Infectious diseases have a huge impact on animal health, production and welfare, and human health. Understanding the role of host genetics in disease spread is important for developing disease control strategies that efficiently reduce infection incidence and risk of epidemics. While heritable variation in disease susceptibility has been targeted in livestock breeding, emerging evidence suggests that there is additional genetic variation in host infectivity, but the potential benefits of including infectivity into selection schemes are currently unknown. A Susceptible-Infected-Recovered epidemiological model incorporating polygenic genetic variation in both susceptibility and infectivity was combined with quantitative genetics selection theory to assess the non-linear impact of genetic selection on field measures of epidemic risk and severity. Response to 20 generations of selection was calculated in large simulated populations, exploring schemes differing in accuracy and intensity. Assuming moderate genetic variation in both traits, 50% selection on susceptibility required seven generations to reduce the basic reproductive number R0 from 7.64 to the critical threshold of <1, below which epidemics die out. Adding infectivity in the selection objective accelerated the decline towards R0 < 1, to 3 generations. Our results show that although genetic selection on susceptibility reduces disease risk and prevalence, the additional gain from selection on infectivity accelerates disease eradication and reduces more efficiently the risk of new outbreaks, while it alleviates delays generated by unfavourable correlations. In conclusion, host infectivity was found to be an important trait to target in future genetic studies and breeding schemes, to help reducing the occurrence and impact of epidemics.