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

• 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.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

 Citation: • • Figure 1.  Inheritance pattern of X-linked recessive disease

Figure 2.  Region of attraction corresponding to wi parameters in Table 3

Figure 3.  Region of attraction corresponding to wi parameters in Table 3 using the two Lyapunov functions.

Figure 4.  Region of attraction of xB = (72.5, 26.5, 68.9) and system's parameters as in Section 5.2.

Figure 5.  Trajectories with different initial conditions and wi in (27)

Figure 6.  Trajectories converging to xB

Figure 7.  Static sensitivity analysis with respect to wi

Table 1.  X-linked recessive inheritance probabilities for sons

 PARENTS SONS father mother healthy$X^A Y$ affected$X^a Y$ $X^A Y$ $X^A X^A$ $1-\gamma$ $\gamma$ $X^A Y$ $X^A X^a$ $\frac{1}{2}(1-\gamma)$ $\frac{1}{2}(1+\gamma)$ $X^A Y$ $X^a X^a$ $0$ $1$ $X^a Y$ $X^A X^A$ $1-\gamma$ $\gamma$ $X^a Y$ $X^A X^a$ $\frac{1}{2}(1-\gamma)$ $\frac{1}{2}(1+\gamma)$ $X^a Y$ $X^a X^a$ $0$ $1$

Table 2.  X-linked recessive inheritance probabilities for daughters

 PARENTS DAUGHTERS father mother healthy$X^A X^A$ carrier$X^A X^a$ affected$X^a X^a$ $X^A Y$ $X^A X^A$ $(1-\gamma)^2$ $2\gamma(1-\gamma)$ $\gamma^2$ $X^A Y$ $X^A X^a$ $\frac{1}{2}(1-\gamma)^2$ $\frac{1}{2}(1+\gamma-2\gamma^2)$ $\frac{1}{2}\gamma(1+\gamma)$ $X^A Y$ $X^a X^a$ $0$ $1-\gamma$ $\gamma$ $X^a Y$ $X^A X^A$ $0$ $1-\gamma$ $\gamma$ $X^a Y$ $X^A X^a$ $0$ $\frac{1}{2}(1-\gamma)$ $\frac{1}{2}(1+\gamma)$ $X^a Y$ $X^a X^a$ $0$ $0$ $1$

Table 3.  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

Table 4.  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
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