1-\gamma$ $\gamma$ $X^A Y$ $X^A X^a$ $\frac{1}{2}(1-\gamma)$ $\frac{1}{2}(1+\gamma)$ $X^A Y$ $X^a X^a$ $0$ $1$ $X^a Y$ $X^A X^A$ $1-\gamma$ $\gamma$ $X^a Y$ $X^A X^a$ $\frac{1}{2}(1-\gamma)$ $\frac{1}{2}(1+\gamma)$ $X^a Y$ $X^a X^a$ $0$ $1$
X-linked recessive inheritance probabilities for daughters
 PARENTS DAUGHTERS father mother healthy$X^A X^A$ carrier$X^A X^a$ affected$X^a X^a$ $X^A Y$ $X^A X^A$ $(1-\gamma)^2$ $2\gamma(1-\gamma)$ $\gamma^2$ $X^A Y$ $X^A X^a$ $\frac{1}{2}(1-\gamma)^2$ $\frac{1}{2}(1+\gamma-2\gamma^2)$ $\frac{1}{2}\gamma(1+\gamma)$ $X^A Y$ $X^a X^a$ $0$ $1-\gamma$ $\gamma$ $X^a Y$ $X^A X^A$ $0$ $1-\gamma$ $\gamma$ $X^a Y$ $X^A X^a$ $0$ $\frac{1}{2}(1-\gamma)$ $\frac{1}{2}(1+\gamma)$ $X^a Y$ $X^a X^a$ $0$ $0$ $1$
 PARENTS DAUGHTERS father mother healthy$X^A X^A$ carrier$X^A X^a$ affected$X^a X^a$ $X^A Y$ $X^A X^A$ $(1-\gamma)^2$ $2\gamma(1-\gamma)$ $\gamma^2$ $X^A Y$ $X^A X^a$ $\frac{1}{2}(1-\gamma)^2$ $\frac{1}{2}(1+\gamma-2\gamma^2)$ $\frac{1}{2}\gamma(1+\gamma)$ $X^A Y$ $X^a X^a$ $0$ $1-\gamma$ $\gamma$ $X^a Y$ $X^A X^A$ $0$ $1-\gamma$ $\gamma$ $X^a Y$ $X^A X^a$ $0$ $\frac{1}{2}(1-\gamma)$ $\frac{1}{2}(1+\gamma)$ $X^a Y$ $X^a X^a$ $0$ $0$ $1$
Parameters values
 N $\gamma$ $w_{13}$ $w_{14}$ $w_{15}$ $w_{23}$ $w_{24}$ $r$ scenario 1 $150$ $10^{-4}$ 0.5 0.45 0.1 1 0.9 11232 scenario 2 150 $10^{-4}$ 0.5 1 0.5 0.5 1 8284 scenario 3 150 $10^{-4}$ 0.63 1.4 0.14 0.7 1.5 2121
 N $\gamma$ $w_{13}$ $w_{14}$ $w_{15}$ $w_{23}$ $w_{24}$ $r$ scenario 1 $150$ $10^{-4}$ 0.5 0.45 0.1 1 0.9 11232 scenario 2 150 $10^{-4}$ 0.5 1 0.5 0.5 1 8284 scenario 3 150 $10^{-4}$ 0.63 1.4 0.14 0.7 1.5 2121
Parameters values for Lyapunov function in (14)
 $\alpha^*$ $\beta^*$ $\mu^*$ $\underline{x}_1$ scenario 1 0.0917 0.9999 0.9072 150 scenario 2 0.2486 0.3729 0.4953 150 scenario 3 0.0247 0.1359 0.1766 63.6
 $\alpha^*$ $\beta^*$ $\mu^*$ $\underline{x}_1$ scenario 1 0.0917 0.9999 0.9072 150 scenario 2 0.2486 0.3729 0.4953 150 scenario 3 0.0247 0.1359 0.1766 63.6
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