# American Institute of Mathematical Sciences

2008, 5(3): 585-599. doi: 10.3934/mbe.2008.5.585

## Calculation of $R_0$ for age-of-infection models

 1 Harvard Graduate School of Education, Harvard University, Cambridge, MA 02138, United States 2 Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2

Received  January 2007 Revised  February 2008 Published  June 2008

We consider age-of-infection epidemic models to describe multiple- stage epidemic models, including treatment. We derive an expression for the basic reproduction number $R_0$ in terms of the distributions of periods of stay in the various compartments. We find that, in the model without treatment, $R_0$ depends only on the mean periods in compartments, and not on the form of the distributions. In treatment models, $R_0$ depends on the form of the dis- tributions of stay in infective compartments from which members are removed for treatment, but the dependence for treatment compartments is only on the mean stay in the compartments. The results give a considerable simplification in the calculation of the basic reproduction number.
Citation: Christine K. Yang, Fred Brauer. Calculation of $R_0$ for age-of-infection models. Mathematical Biosciences & Engineering, 2008, 5 (3) : 585-599. doi: 10.3934/mbe.2008.5.585
 [1] Divine Wanduku. Finite- and multi-dimensional state representations and some fundamental asymptotic properties of a family of nonlinear multi-population models for HIV/AIDS with ART treatment and distributed delays. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021005 [2] Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345 [3] Xin Guo, Lexin Li, Qiang Wu. Modeling interactive components by coordinate kernel polynomial models. Mathematical Foundations of Computing, 2020, 3 (4) : 263-277. doi: 10.3934/mfc.2020010 [4] Urszula Ledzewicz, Heinz Schättler. On the role of pharmacometrics in mathematical models for cancer treatments. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 483-499. doi: 10.3934/dcdsb.2020213 [5] P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178 [6] Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256 [7] Zhimin Li, Tailei Zhang, Xiuqing Li. Threshold dynamics of stochastic models with time delays: A case study for Yunnan, China. Electronic Research Archive, 2021, 29 (1) : 1661-1679. doi: 10.3934/era.2020085 [8] Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 603-632. doi: 10.3934/dcdsb.2020260 [9] Shigui Ruan. Nonlinear dynamics in tumor-immune system interaction models with delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 541-602. doi: 10.3934/dcdsb.2020282 [10] Eric Foxall. Boundary dynamics of the replicator equations for neutral models of cyclic dominance. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1061-1082. doi: 10.3934/dcdsb.2020153 [11] Yanan Li, Zhijian Yang, Na Feng. Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021018 [12] Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168 [13] Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460 [14] Yen-Luan Chen, Chin-Chih Chang, Zhe George Zhang, Xiaofeng Chen. Optimal preventive "maintenance-first or -last" policies with generalized imperfect maintenance models. Journal of Industrial & Management Optimization, 2021, 17 (1) : 501-516. doi: 10.3934/jimo.2020149 [15] Xueli Bai, Fang Li. Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3075-3092. doi: 10.3934/dcds.2020035 [16] Wenbin Lv, Qingyuan Wang. Global existence for a class of Keller-Segel models with signal-dependent motility and general logistic term. Evolution Equations & Control Theory, 2021, 10 (1) : 25-36. doi: 10.3934/eect.2020040 [17] Ching-Hui Wang, Sheng-Chen Fu. Traveling wave solutions to diffusive Holling-Tanner predator-prey models. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021007 [18] Yu-Jhe Huang, Zhong-Fu Huang, Jonq Juang, Yu-Hao Liang. Flocking of non-identical Cucker-Smale models on general coupling network. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1111-1127. doi: 10.3934/dcdsb.2020155 [19] Yancong Xu, Lijun Wei, Xiaoyu Jiang, Zirui Zhu. Complex dynamics of a SIRS epidemic model with the influence of hospital bed number. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021016 [20] Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

2018 Impact Factor: 1.313