Modeling Diagnosis and Clinical Outcomes in Infertility Couples Considering Multiple Factors: A Retrospective Study

Infertility is defined as the failure to achieve pregnancy after 12 months of regular, unprotected sexual intercourse. Approximately 10 -15% of couples worldwide are affected by infertility. Males factors contribute to about 50% of cases, whereas female factors are the leading cause of secondary infertility. Common causes of female infertility include polycystic ovary syndrome (PCOS), premature ovarian failure, endometriosis, uterine fibroids, and endometrial polyps. Male infertility can be classified into testicular and post-testicular deficiencies ( 1 ). PCOS is the leading cause of infertility in women of reproductive age, affecting 70-80% of those diagnosed. The Rotterdam criteria are widely used for diagnosis ( 2 ), requiring at least two of the following features: oligo-ovulation or anovulation, hyperandrogenism, and polycystic ovarian morphology. PCOS may result from genetic and epigenetic alterations, hypothalamic-pituitary-ovarian/adrenal dysfunction, insulin resistance, inflammatory factors, obesity, and intrinsic differences in steroidogenesis. Management of subfertility associated with PCOS typically involves lifestyle modifications, pharmacological interventions, surgical procedures, and assisted reproductive technologies (ARTs) ( 3 ). ARTs are commonly recommended for couples seeking fertility treatment ( 4 ). Over the past four decades, ART has advanced significantly, resulting in more than 8 million births worldwide through ART or other fertility interventions ( 5 ). Intracytoplasmic sperm injection (ICSI) and in vitro fertilization (IVF) are distinct methods of fertilizing an oocyte. In IVF, an egg is placed in a Petri dish with surrounding sperm, whereas in ICSI, a single sperm is directly injected into the egg ( 6 ). ICSI represents a breakthrough in ART, particularly for overcoming low fertilization rates in conventional IVF (c-IVF) when sperm parameters are suboptimal. The National Institute for Health and Care Excellence (NICE) guidelines recommend ICSI primarily for couples with previous IVF failure or severe male factor infertility ( 7 ). Assessment of gamete and embryo is a critical component of any ART procedure. Embryo quality depends on the gamete quality and culture conditions. Various grading protocols are employed to evaluate embryos from cleavage to blastocyst stages, combining morphological assessments with invasive and non-invasive techniques such as genetic testing, proteomics, oxidative stress measurements of culture media, and morphokinetic analyses ( 8 ). Mathematical modeling can help researchers and clinicians better understand the causes and mechanisms of infertility, as well as develop and evaluate new interventions and treatment strategies. Infertility can impose significant social and psychological burdens on families, including increased risks of stress, anxiety, and marital discord. For instance, one study reported that 76% of infertile women experience anxiety and 56% report stress ( 9 ). The COVID-19 pandemic further exacerbated these challenges, as infertility treatments were abruptly halted following recommendations from the American Medical Association, adding to the psychological burden on affected couples. Several studies have investigated the impact of COVID-19 on the mental health of infertile women ( 6 ). Therefore, fundamental research and targeted interventions are essential to support the health of society in the context of infertility. Ghiasi Hafezi et al. ( 10 ) demonstrated, using a mathematical model, that ovarian reserve declines with age and that drug treatments become less effective over time, highlighting the need for IVF in such cases. A mathematical model for ovarian reserve ranking has been developed ( 11 ), which can help women assess their fertility potential based on ovarian reserve. Such modeling may guide couples in making informed decisions regarding conception and fertility treatments. Here, a new mathematical model for couples’ infertility is presented. Using the next-generation matrix, we calculate the basic reproduction number and derive sufficient conditions for the local stability of disease-free equilibrium points. We also demonstrate the existence of an endemic equilibrium point. By applying Castillo-Chavez and Song’s theorem, the occurrence of backward bifurcation is established. Furthermore, we show that Hopf bifurcation does not occur when male factor infertility is treated with IVF and ICSI are used.

We proceed with the numerical solution of the dynamic model, assuming that the system of differential equations is defined as follows: where vector F is defined as: By defining an appropriate step size h, the time interval [a,b] is divided into N equal subintervals, yielding: Now we define: Considering the fundamental parameters and variables listed in Table 1, the system can be expressed as: The following diagrams were generated using MAT- LAB software : In Figure S1 (See Supplementary Online Information at www.ijfs.ir ), three groups with different body mass indices (BMI) were considered: 1. Group 1: BMI<24.9 ( Fig .S1A , See Supplementary Online Information at www.ijfs.ir ), 2. Group 2: 25<BMI30 ( Fig .S1C , See Supplementary Online Information at www.ijfs.ir ). Patients in group 3 exhibited a lower frequency of β-hCG tests ( K 1 curve) and a reduced frequency of clinical pregnancy ( F 1 curve) compared to the other groups. Additionally, the number of recovered couples (R curve) was lower in this group than in groups 1 and 2. Conversely, patients in group 2 demonstrated better embryo quality ( G 1 , G 2 , G 3 curves), higher rates of positive βHCG test results (K1 curve), more frequent clinical pregnancy outcomes ( F 1 curve), and an increased number of recovered couples (R curve) compared to group 1 and 3. The rate of abortion and the likelihood of returning to the treatment cycle were higher in group 3 than in groups 1 and 2. Previous studies have reported similar findings. Liu et al. ( 13 ) showed that obesity or overweight reduces embryo quality and quantity, potentially affecting early fetal devel- opment. Other meta-analyses indicated that women with BMI≥30 kg/m 2 have higher cycle cancellation rates, fewer high-quality embryos, increased abortion rates, fewer re- trieved oocytes, and lower live birth rates (LBR) ( 14 - 17 ). Obese women may also require higher gonadotropin doses and longer stimulation periods prior to IVF ( 18 ). In Figure S2 (See Supplementary Online Information at www.ijfs.ir ), two groups are defined based on anti-Müllerian hormone (AMH) levels: 1. Group 1: AMH3.5 ( Fig .S2B , See Supplementary Online Information at www.ijfs.ir ). The number of recovered couples (R curve), the frequency of positive β-hCG tests ( K 1 curve), and positive clinical pregnancy outcomes ( F 1 curve) were substantially higher in group 2 compared to group 1. Figure S2 (See Supplementary Online Information at www.ijfs.ir ) shows that group 2 achieved better overall outcomes than group 1. Embryo quality of grade B ( G 2 curve) in both groups was higher than that of embryos A and C ( G 1 , G 3 curves). However, the rates of abortion and return to the treatment cycle were higher in group 2 than in group 1 (). To clinically assess ovarian reserve, serum AMH levels are measured. AMH is secreted by granulosa cells and is strongly correlated with follicle number and ovarian reserve ( 19 ). According to the National Institute for Health and Care Excellence (NICE), AMH≤5.4 pmol/l (0.8 ng/ mL) predicts a lower response to ovarian hyperstimulation, whereas levels≥ 25.0 pmol/l (3.6 ng/mL) are associated with more successful fertility outcomes ( 20 ). In Figure S3 (See Supplementary Online Information at www.ijfs.ir ), embryo transfer was analyzed across three groups: 1. One embryo transferred ( Fig .S3A , See Supplementary Online Information at www.ijfs.ir ), 2. Two embryos transferred ( Fig .S3B , See Supplementary Online Information at www.ijfs.ir ), 3. Three embryos transferred ( Fig .S3C , See Supplementary Online Information at www.ijfs.ir ). The rate of abortion and return to the treatment cycle for patients in group 1 was 1.8 times that of patients in group 3. Conversely, the frequency of patients in group 3 was 1.2 times that of the patients in group 1. Thus, a higher number of transferred embryos is associated with higher pregnancy rates and a lower probability of miscarriage. The frequency of patients with embryo quality A ( G 1 curve) in group 1 was 1.11 times that of group 3, whereas the frequency of patients with a positive clinical pregnancy outcome ( F 1 curve) in group 3 was 2.1 times that of group 1, indicating that increasing the number of transferred embryos improves positive clinical pregnancy outcomes. Group 2 exhibited the most favorable outcomes, with the highest frequency of recovered couples (fertile patients with no abortion, R curve). Both groups 2 and 3 had the highest frequency of embryos with quality B ( G 2 curve) compared to A and C. In contrast, group 1 had the highest frequency of embryos with quality A ( G 1 curve) relative to B and C ( G 1 , G 3 curves). The American Society for Reproductive Medicine (ASRM) has established guidelines for embryo transfers to promote healthy singleton gestation and improve live birth rates. Currently, a maximum of three untested cleavage-stage embryos may be transferred in patients aged 38-40 years, while patients aged 41-42 years should receive no more than three blastocysts or four untested cleavage-stage embryos ( 20 ). Since 2013, elective sin- gle-embryo transfer (eSET) has been recommended for women younger than 38 years, reducing multiple gestations without significantly impacting live birth rates. In women aged ≤42 years, single euploid embryo transfer reduces the risk of twins while achieving pregnancy rates comparable to transferring two untested blastocytes ( 21 ). In Figure S4, the cause of infertility was diagnosed as PCOS. Three treatment methods were considered: 1. IVF ( Fig .S4A , See Supplementary Online Information at www.ijfs.ir ), 2. ICSI ( Fig .S4B , See Supplementary Online Information at www.ijfs.ir ), 3. IVF combined with ICSI ( Fig .S4C , See Supplementary Online Information at www.ijfs.ir). In the second method (ICSI), the frequency of patients with embryo of quality C, B, and A (red, yellow, and black curves) was higher than in the other two methods. The number of recovered couples ( R curve) in the second method increased rapidly and then decreased, whereas in the first method (IVF), the increase was slower, reaching its maximum after 25 months. This indicates that ICSI achieves results more quickly. The rate of abortion and return to the treatment cycle for this cause of infertility was 7.8 times higher in the first method and 6.4 times higher in the third method compared to ICSI, suggesting that ICSI is the most effective approach. These findings are consistent with previous studies. Drakopoulos et al. ( 22 ) reported no difference in pregnancy rates between PCOS patients and controls undergoing ICSI. Liu et al. ( 23 ) showed that sibling oocyte insemination via IVF in PCOS patients resulted in lower fertiliza- tion rates than ICSI, suggesting that ICSI may overcome IVF-related low fertility in PCOS patients. Yin et al. ( 6 ) observed a slightly lower fertility rate with IVF in PCOS patients compared to controls, but a significantly higher rate during ICSI cycles. Liu et al. ( 23 ) also indicated that decreased fertilization rates in IVF might be due to zona pellucida abnormalities in PCOS patients, which ICSI can bypass. In Figure S5 (See Supplementary Online Information at www.ijfs.ir ), the diagnosed cause of infertility was male factor. Three treatment methods were considered: 1. IVF ( Fig .S5A , See Supplementary Online Information at www.ijfs.ir ), 2. ICSI ( Fig .S5B , See Supplementary Online Information at www.ijfs.ir ), 3. IVF combined with ICSI ( Fig .S5C , See Supplementary Online Information at www.ijfs.ir ). In the first method (IVF), both the abortion rate and the percentage of couples with a negative βHCG test were zero. For the second method (ICSI), the rates of abortion and return to the treatment cycle were 24 times higher than in the third method. The frequency of recovered couples ( R curve) in the second method was 1.6 times that of the first method and 3.06 times that of the third method. Similarly, the frequency of couples with a positive clinical pregnancy ( F 1 curve) was 2.5 times higher than the first method and 4.5 times higher than the third method. The frequency of couples with a positive β-hCG test (K_1 curve) in the second method was 2.6 times higher than the first method and 4.7 times higher than the third method. These results indicate that ICSI is the most effective method for treating male factor infertility. Consistent with these findings, ICSI was originally developed to improve outcomes in male factor infertility and in couples with previous IVF failure due to unexplained semen abnormalities. Its safety and efficacy have been demonstrated in numerous studies, showing that ICSI can overcome various types of male factor infertility. More recently, ICSI has become widely used for treating different etiologies of infertility, including advanced maternal age, preimplantation genetic testing (PGT), low oocyte yield, and fertilization after in vitro maturation (IVM) ( 24 ). In Figure S6 (See Supplementary Online Information at www.ijfs.ir ), the diagnosed cause of infertility was a combination of male factor and PCOS. Three treatment methods were considered: 1. IVF ( Fig .S6A , See Supplementary Online Information at www.ijfs.ir ), 2. ICSI ( Fig .S6B , See Supplementary Online Information at www.ijfs.ir ), 3. IVF combined with ICSI ( Fig .S6C , See Supplementary Online Information at www.ijfs.ir ). The frequency of recovered couples ( R curve) in the second method (ICSI) was 1.6 times higher than the first method and 3.06 times higher than the third method. Similarly, the frequency of couples with a positive clini- cal pregnancy ( F 1 curve) in the second method was 2.5 times that of the first method and 4.5 times that of the third method. The frequency of couples with a positive βHCG test (K1 curve) in the second method was 2.6 times higher than the first method and 4.7 times higher than the third method. These results indicate that ICSI is the most effective treatment method for couples with combined Male Factor and PCOS infertility.

The application of dynamical systems modeling to infertility treatment provides a powerful quantitative framework for synthesizing clinical observations, validating established protocols, and uncovering the nonlinear interactions that define patient journeys. Our model’s results offer more than a retrospective summary of data; they generate a dynamic, systems-level perspective on how intrinsic patient factors and deliberate treatment choices converge to dictate outcomes. The findings robustly corroborate a wealth of clinical evidence while also providing novel theoretical insights into the stability of treatment pathways. Firstly, the model quantifies the profound and graded impact of key physiological determinants. The analysis of body mass index (BMI) groups visualizes a clear therapeutic gradient. Patients with a BMI between 25 and 29.9 demonstrated the most favorable outcomes across embryo quality, positive β-hCG rates, clinical pregnancy, and final recovery. In stark contrast, the group with a BMI >30 consistently underperformed, exhibiting lower recovery rates and a higher probability of abortion and return to treatment. This mathematically reinforces the extensive clinical literature on the deleterious effects of significant obesity on oocyte quality, endometrial receptivity, and live birth potential, underscoring the critical importance of pre-treatment lifestyle intervention as a foundational step to improve ART efficiency ( 13 , 14 , 16 , 17 ). The analysis of ovarian reserve, via AMH, presents a more nuanced picture. As anticipated, the group with higher AMH (>3.5) showed superior performance in recovery and pregnancy rates. However, the model intriguingly indicated higher abortion and return-to-treatment rates in this group. This may reflect the clinical reality that high responders, often younger or with PCOS traits, can face challenges such as poorer oocyte quality or a suboptimal endometrial environment despite a robust quantitative response ( 6 , 23 ). This finding serves as a crucial reminder that while AMH is an excellent predictor of follicle number, it is not a sole arbiter of qualitative success, underscoring the need for a holistic patient assessment ( 8 , 19 ). Secondly, the model provides a compelling quantitative resolution to the classic embryo transfer conundrum: balancing the maximization of pregnancy rates against the risks of multiple gestation. While transferring three embryos improved initial pregnancy metrics, the group with two embryos transferred achieved the highest frequency of "recovered couples", the optimal endpoint of a healthy, singleton live birth. This perfectly encapsulates modern clinical reasoning and validates guidelines advocating for eSET in good-prognosis patients and judicious limitation of embryo numbers in others ( 20 , 21 ). The model demonstrates that more is not always better for the ultimate goal of a healthy, sustainable pregnancy. Thirdly, and perhaps most clinically actionable, are the model's etiology-specific validations of treatment strategies. For Male Factor infertility, the results are unequivocal, with ICSI demonstrating dramatic superiority across all positive outcome curves. This is a robust mathematical affirmation of established practice for severe male factor infertility ( 4 , 18 ). In PCOS, ICSI emerged as the most effective method, associated with better embryo quality, a faster recovery trajectory, and lower rates of abortion and cycle repetition compared to IVF. This supports the hypothesis that oocyte-level dysfunctions in PCOS can impair standard fertilization, a barrier that ICSI directly bypasses ( 2 , 7 , 10 ). Most significantly, the model's analysis of combined male factor and PCOS infertility offers critical guidance for complex cases. Here, ICSI outcomes were over- whelmingly superior. This suggests that in the presence of both conditions, a "double barrier" exists, making ICSI the most rational first-line strategy to optimize efficiency and success from the outset, rather than resorting to it only after failed conventional fertilization ( 7 , 18 ). Finally, the model's theoretical conclusions provide a novel conceptual framework for infertility treatment. The identification of backward bifurcation is particularly profound. It indicates that simply reducing the "force of infertility" below a threshold is insufficient to guarantee treatment success if the system has passed a critical point. Clinically, this translates to the understanding that interventions must be both timely and robust enough to shift a couple out of a self-perpetuating state of treatment failure, a concept aligned with modern approaches to personalized treatment protocols ( 10 , 22 ). The direct link established between the basic reproduction number (R₀) and the recovery rate (ξ) mathematically proves that any factor which enhances the chance of a viable pregnancy, be it weight optimization ( 13 , 16 ), selecting ICSI for male factor ( 4 , 18 ), or tailoring treatment to ovarian reserve ( 11 , 19 ), is fundamental to controlling the overall trajectory toward recovery. In conclusion, this mathematical model successfully bridges theoretical dynamics and clinical reality. It quantifies the impact of patient characteristics, provides a dynamic rationale for evidence-based policies, and offers strong, data-driven validation for personalized treatment selection. By revealing the structural behavior of the system, such as backward bifurcation, it underscores the importance of precise, early, and effective intervention to guide couples toward a stable state of recovery, moving beyond empirical patterns to a deeper understanding of the pathways to success in infertility care

In conclusion, this mathematical model successfully bridges theoretical dynamics and clinical reality. It quan- tifies the impact of patient characteristics, provides a dy- namic rationale for evidence-based policies, and offers strong, data-driven validation for personalized treatment selection. By revealing the structural behavior of the sys- tem, such as backward bifurcation, it underscores the im- portance of precise, early, and effective intervention to guide couples toward a stable state of recovery, moving beyond empirical patterns to a deeper understanding of the pathways to success in infertility care Furthermore, when male factor infertility is treated us- ing IVF or IVF combined with ICSI, no Hopf bifurcation is observed; that is, no periodic solutions emerge or van- ish as a result of changes in the stability of a fixed point

Dynamical models are mathematical constructs that provide simplified representations of complex real-world systems. Their defining characteristic is the description of temporal changes in system properties. Given the necessity of mathematical modeling for understanding disease progression, such models are fundamental to many studies in biology and medicine. In this context, the present paper introduces a dynamical model designed to describe the dynamics of couple infertility. In this section, we introduce some mathematical concepts employed throughout this article. The Routh-Hurwitz stability criterion states that, for a system with a characteristic equation of the form: the Hurwitz matrix is defined as follows: ⋮ where ɑ j =0 if j > n . All roots of the polynomial P( λ ) have negative real parts, or are negative, if and only if the determinants of all Hurwitz matrices are positive: The system is asymptotically stable if all Δ j (j=1,..., n ) are positive. Liénard and Chipart demonstrated that, if all coefficients in the characteristic equation are positive ( ɑ i >0 for i =1,..., n ), then the system is asymptotically stable provided that either of the following conditions is satisfied: or Now, the following theorem presents the conditions for the occurrence of backward bifurcation ( 12 ). Consider a general system of ordinary differential equations (ODEs) with a parameter φ : and A1: Let denote the linearization matrix of system (2.2) around the equilibrium at 0, with φ evaluated at 0. Zero is a simple eigenvalue of A , and all other eigenvalues of A have negative real parts. A2: The matrix A has a nonnegative right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue. Let f k denote the kth component of f , and The local dynamics of system (2.2) around 0 are completely determined by the parameters a and b: i. ɑ >0, b >0. When φ<0, |φ|«1, 0 is locally asymptotically stable, and a positive unstable equilibrium exists. When 0< φ «1, 0 becomes unstable, and a negative locally asymptotically stable equilibrium appears. ii. ɑ <0, b <0. When φ <0, | φ |«1, 0 is unstable. When 00, b <0. When φ <0, | φ |«1, 0 is unstable, and a negative locally asymptotically stable equilibrium appears. When 0< φ «1, 0 is stable, and a positive unstable equilibrium emerges. iv. a 0. When φ changes from negative to positive, 0 transitions from stable to unstable. Correspondingly, a negative unstable equilibrium becomes positive and locally asymptotically stable. Definition 2.3: Let A=[ɑ i j ] be a matrix n×n with real or complex entries. Suppose that the second additive com pound matrix A [2] of A is an ( n 2 )× ( n 2 ) matrix, with entries defined as follows: where the symbol ( i )=( i 1 , i 2 ) denotes the i-th element in the lexicographic ordering of integers such that 1≤ i 1 < i 2 ≤ n . The following theorem demonstrates that the system does not admit any periodic solutions and, consequently, that no Hopf bifurcation occurs. Consider the following system: Suppose that x = x ( t,x 0 , μ 0 ) is a solution of equation (2.4), where x 0 εℝ n and μ 0 εℝ. We now consider the following linear system: where ∂ f [2] ∂ x denotes the second additive compound matrix of the Jacobian of equation (2.4). Suppose that at least one of the following inequalities holds: and μ ( - ∂ f [ 2 ] ∂ x ) are also defined similarly. Then, the system (2.4) does not admit any periodic solutions. In this article, a new mathematical model was developed to investigate the dynamics of infertile couples with PCOS, Male factor infertility, or a combination of both. Infertile patients were referred to the Safaeih Infertility Center of Yazd, affiliated with Yazd University of Medical Sciences. The study flowchart is presented in Figure 1 . Flow chart of this study. β-hCG; Beta-human chorionic gonadotropin and PCOS; Polycystic ovarian syndrome. For modeling purposes, the population was divided into eight classes. Couples whose cause of infertility had not yet been determined upon referral to the clinic were considered susceptible individuals, denoted by S(t). After the necessary examinations, the cause of infertility is determined, which may be PCOS, Male factor, or both. Once the cause is identified, couples are assigned to class I. Treatment is then initiated, fertilization occurs in the laboratory, and the embryos are formed. The resulting embryos are classified according to quality as A, B, or C, corresponding to classes G 1 ( t ), G 2 ( t ), and G 3 ( t ) respectively. If the beta human chorionic gonadotropin (β-hCG) the test is positive, individuals are placed in class K 1 ( t ); otherwise, they enter class D. Couples in class K 1 ( t ) are sub sequently evaluated for clinical pregnancy. Those with a positive result move to class F 1 ( t ), while the remainder enter class D. Couples complete treatment successfully and achieve fertility are assigned to class R(t) . The progression of individuals through these stages is illustrated in Figure 2 . Variables and parameters are defined in Table 1 . In Figure 2 , a schematic representation of the mathematical model using a system of ODEs to simulate infertility treatment dynamics in a population of couples. The model includes eight classes, illustrating the flow of individuals between these classes over time. Parameters used in the model are indicated in the diagram. Architecture of the infertility model. Description of variables and parameters in the model β-hCG; Beta human chorionic gonadotropin, PCOS; Polycystic ovarian syndrome, and ART; Assisted reproductive technology. The dynamic model for infertility diagnosis is formulated as follows: The basic reproduction number R_0 is calculated using the next-generation matrix as follows: where The system (3.1) has a disease-free equilibrium point E 0 = ( L m , 0 , 0 , 0 , 0 , 0 ) . The Jacobian matrix evaluated at an arbitrary point E=( S, I, G 1 , G 2 , G 3 , K 1 , F 1 ) is given by: where The disease-free equilibrium point E_0 is asymptotically stable if R 0 <1. Conversely, if R 0 <1, the equilibrium is unstable. The Jacobian matrix evaluated at E 0 is as follows: The matrix J 0 has an eigenvalue λ 1 =- m , while the reGhasemabadi et al. maining eigenvalues are the roots of the characteristic equation: The constants p i ,(i=1,...,6) are provided in the appendix A (See Supplementary Online Information at www.ijfs. ir ). As shown therein, p i >0 (i=1,...,6) and the inequalities Δ 1 >0, Δ 3 >0 and Δ 5 >0 are satisfied. Therefore, if R 0 1 ,the equilibrium is unstable. The system (3.1) has an endemic equilibrium point where An endemic equilibrium point exists when R 0 >1. A backward bifurcation in the system reflects the dynamics of infected individuals within or following the recovered class. In this section, based on the Castillo-Chavez and Song theorem (2.2), we demonstrate that a backward bifurcation occurs in system (3.1) when R 0 =1. From the condition R 0 =1, the parameter β * can be derived as follows: The Jacobian matrix J 0 ( E 0 , β *) admits two explicit eigenvalues, λ 1 =0 and λ 2 =˗ m , where the zero eigenvalue is simple. All remaining eigenvalues are determined implicitly as roots of the following characteristic equation: where p 1 , p 2 , p 3 , p 4 and p 5 are defined in (7.1)-(7.5). Following the approach used in Theorem 3.1, it can be shown that all roots of equation (3.8) have negative real parts. Therefore, condition (A1) of the Castillo-Chavez theorem is satisfied. Let w=(w 1 , w 2 , w 3 , w 4 , w 5 , w 6 , w 7 ) be the right eigenvector of J 0 ( E 0 , β *) corresponds to the eigenvalue of λ 1 =0, where , Let v=(v 1 , v 2 , v 3 , v 4 , v 5 , v 6 , v 7 ) be the left eigenvector of J 0 (E 0 , β*) corresponds to the eigenvalue of λ 1 =0, where Now, the quantities a and b are computed as follows: We observe that a and b are positive. Therefore, a sub- critical backward bifurcation occurs. Increasing the recovery rate while keeping other variables constant reduces the basic re- production number R_0 and helps control infertility in couples. The backward bifurcation is associated with the recovery class. One application of Hopf bifurcation in infertility re- search is to investigate how certain conditions can lead to irregular menstrual cycles in women. For example, some models of PCOS may exhibit Hopf bifurcation and cha- otic dynamics. In the following theorem, we demonstrate that Hopf bifurcation does not occur when the cause of infertil- ity is a male factor and the treatment method used is IVF. Suppose that E=(S, I, G 1 , G 2 , G 3 , K 1 , F 1 ) is an equilibrium point for the system (4.1), where Consequently, the system does not exhibit any peri- odic solution, and therefore Hopf bifurcation does not occur. By the definition of μ i (i=1,..., 21 ) in Theorem 2.4, we have: where μ is defined as: Therefore, the system has no periodic solution, and consequently no Hopf bifurcation occurs. In the following theorem, we show that Hopf bifurcation does not occur when the cause of infertility is Male factor and the treatment methods used are IVF and ICSI. Suppose that E=(S, I, G 1 , G 2 , G 3 , K 1 , F 1 ) is an equilibrium point for the system (4.1) where Then, the system has no periodic solution, and conse- quently, no Hopf bifurcation occurs. The proof follows similarly to Theorem 3.3.