Combination therapies have shown remarkable success in preventing the evolution of resistance to multiple drugs, including HIV, tuberculosis, and cancer. Nevertheless, the rise in drug resistance still remains an important challenge. The capability to accurately predict the emergence of resistance, either to one or multiple drugs, may help to improve treatment options. Existing theoretical approaches often focus on exponential growth laws, which may not be realistic when scarce resources and competition limit growth. In this work, we study the emergence of single and double drug resistance in a model of combination therapy of two drugs. The model describes a sensitive strain, two types of single-resistant strains, and a double-resistant strain. We compare the probability that resistance emerges for three growth laws: exponential growth, logistic growth without competition between strains, and logistic growth with competition between strains. Using mathematical estimates and numerical simulations, we show that between-strain competition only affects the emergence of single resistance when resources are scarce. In contrast, the probability of double resistance is affected by between-strain competition over a wider space of resource availability. This indicates that competition between different resistant strains may be pertinent to identifying strategies for suppressing drug resistance, and that exponential models may overestimate the emergence of resistance to multiple drugs. A by-product of our work is an efficient strategy to evaluate probabilities of single and double resistance in models with multiple sequential mutations. This may be useful for a range of other problems in which the probability of resistance is of interest.

组合疗法在预防对多种药物（包括艾滋病毒，结核病和癌症）的耐药性发展方面已显示出显著成功。然而，耐药性的升高仍然是一个重要的挑战。准确预测对一种或多种药物产生抗药性的能力可能有助于改善治疗选择。现有的理论方法通常关注指数增长定律，当稀缺的资源和竞争限制增长时，这可能是不现实的。在这项工作中，我们研究了两种药物联合治疗模型中单药耐药和双药耐药的出现。该模型描述了敏感菌株，两种单抗菌株和双抗菌株。我们比较了三种增长规律出现阻力的可能性：指数增长，没有品系间竞争的逻辑增长，以及品系间竞争的逻辑增长。使用数学估计和数值模拟，我们表明，当资源稀缺时，品系间竞争只会影响单个抗性的出现。相反，在更广泛的资源可用性空间中，两次抗争的可能性受到菌株间竞争的影响。这表明不同耐药菌株之间的竞争可能与确定抑制耐药性的策略有关，并且指数模型可能高估了对多种药物的耐药性。我们工作的副产品是在具有多个连续突变的模型中评估单抗和双抗的概率的有效策略。