# American Institute of Mathematical Sciences

August  2018, 23(6): 2071-2090. doi: 10.3934/dcdsb.2018226

## Epidemic dynamics on complex networks with general infection rate and immune strategies

 1 College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian, 350116, China 2 Mathematics and Science College, Shanghai Normal University, Shanghai, 200234, China

* Corresponding author: Shouying Huang

Received  March 2016 Revised  April 2018 Published  August 2018 Early access  July 2018

Fund Project: This work is supported by the Natural Science Foundation of China (NSFC) under Grant No.11771295 and 11601336, the Natural Science Foundation of Fujian Province under Grant No.2018J01664, the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, and Shanghai Gaofeng Project for University Academic Program Development.

This paper mainly aims to study the influence of individuals' different heterogeneous contact patterns on the spread of the disease. For this purpose, an SIS epidemic model with a general form of heterogeneous infection rate is investigated on complex heterogeneous networks. A qualitative analysis of this model reveals that, depending on the epidemic threshold $R_0$, either the disease-free equilibrium or the endemic equilibrium is globally asymptotically stable. Interestingly, no matter what functional form the heterogeneous infection rate is, whether the disease will disappear or not is completely determined by the value of $R_0$, but the heterogeneous infection rate has close relation with the epidemic threshold $R_0$. Especially, the heterogeneous infection rate can directly affect the final number of infected nodes when the disease is endemic. The obtained results improve and generalize some known results. Finally, based on the heterogeneity of contact patterns, the effects of different immunization schemes are discussed and compared. Meanwhile, we explore the relation between the immunization rate and the recovery rate, which are the two important parameters that can be improved. To illustrate our theoretical results, the corresponding numerical simulations are also included.

Citation: Shouying Huang, Jifa Jiang. Epidemic dynamics on complex networks with general infection rate and immune strategies. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2071-2090. doi: 10.3934/dcdsb.2018226
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##### References:
The time series of $I(t)$ with different forms of infection rate
The densities of infected nodes with different degrees. The lines from bottom to top are $I_1(t), I_{10}(t), I_{20}(t), \cdots, I_{90}(t), I_n(t)$
The influence of initial conditions on the density of $I_{30}(t)$
The time evolutions of $I(t)$ with uniform immunization and targeted immunization for a given recovery rate $\gamma = 0.05$
Comparison of the effectiveness of different immunization schemes: uniform immunization with $\sigma = 0.0026$, targeted immunization with $\bar{\sigma} = 0.0026$ (i.e., $k_c = 25$) and acquaintance immunization with $q_{_0} = 0.35$. Here the parameters and initial value are the same as those of Fig. 1(b)
The time evolutions of $I(t)$ for different recovery rate $\gamma$ under the given acquaintance immunization rate $q_{_0}$ and high-risk immunization rate $v_{_0}$. Here, the other parameters and initial value are the same as those of Fig. 4
Effectiveness of high-risk immunization schemes with different value of $v_{_0}$. Here the parameters and initial value are the same as those of Fig. 1(b), except for $n = 30$
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