Start by assuming that only 1 dog was sick. The owner of the sick dog would see that 9 other dogs are healthy and comes to the conclusion that his own dog is sick (given the information that there is at least 1 dog sick). He would kill his dog on the first night which leads to a contradiction that the killing happens on the 4th night.
Assume now that there are 2 sick dogs. Without loss of generality between the two owners of the sick dog, the sick dog owner would see that there is 1 other sick dog among the nine other dogs. However, both sick dog owners do not know for a fact whether their own dogs are sick. Hence no killing happens on the first night. Again without loss of generality, after the first night the sick dog owner knows that if his own dog was healthy, the other sick dog owner would have killed his own dog on the first night as in the first scenario. Hence the sick dog owner comes to the conclusion that his own dog must also be sick and both dogs are killed on the second night.
By inductive reasoning, on the 4th night, 4 dogs are killed by their owners.