Moral evil began with the rejection of God by mankind at its origin. God warned our first parents not to eat from the tree of knowledge of good and evil thus, "You shall not eat from it or touch it, or you will die." They went ahead and decided to live by their own rules followed by its consequences, "Because from it you were taken; For you are dust, And to dust you shall return."(Gn 3:19) "What man lost by the Fall was his original specific nature. ...The total organism which had been taken up into his spiritual life was allowed to fall back into the merely natural condition from which, at his making, it had been raised." (C.S. Lewis, The Problem of Pain, HarperSanFrancisco, 1940, p. 78) And so from that day on man experiences death by aging, sickness, animal and human predation, and natural disasters. Note: in Genesis God does not specify how man would die but that he would die. The immediate cause of human death is not God but man.
As with regard to natural disasters much of the death occurs because of what we have done and do. We built cities and rebuild them on known fault zones such as San Francisco. Naples is within striking distance of Mt. Vesuvius near which will eventually explode and cause much death. Nine percent of the world population is within striking distance of n exploring volcanos. Again since we like the sea we build close to its shores within reach of devastating tsunamis. Knowing or unknowingly we cause global warming with all its devastating effects on the planet.
So, why does God allow natural disasters?
"For one thing, such events shake our confidence in this life and force us to think about eternity. Churches are usually filled after disasters as people realize how tenuous their lives really are and how life can be taken away in an instant. What we do know is this: God is good! Many amazing miracles occurred during the course of natural disasters that prevented even greater loss of life. Natural disasters cause millions of people to reevaluate their priorities in life. Hundreds of millions of dollars in aid is sent to help the people who are suffering. Christian ministries have the opportunity to help, minister, counsel, pray, and lead people to saving faith in Christ! God can, and does, bring great good out of terrible tragedies." (Website: GotQuestions.org) The eternal salvation of man is the aim of God in allowing natural disasters.