Jain Shikha, Kumar Sachin
Department of Mathematics, University of Delhi, Delhi, New Delhi 110007 India.
Eur Phys J Plus. 2021;136(4):439. doi: 10.1140/epjp/s13360-021-01390-3. Epub 2021 Apr 23.
Consideration of every important aspect while modeling a disease makes the model more precise and the disease eradication strategy more powerful. In the present paper, we analyze the importance of innate immunity on SEIS modeling. We propose an SEIS model with Holling type II and type III functions representing innate immunity. We find the existence and stability conditions for the equilibria. When innate immunity is in the form of Holling type II function, the disease-free equilibrium exists for reproduction number less than unity and is locally asymptotically stable, and supercritical transcritical (forward) as well as subcritical transcritical (backward) bifurcation may occur where the contact rate acts as the bifurcation parameter. Hence, disease-free equilibrium need not be globally stable. For reproduction number greater than unity unique endemic equilibrium exists which is locally asymptotically stable. The global stability conditions for the same are deduced with the help of Lozinski measure. When innate immunity is considered a Holling type III function, the disease-free equilibrium point exists for reproduction number less than unity and is locally as well as globally stable. The existence of either unique or multiple endemic equilibria is found when reproduction number is greater than unity, and there exists at least one locally asymptotically stable equilibrium point and bistability can also be encountered. The conditions for the existence of Andronov-Hopf bifurcation are deduced for both cases. Moreover, we observe that ignoring innate immunity annihilates the possibility of Andronov-Hopf bifurcation. Numerical simulation is performed to validate the mathematical findings. Comparing the obtained results to the case when innate immunity is ignored, it is deduced that ignoring it ends the possibility of backward bifurcation, Andronov-Hopf bifurcation as well as the existence of multiple equilibria, and it also leads to the prediction of higher infection than the actual which may deflect the accuracy of the model to a high extent. This would further lead to false predictions and inefficient disease control strategies which in turn would make disease eradication a difficult and more expensive task.
在对疾病进行建模时考虑到每一个重要方面会使模型更加精确，疾病根除策略更加强有力。在本文中，我们分析了固有免疫在SEIS模型中的重要性。我们提出了一个具有表示固有免疫的Holling II型和III型函数的SEIS模型。我们找到了平衡点的存在性和稳定性条件。当固有免疫呈Holling II型函数形式时，对于小于1的繁殖数，无病平衡点存在且局部渐近稳定，并且当接触率作为分岔参数时，可能会出现超临界跨临界（向前）以及亚临界跨临界（向后）分岔。因此，无病平衡点不一定是全局稳定的。对于大于1的繁殖数，存在唯一的地方病平衡点，它是局部渐近稳定的。借助Lozinski测度推导出了其全局稳定性条件。当固有免疫被视为Holling III型函数时，对于小于1的繁殖数，无病平衡点存在且局部和全局稳定。当繁殖数大于1时，发现存在唯一或多个地方病平衡点，并且至少存在一个局部渐近稳定的平衡点，还可能出现双稳性。推导了两种情况下安德罗诺夫 - 霍普夫分岔存在的条件。此外，我们观察到忽略固有免疫消除了安德罗诺夫 - 霍普夫分岔的可能性。进行了数值模拟以验证数学结果。将所得结果与忽略固有免疫的情况进行比较，推断出忽略它会消除向后分岔、安德罗诺夫 - 霍普夫分岔以及多个平衡点存在的可能性，并且还会导致预测的感染率高于实际情况，这可能在很大程度上使模型的准确性产生偏差。这将进一步导致错误的预测和低效的疾病控制策略，并进而使疾病根除成为一项困难且成本更高的任务。
