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Hello and thank you for using JustAnswer. I am sorry that your husband has a low libido and does not want to engage in sexual intercourse with you. I am not sure why his doctors refuse to do any tests on him, but because this issue started to occur around the time of his vasectomy leads one to believe that this is a physical issue and may not be a mental health issue. Your husband needs to find a urologist to perform the adequate tests on him including testing for blood flow to the penis and testicles and hormone levels. If physically he is fine, than his doctor may be able to prescribe either Wellbutrin or Requip which are medications that can increase Dopamine levels and increase sexual libido as well. I do agree that your husband does need to lose weight though as obesity can cause sexual dysfunction issues including low libido. If your husband starts to do more cardio and weight training this will also increase levels of testosterone which will increase libido. Ultimately though he needs to find a doctor, preferably a Urologist, to perform the appropriate tests to determine what can be the cause of these issues and then to approach treatment accordingly.
Sorry I misunderstood, I though you stated that this issue started after the vasectomy. Because this has been an ongoing issue than it could be a mental health issue, I would recommend that he see a psychiatrist for the medications I recommended and also that you both consider going to a Sex Therapist for Couples Therapy. Here is a good link to finding a Certified Sex Therapist in your area. This website means that they accomplished post-graduate training in Sex Therapy, so they went beyond the qualifications of their professional degree.