Antiviral treatment remains one of the key pharmacological interventions against influenza pandemic. However, widespread use of antiviral drugs brings with it the danger of drug resistance evolution. To assess the risk of the emergence and diffusion of resistance, in this paper, we develop a diffusive influenza model where influenza infection involves both drug-sensitive and drug-resistant strains. We first analyze its corresponding reaction model, whose reproduction numbers and equilibria are derived. The results show that the sensitive strains can be eliminated by treatment. Then, we establish the existence of the three kinds of traveling waves starting from the disease-free equilibrium, i.e., semi-traveling waves, strong traveling waves and persistent traveling waves, from which we can get some useful information (such as whether influenza will spread, asymptotic speed of propagation, the final state of the wavefront). On the other hand, we discuss three situations in which semi-traveling waves do not exist. When the control reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{C}$$\end{document}RC is larger than 1, the conditions for the existence and nonexistence of traveling waves are determined completely by the reproduction numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{SC}$$\end{document}RSC, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{RC}$$\end{document}RRC and the wave speed c. Meanwhile, we give an interval estimation of minimal wave speed for influenza transmission, which has important guiding significance for the control of influenza in reality. Our findings demonstrate that the control of influenza depends not only on the rates of resistance emergence and transmission during treatment, but also on the diffusion rates of influenza strains, which have been overlooked in previous modeling studies. This suggests that antiviral treatment should be implemented appropriately, and infected individuals (especially with the resistant strain) should be tested and controlled effectively. Finally, we outline some future directions that deserve further investigation.

中文翻译：

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多菌株扩散流感模型的行波和最小波速估计

抗病毒治疗仍然是对抗流感大流行的关键药物干预措施之一。然而，抗病毒药物的广泛使用也带来了耐药性进化的危险。为了评估耐药性出现和扩散的风险，在本文中，我们开发了一个扩散流感模型，其中流感感染涉及药物敏感株和耐药株。我们首先分析其对应的反应模型，推导出其再生数和平衡。结果表明，敏感菌株可通过处理消除。然后，我们从无病平衡出发，建立了三种行波的存在性，即半行波、强行波和持续行波，从中我们可以得到一些有用的信息（例如流感是否会传播、渐进传播速度、波前的最终状态）。另一方面，我们讨论不存在半行波的三种情况。当控制复制数 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength {\oddsidemargin}{-69pt} \begin{document}$$R_{C}$$\end{document}RC 大于 1，行波存在和不存在的条件完全由再生数决定 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{SC}$$\end{document}RSC, \documentclass[12pt]{minimal} \usepackage {amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R_{RC}$$\end{document}RRC 和波速 c. 同时，我们给出了流感传播最小波速的区间估计，对现实中的流感控制具有重要的指导意义。我们的研究结果表明，流感的控制不仅取决于治疗期间耐药性的出现和传播率，还取决于流感毒株的扩散率，而这在之前的建模研究中被忽视了。这表明应适当实施抗病毒治疗，并应有效检测和控制感染者（尤其是耐药株）。最后，我们概述了一些值得进一步研究的未来方向。应有效检测和控制感染者（尤其是耐药株）。最后，我们概述了一些值得进一步研究的未来方向。应有效检测和控制感染者（尤其是耐药株）。最后，我们概述了一些值得进一步研究的未来方向。