Department of Mathematics, Central University of Rajasthan, Bandar Sindri, Kishangarh, Ajmer, Rajasthan, 305817, India.
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada.
J Math Biol. 2024 Mar 20;88(4):45. doi: 10.1007/s00285-024-02068-x.
COVID-19 is a respiratory disease triggered by an RNA virus inclined to mutations. Since December 2020, variants of COVID-19 (especially Delta and Omicron) continuously appeared with different characteristics that influenced death and transmissibility emerged around the world. To address the novel dynamics of the disease, we propose and analyze a dynamical model of two strains, namely native and mutant, transmission dynamics with mutation and imperfect vaccination. It is also assumed that the recuperated individuals from the native strain can be infected with mutant strain through the direct contact with individual or contaminated surfaces or aerosols. We compute the basic reproduction number, , which is the maximum of the basic reproduction numbers of native and mutant strains. We prove the nonexistence of backward bifurcation using the center manifold theory, and global stability of disease-free equilibrium when , that is, vaccine is effective enough to eliminate the native and mutant strains even if it cannot provide full protection. Hopf bifurcation appears when the endemic equilibrium loses its stability. An intermediate mutation rate leads to oscillations. When increases over a threshold, the system regains its stability and exhibits an interesting dynamics called endemic bubble. An analytical expression for vaccine-induced herd immunity is derived. The epidemiological implication of the herd immunity threshold is that the disease may effectively be eradicated if the minimum herd immunity threshold is attained in the community. Furthermore, the model is parameterized using the Indian data of the cumulative number of confirmed cases and deaths of COVID-19 from March 1 to September 27 in 2021, using MCMC method. The cumulative cases and deaths can be reduced by increasing the vaccine efficacies to both native and mutant strains. We observe that by considering the vaccine efficacy against native strain as 90%, both cumulative cases and deaths would be reduced by 0.40%. It is concluded that increasing immunity against mutant strain is more influential than the vaccine efficacy against it in controlling the total cases. Our study demonstrates that the COVID-19 pandemic may be worse due to the occurrence of oscillations for certain mutation rates (i.e., outbreaks will occur repeatedly) but better due to stability at a lower infection level with a larger mutation rate. We perform sensitivity analysis using the Latin Hypercube Sampling methodology and partial rank correlation coefficients to illustrate the impact of parameters on the basic reproduction number, the number of cumulative cases and deaths, which ultimately sheds light on disease mitigation.
新型冠状病毒肺炎（COVID-19）是一种由 RNA 病毒引起的呼吸道疾病，容易发生突变。自 2020 年 12 月以来，COVID-19 的变异株（尤其是德尔塔和奥密克戎）不断出现，具有不同的特征，在全球范围内出现了影响死亡率和传染性的变异株。为了应对疾病的新动态，我们提出并分析了一个包含两种病毒株（原生株和突变株）的动力学模型，该模型考虑了突变和不完全疫苗接种的传播动力学。此外，我们还假设从原生株中康复的个体可以通过与个体或受污染的表面或气溶胶的直接接触而感染突变株。我们计算了基本繁殖数， ，这是原生株和突变株基本繁殖数的最大值。我们使用中心流形理论证明了不存在反向分支，并且当 时，无病平衡点是全局稳定的，也就是说，即使疫苗不能提供完全保护，它也足以消除原生株和突变株。当地方病平衡点失去稳定性时，会出现霍普夫分支。中间突变率 导致了震荡。当 增加到一个阈值以上时，系统恢复稳定，并表现出一种有趣的动力学，称为地方病泡沫。导出了疫苗诱导的群体免疫的解析表达式。群体免疫阈值的流行病学意义是，如果在社区中达到最小的群体免疫阈值，那么疾病可能会被有效根除。此外，该模型使用 2021 年 3 月 1 日至 9 月 27 日印度 COVID-19 确诊病例和死亡人数的累积数据进行参数化，使用 MCMC 方法。通过提高对原生株和突变株的疫苗效力，可以减少累积病例和死亡人数。我们观察到，当考虑对原生株的疫苗效力为 90%时，累积病例和死亡人数将减少 0.40%。结论是，提高对突变株的免疫力比对它的疫苗效力更能控制总病例数。我们的研究表明，由于某些突变率（即，疫情将反复发生）导致的振荡，COVID-19 大流行可能会更严重，但由于较低的感染水平和较大的突变率而更加稳定。我们使用拉丁超立方抽样方法和部分秩相关系数进行敏感性分析，以说明参数对基本繁殖数、累积病例和死亡人数的影响，最终为疾病缓解提供了启示。
