Following infection to a host, some pathogens repeatedly alter their antigen expression, and thereby escape the immune defense (antigen drift/switching). This paper examines the evolutionarily stable mutation rate of pathogens which maximizes the stationary pathogen density in a host. Assumptions are: (i) most mutations are deleterious but a minor fraction, p, of mutations can contribute to the alternation of antigenic property of the pathogen; and. (ii) potential antigen types can be indexed in a one-dimensional lattice (the stepping-stone model). The model reveals that: (a) if the mutation rate is higher than a threshold mu(c) = R0/(1-p), where R0 is the per capita growth rate of pathogen before the immune system is activated, pathogens cannot maintain themselves because too many progeny are lost by lethal mutations; (b) if the mutation rate lies between zero and mu(c), the system converges to a traveling wave of antigen variants with a constant wave speed; (c) the evolutionarily stable mutation rate microESS is unexpectedly high: more than 0.25 per genome per replication even if most mutations are lethal. Hence more than a fourth of progeny are born defective in the evolutionarily stable state; (d) the microESS is even higher if multiple infections by pathogens are common. The paper also studies the evolutionarily stable mutation rate if every mutant antigen belongs to a different type (the infinite allele model), and the evolution of antigen switching between a finite number of antigen variants stored in the pathogen genome.