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The worldwide healthcare and economic crisis caused by the COVID-19 pandemic highlights the need for a deeper understanding of investing in the mitigation of epidemic risks. To address this, we built a mathematical model to optimize investments into two types of measures for mitigating the risks of epidemic propagation: prevention/containment measures and treatment/recovery measures. The new model explicitly accounts for the characteristics of networks of individuals, as a critical element of epidemic propagation. Subsequent analysis shows that, to combat an epidemic that can cause significant negative impact, optimal investment in either category increases with a higher level of connectivity and intrinsic loss, but it is limited to a fraction of that total potential loss. However, when a fixed and limited mitigation investment is to be apportioned among the two types of measures, the optimal proportion of investment for prevention and containment increases when the investment limit goes up, and when the network connectivity decreases. Our results are consistent with existing studies and can be used to properly interpret what happened in past pandemics as well as to shed light on future and ongoing events such as COVID-19.