Word must have gotten out about the healing of Peter's mother-in-law because by evening Jesus was being flooded with cases of demon possession and illness. "He cast out the spirits with a word, and healed all who were ill"--implying that not all cases of sickness were believed to be caused by demons. The caricature of a person who believes in demons is that they automatically think all illness and misfortune can be blamed on them, but I don't think that wasn't even true of these first century folks. The Gospel writers link some sicknesses to demons but are silent about the cause of others. Whatever the case, we know from this Gospel account that demons do exist whether they choose to manifest their presence to us through illness or not. In our secular, materialist culture they probably prefer do their work more covertly, since people's unbelief allows them to get away with even greater evils than simply possessing individuals or inflicting them with illness.
Most Christians I know seem to fall into two extremes regarding demons. Either they are dismissive of the idea of demons, or they are overly obsessed with them. But here in this passage Jesus "cast out the spirits with a word," showing that he both acknowledges the reality of their evil work and is able to exercise his absolute power over them. By his continual confrontation with demons throughout his ministry, Jesus shows time and again that he views them as an enemy and a threat to the lost sheep he is seeking to gather to himself. Jesus does not ignore them or take them lightly. At the same time he is capable of utterly dominating them. There is no contest: a single word from his mouth sends them fleeing. He teaches us that we should acknowledge the presence of demons around us but there's no need to get fanatical about it. If we only hide ourselves in Jesus, we should be perfectly safe.
These stories of demon possessions and healings are designed to draw us into our own encounter with Jesus and see ourselves in the people he heals. At one time we were in bondage to Satan, but Jesus set us free with a command. At one time we were diseased with our sins and transgressions, but a healing touch from Jesus restored us to the health of new life. This is why Isaiah 53:4 is quoted here: "He himself took our infirmities and carried away our diseases." Because if you look at the original passage, the next verse (v. 5) goes on to say, "He was pierced through for our transgressions and he was crushed for our iniquities." The smooth transition Isaiah makes from infirmities and diseases to transgressions and iniquities shows that one is meant to symbolize the other. When Jesus heals a sick person in the Gospel accounts, the real malady is that they are sick with sin. Physical blindness is also spiritual blindness; physical deafness is also spiritual deafness; physical paralysis is also spiritual paralysis. In Isaiah's words Jesus carries those infirmities and diseases away, meaning he bears our sin all way to the cross where he is pierced through and crushed for them. It is a wonderful, seemingly impossible exchange. He takes our sickness, we take his health. He takes our sin, we take his righteousness. "[God] made him who knew no sin to be sin on our behalf, that we might become the righteousness of God in him" (2 Corinthians 5:21).