ホスト体内におけるウイルスの進化によって、ある量が特定の方向に変化するという 命題が、広い範囲で成りたつ．つまり突然変異が広がり元のタイプに加わったり置き 換わったりするときには、未感染細胞の数は単調に減少し、全感染力もしくは全細胞 ビルレンスが増大する．これは、交叉免疫がなければ、免疫が感染細胞を殺す場合と 感染効率を低下させる場合の両方で成立する．交叉免疫があると反例を作ることがで きるが、このことは病気がより重篤になる程度が頭打ちになることを示唆する．免疫 が感染細胞を殺すモデルでは、リアプノフ関数を用いて大域安定が証明ができる．

Fisher's fundamental theorem states that the average fitness of a population increases due to natural selection. This is true if fitness is constant, but does not hold for general frequency dependent fitness. Here we analyze models for the evolutionary dynamics of viral or other infectious agents within a host. We study how the invasion of a new strain affects the composition and diversity of the viral population. We show that under strain specific immunity the equilibrium abundance of uninfected cells declines during viral evolution. In addition, for cytotoxic immunity the absolute force of infection, and for non-cytotoxic immunity the absolute cellular virulence increases during viral evolution. We prove global stability by means of Lyapunov functions. Our сfundamental theorem of immune selection・ does not hold for general cross-reactive immune responses, which introduce frequency dependent selection among viral strains. Therefore, appropriate cross-reactive immunity can lead to a viral evolution within a host which limits the extent of the disease.