• Previous Article
    Optimization of a control law to synchronize first-order dynamical systems on Riemannian manifolds by a transverse component
  • DCDS-B Home
  • This Issue
  • Next Article
    Effects of fear and anti-predator response in a discrete system with delay
doi: 10.3934/dcdsb.2021037
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Analysis of an age-structured model for HIV-TB co-infection

1. 

School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070, China

2. 

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, China

* Corresponding author: Zhong-Kai Guo and Hai-Feng Huo

Received  August 2020 Revised  November 2020 Early access January 2021

Fund Project: This work is supported by the National Natural Science Foundation of China (11861044 and 11661050), the HongLiu first-class disciplines Development Program of Lanzhou University of Technology, the Youth Science Fund of Lanzhou Jiaotong University(1200060930), and the Scientific Research Foundation of Lanzhou Jiaotong University(1520020410)

According to the report of the WHO, there is a strong relationship between AIDS and tuberculosis (TB). Therefore, it is very important to study how to control TB in the context of the global AIDS epidemic. In this paper, we establish an age structured mathematical model of HIV-TB co-infection to study the transmission dynamics of this co-infection, and consider awareness in the modeling. We give the basic reproduction numbers for each of the two diseases and find four equilibria, namely, disease-free equilibrium, TB-free equilibrium, HIV-free equilibrium and endemic disease equilibrium. Then we discuss the local stability of the equilibria according to the range of values of the two basic reproduction numbers, and find the endemic equilibrium is unstable. We also discuss the global stability of the disease-free equilibrium and the TB-free equilibrium. Based on the new HIV-positive cases and TB cases data in China, the best-fit parameter values and initial values of the model are identified by the MCMC algorithm. Then we perform uncertainty and sensitivity analysis to identify the parameters that have significant impact on the basic reproduction number $ \mathcal{R}_{T} $. Finally, combined with the established model, we give some measures that may help China achieve the goal of WHO of reducing the incidence of TB by 80% by 2030 compared to 2015.

Citation: Zhong-Kai Guo, Hai-Feng Huo, Hong Xiang. Analysis of an age-structured model for HIV-TB co-infection. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021037
References:
[1]

World Health Organization, Available from: https://www.who.int/health-topics/hiv-aids/. Google Scholar

[2]

Centers for Disease Control and Prevention, Available from: https://www.cdc.gov/tb/. Google Scholar

[3]

World Health Organization, Available from: https://www.who.int/tb/en/. Google Scholar

[4]

E. M. C. D'AgataP. MagalS. Ruan and G. Webb, Asymptotic behavior in nosocomial epidemic models with antibiotic resistance, Differential Integral Equations, 19 (2006), 573-600.   Google Scholar

[5]

F. B. Agusto and A. Adekunle, Optimal control of a two-strain tuberculosis-HIV/AIDS co-infection model, Biosystems, 119 (2014), 20-44.  doi: 10.1016/j.biosystems.2014.03.006.  Google Scholar

[6]

C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model, Discrete and Continuous Dynamical Systems - Series B, 18 (2013), 1999-2017.  doi: 10.3934/dcdsb.2013.18.1999.  Google Scholar

[7]

S. Gakkhar and N. Chavda, A dynamical model for HIV-TB co-infection, Applied Mathematics and Computation, 218 (2012), 9261-9270.  doi: 10.1016/j.amc.2012.03.004.  Google Scholar

[8]

I. GhoshP. K. TiwariS. SamantaI. M. ElmojtabaN. Al-Salti and J. Chattopadhyay, A simple SI-type model for HIV/AIDS with media and self-imposed psychological fear, Mathematical Biosciences, 306 (2018), 160-169.  doi: 10.1016/j.mbs.2018.09.014.  Google Scholar

[9]

Z.-K. GuoH.-F. Huo and H. Xiang, Global dynamics of an age-structured malaria model with prevention, Mathematical Biosciences and Engineering, 16 (2019), 1625-1653.  doi: 10.3934/mbe.2019078.  Google Scholar

[10]

H. HaarioM. LaineA. Mira and E. Saksman, Dram: Efficient adaptive MCMC, Statistics and Computing, 16 (2006), 339-354.  doi: 10.1007/s11222-006-9438-0.  Google Scholar

[11]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM Journal on Mathematical Analysis, 20 (1989), 388-395.  doi: 10.1137/0520025.  Google Scholar

[12]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giadini Editori e Stampatori, Pisa, 1994. Google Scholar

[13]

D. Kirschner, Dynamics of co-infection with M. tuberculosis and HIV-1, Theoretical Population Biology, 55 (1999), 94-109.  doi: 10.1006/tpbi.1998.1382.  Google Scholar

[14]

P. MagalC. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Applicable Analysis, 89 (2010), 1109-1140.  doi: 10.1080/00036810903208122.  Google Scholar

[15]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM Journal on Mathematical Analysis, 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

[16]

A. MallelaS. Lenhart and N. K. Vaidya, HIV-TB co-infection treatment: Modeling and optimal control theory perspectives, Journal of Computational and Applied Mathematics, 307 (2016), 143-161.  doi: 10.1016/j.cam.2016.02.051.  Google Scholar

[17]

S. MarinoI. B. HogueC. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, Journal of Theoretical Biology, 254 (2008), 178-196.  doi: 10.1016/j.jtbi.2008.04.011.  Google Scholar

[18]

M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3.  Google Scholar

[19]

E. MassadM. N. BurattiniF. A. B. CoutinhoH. M. Yang and S. M. Raimundo, Modeling the interaction between AIDS and tuberculosis, Mathematical and Computer Modelling, 17 (1993), 7-21.  doi: 10.1016/0895-7177(93)90013-O.  Google Scholar

[20]

C. M. A. Pinto and A. R. M. Carvalho, New findings on the dynamics of HIV and TB coinfection models, Applied Mathematics and Computation, 242 (2014), 36-46.  doi: 10.1016/j.amc.2014.05.061.  Google Scholar

[21]

M. SamsuzzohaM. Singh and D. Lucy, Uncertainty and sensitivity analysis of the basic reproduction number of a vaccinated epidemic model of influenza, Applied Mathematical Modelling, 37 (2013), 903-915.  doi: 10.1016/j.apm.2012.03.029.  Google Scholar

[22]

S. C. Shiboski and N. P. Jewell, Statistical analysis of the time dependence of HIV infectivity based on partner study data, Journal of the American Statistical Association, 87 (1992), 360-372.  doi: 10.1080/01621459.1992.10475215.  Google Scholar

[23]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Society, Providence, 118 2011. doi: 10.1090/gsm/118.  Google Scholar

[24]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.  Google Scholar

[25]

National Bureau of Statistics of China, Available from: http://www.stats.gov.cn/. Google Scholar

[26]

Chinese Center for Disease Control and Prevention, Available from: http://www.chinacdc.cn/. Google Scholar

show all references

References:
[1]

World Health Organization, Available from: https://www.who.int/health-topics/hiv-aids/. Google Scholar

[2]

Centers for Disease Control and Prevention, Available from: https://www.cdc.gov/tb/. Google Scholar

[3]

World Health Organization, Available from: https://www.who.int/tb/en/. Google Scholar

[4]

E. M. C. D'AgataP. MagalS. Ruan and G. Webb, Asymptotic behavior in nosocomial epidemic models with antibiotic resistance, Differential Integral Equations, 19 (2006), 573-600.   Google Scholar

[5]

F. B. Agusto and A. Adekunle, Optimal control of a two-strain tuberculosis-HIV/AIDS co-infection model, Biosystems, 119 (2014), 20-44.  doi: 10.1016/j.biosystems.2014.03.006.  Google Scholar

[6]

C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model, Discrete and Continuous Dynamical Systems - Series B, 18 (2013), 1999-2017.  doi: 10.3934/dcdsb.2013.18.1999.  Google Scholar

[7]

S. Gakkhar and N. Chavda, A dynamical model for HIV-TB co-infection, Applied Mathematics and Computation, 218 (2012), 9261-9270.  doi: 10.1016/j.amc.2012.03.004.  Google Scholar

[8]

I. GhoshP. K. TiwariS. SamantaI. M. ElmojtabaN. Al-Salti and J. Chattopadhyay, A simple SI-type model for HIV/AIDS with media and self-imposed psychological fear, Mathematical Biosciences, 306 (2018), 160-169.  doi: 10.1016/j.mbs.2018.09.014.  Google Scholar

[9]

Z.-K. GuoH.-F. Huo and H. Xiang, Global dynamics of an age-structured malaria model with prevention, Mathematical Biosciences and Engineering, 16 (2019), 1625-1653.  doi: 10.3934/mbe.2019078.  Google Scholar

[10]

H. HaarioM. LaineA. Mira and E. Saksman, Dram: Efficient adaptive MCMC, Statistics and Computing, 16 (2006), 339-354.  doi: 10.1007/s11222-006-9438-0.  Google Scholar

[11]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM Journal on Mathematical Analysis, 20 (1989), 388-395.  doi: 10.1137/0520025.  Google Scholar

[12]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giadini Editori e Stampatori, Pisa, 1994. Google Scholar

[13]

D. Kirschner, Dynamics of co-infection with M. tuberculosis and HIV-1, Theoretical Population Biology, 55 (1999), 94-109.  doi: 10.1006/tpbi.1998.1382.  Google Scholar

[14]

P. MagalC. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Applicable Analysis, 89 (2010), 1109-1140.  doi: 10.1080/00036810903208122.  Google Scholar

[15]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM Journal on Mathematical Analysis, 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

[16]

A. MallelaS. Lenhart and N. K. Vaidya, HIV-TB co-infection treatment: Modeling and optimal control theory perspectives, Journal of Computational and Applied Mathematics, 307 (2016), 143-161.  doi: 10.1016/j.cam.2016.02.051.  Google Scholar

[17]

S. MarinoI. B. HogueC. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, Journal of Theoretical Biology, 254 (2008), 178-196.  doi: 10.1016/j.jtbi.2008.04.011.  Google Scholar

[18]

M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3.  Google Scholar

[19]

E. MassadM. N. BurattiniF. A. B. CoutinhoH. M. Yang and S. M. Raimundo, Modeling the interaction between AIDS and tuberculosis, Mathematical and Computer Modelling, 17 (1993), 7-21.  doi: 10.1016/0895-7177(93)90013-O.  Google Scholar

[20]

C. M. A. Pinto and A. R. M. Carvalho, New findings on the dynamics of HIV and TB coinfection models, Applied Mathematics and Computation, 242 (2014), 36-46.  doi: 10.1016/j.amc.2014.05.061.  Google Scholar

[21]

M. SamsuzzohaM. Singh and D. Lucy, Uncertainty and sensitivity analysis of the basic reproduction number of a vaccinated epidemic model of influenza, Applied Mathematical Modelling, 37 (2013), 903-915.  doi: 10.1016/j.apm.2012.03.029.  Google Scholar

[22]

S. C. Shiboski and N. P. Jewell, Statistical analysis of the time dependence of HIV infectivity based on partner study data, Journal of the American Statistical Association, 87 (1992), 360-372.  doi: 10.1080/01621459.1992.10475215.  Google Scholar

[23]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Society, Providence, 118 2011. doi: 10.1090/gsm/118.  Google Scholar

[24]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.  Google Scholar

[25]

National Bureau of Statistics of China, Available from: http://www.stats.gov.cn/. Google Scholar

[26]

Chinese Center for Disease Control and Prevention, Available from: http://www.chinacdc.cn/. Google Scholar

Figure 1.  Estimated number of deaths from HIV/AIDS and TB in 2017. Deaths from TB among HIV-positive people are shown in grey
Figure 2.  Flowchart of the TB-HIV co-infection
Figure 3.  Data fitting: (a) the fitting results of the number of new TB cases reported from 2005 to 2017; (b)the fitting results of the number of new HIV-positive cases reported from 2005 to 2017. The solid black line represents the fitted data, and the red dots represent the actual data. The areas from the darkest to the lightest correspond to the 50%, 90%, 95% and 99% posterior limits of the model uncertainty
Figure 4.  The distribution histogram of the basic reproduction number $ \mathcal{R}_{T} $
Figure 5.  The PRCC values
Figure 6.  The effect of changes in $ m_{5} $ on the number of new TB cases
Figure 7.  The effect of changes in $ m_{4} $ on the number of new TB cases
Figure 8.  The effect of changes in $ \beta_{T} $ on the number of new TB cases
Figure 9.  The effect of changes in $ m_{5} $, $ \beta_{T} $ and $ m_{4} $ on the number of new TB cases
Table 1.  Description of parameters of the model $ (1) $
Parameters Description
$ \Lambda $ the recruitment rate of the susceptible class
$ \mu $ the natural death rate of the population
$ \beta(a) $ the transmission coefficient of HIV class to susceptible class
$ \beta_{T} $ the transmission coefficient of active TB class to susceptible class
$ \delta_{i}(a) $ the death rate due to HIV
$ \sigma(\theta) $ the rate at which latent class progress into infectious class
$ \mu_{T} $ the death rate due to TB
$ \mu_{c} $ the death rate due to co-infection
$ \alpha $ the rate at which treatment infectious individuals progress into
susceptible class
$ \delta(\theta) $ the rate at which treatment latent individuals progress into
susceptible class
$ \delta_{T}(a) $ the transmission coefficient of TB infectious class to HIV class
Parameters Description
$ \Lambda $ the recruitment rate of the susceptible class
$ \mu $ the natural death rate of the population
$ \beta(a) $ the transmission coefficient of HIV class to susceptible class
$ \beta_{T} $ the transmission coefficient of active TB class to susceptible class
$ \delta_{i}(a) $ the death rate due to HIV
$ \sigma(\theta) $ the rate at which latent class progress into infectious class
$ \mu_{T} $ the death rate due to TB
$ \mu_{c} $ the death rate due to co-infection
$ \alpha $ the rate at which treatment infectious individuals progress into
susceptible class
$ \delta(\theta) $ the rate at which treatment latent individuals progress into
susceptible class
$ \delta_{T}(a) $ the transmission coefficient of TB infectious class to HIV class
Table 2.  New TB cases and HIV-positive cases from 2005 to 2017 in China (persons)
Year 2005 2006 2007 2008 2009 2010 2011
TB cases 1,259,308 1,127,571 1,163,959 1,169,540 1,076,938 991,350 953,275
HIV-positive cases 30,887 38,262 42,633 51,525 57,473 61,622 73,196
Year 2012 2013 2014 2015 2016 2017
TB cases 951,508 904,434 889,381 864,015 836,236 835,193
HIV-positive cases 100,328 105,784 119,193 132,016 142,124 152,746
Year 2005 2006 2007 2008 2009 2010 2011
TB cases 1,259,308 1,127,571 1,163,959 1,169,540 1,076,938 991,350 953,275
HIV-positive cases 30,887 38,262 42,633 51,525 57,473 61,622 73,196
Year 2012 2013 2014 2015 2016 2017
TB cases 951,508 904,434 889,381 864,015 836,236 835,193
HIV-positive cases 100,328 105,784 119,193 132,016 142,124 152,746
Table 3.  The parameters values and initial values of the model $ (3) $
Parameter Mean Std 95% CI Source
$ \Lambda $ 16439333 - - [25]
$ \mu $ 1/74.7 - - [25]
$ S(0) $ 1307560000 - - [25]
$ \alpha $ 0.002489009 0.000296466 [0.002483197, 0.00249482] MCMC
$ m_{5} $ 0.073713359 0.002394865 [0.073666415, 0.073760303] MCMC
$ \delta $ 0.054719731 0.002310545 [0.05467444, 0.054765022] MCMC
$ m_{3} $ 3.07 $ \times 10^{-9} $ 7.63$ \times 10^{-11} $ [3.070478$ \times 10^{-9} $, 3.073469$ \times 10^{-9} $ ] MCMC
$ \beta $ 1.591747005 0.02403578 [1.591275856, 1.592218155] MCMC
$ \beta_{T} $ 1.49584$ \times 10^{-9} $ 2.29066$ \times 10^{-11} $ [ 1.495393$ \times 10^{-9} $, 1.496291$ \times 10^{-9} $] MCMC
$ m_{1} $ 0.085041492 0.002759644 [0.084987398, 0.085095587] MCMC
$ \delta_{i} $ 0.921801960 0.036083393 [0.921094653, 0.922509267] MCMC
$ m_{2} $ 1.405989 $ \times 10^{-9} $ 9.99339$ \times 10^{-11} $ [1.404030$ \times 10^{-9} $, 1.407948$ \times 10^{-9} $] MCMC
$ m_{4} $ 0.025716455 0.000489538 [0.025706859, 0.025726051] MCMC
$ \sigma $ 0.010987723 0.000575755 [0.010976437, 0.010999009] MCMC
$ \mu_{T} $ 0.894619519 0.0263988 [0.89410205, 0.895136989] MCMC
$ \mu_{c} $ 0.000943614 0.00004225 [0.000942785, 0.000944442] MCMC
$ e(0) $ 154136723 566182 [154125625, 154147821] MCMC
$ i(0) $ 2566318 597 [2566306, 2566329] MCMC
$ t_{c}(0) $ 7433129 289907 [7427446, 7438811] MCMC
$ I_{t}(0) $ 14052275 30076 [14051685, 14052864] MCMC
$ I_{c}(0) $ 27800 1803 [27765, 27835] MCMC
$ t_{s}(0) $ 77090 4091 [77010, 77170] MCMC
Parameter Mean Std 95% CI Source
$ \Lambda $ 16439333 - - [25]
$ \mu $ 1/74.7 - - [25]
$ S(0) $ 1307560000 - - [25]
$ \alpha $ 0.002489009 0.000296466 [0.002483197, 0.00249482] MCMC
$ m_{5} $ 0.073713359 0.002394865 [0.073666415, 0.073760303] MCMC
$ \delta $ 0.054719731 0.002310545 [0.05467444, 0.054765022] MCMC
$ m_{3} $ 3.07 $ \times 10^{-9} $ 7.63$ \times 10^{-11} $ [3.070478$ \times 10^{-9} $, 3.073469$ \times 10^{-9} $ ] MCMC
$ \beta $ 1.591747005 0.02403578 [1.591275856, 1.592218155] MCMC
$ \beta_{T} $ 1.49584$ \times 10^{-9} $ 2.29066$ \times 10^{-11} $ [ 1.495393$ \times 10^{-9} $, 1.496291$ \times 10^{-9} $] MCMC
$ m_{1} $ 0.085041492 0.002759644 [0.084987398, 0.085095587] MCMC
$ \delta_{i} $ 0.921801960 0.036083393 [0.921094653, 0.922509267] MCMC
$ m_{2} $ 1.405989 $ \times 10^{-9} $ 9.99339$ \times 10^{-11} $ [1.404030$ \times 10^{-9} $, 1.407948$ \times 10^{-9} $] MCMC
$ m_{4} $ 0.025716455 0.000489538 [0.025706859, 0.025726051] MCMC
$ \sigma $ 0.010987723 0.000575755 [0.010976437, 0.010999009] MCMC
$ \mu_{T} $ 0.894619519 0.0263988 [0.89410205, 0.895136989] MCMC
$ \mu_{c} $ 0.000943614 0.00004225 [0.000942785, 0.000944442] MCMC
$ e(0) $ 154136723 566182 [154125625, 154147821] MCMC
$ i(0) $ 2566318 597 [2566306, 2566329] MCMC
$ t_{c}(0) $ 7433129 289907 [7427446, 7438811] MCMC
$ I_{t}(0) $ 14052275 30076 [14051685, 14052864] MCMC
$ I_{c}(0) $ 27800 1803 [27765, 27835] MCMC
$ t_{s}(0) $ 77090 4091 [77010, 77170] MCMC
[1]

Georgi Kapitanov. A double age-structured model of the co-infection of tuberculosis and HIV. Mathematical Biosciences & Engineering, 2015, 12 (1) : 23-40. doi: 10.3934/mbe.2015.12.23

[2]

Zindoga Mukandavire, Abba B. Gumel, Winston Garira, Jean Michel Tchuenche. Mathematical analysis of a model for HIV-malaria co-infection. Mathematical Biosciences & Engineering, 2009, 6 (2) : 333-362. doi: 10.3934/mbe.2009.6.333

[3]

Kazeem Oare Okosun, Robert Smith?. Optimal control analysis of malaria-schistosomiasis co-infection dynamics. Mathematical Biosciences & Engineering, 2017, 14 (2) : 377-405. doi: 10.3934/mbe.2017024

[4]

Salihu Sabiu Musa, Nafiu Hussaini, Shi Zhao, He Daihai. Dynamical analysis of chikungunya and dengue co-infection model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1907-1933. doi: 10.3934/dcdsb.2020009

[5]

A. M. Elaiw, N. H. AlShamrani. Global stability of HIV/HTLV co-infection model with CTL-mediated immunity. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021108

[6]

Yijun Lou, Li Liu, Daozhou Gao. Modeling co-infection of Ixodes tick-borne pathogens. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1301-1316. doi: 10.3934/mbe.2017067

[7]

Jinliang Wang, Xiu Dong. Analysis of an HIV infection model incorporating latency age and infection age. Mathematical Biosciences & Engineering, 2018, 15 (3) : 569-594. doi: 10.3934/mbe.2018026

[8]

Yuming Chen, Junyuan Yang, Fengqin Zhang. The global stability of an SIRS model with infection age. Mathematical Biosciences & Engineering, 2014, 11 (3) : 449-469. doi: 10.3934/mbe.2014.11.449

[9]

Suxia Zhang, Xiaxia Xu. A mathematical model for hepatitis B with infection-age structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1329-1346. doi: 10.3934/dcdsb.2016.21.1329

[10]

Suxia Zhang, Hongbin Guo, Robert Smith?. Dynamical analysis for a hepatitis B transmission model with immigration and infection age. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1291-1313. doi: 10.3934/mbe.2018060

[11]

C. Connell McCluskey. Global stability for an $SEI$ model of infectious disease with age structure and immigration of infecteds. Mathematical Biosciences & Engineering, 2016, 13 (2) : 381-400. doi: 10.3934/mbe.2015008

[12]

Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859

[13]

Yuhua Zhu. A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method. Networks & Heterogeneous Media, 2019, 14 (4) : 677-707. doi: 10.3934/nhm.2019027

[14]

C. Connell McCluskey. Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes. Mathematical Biosciences & Engineering, 2012, 9 (4) : 819-841. doi: 10.3934/mbe.2012.9.819

[15]

Expeditho Mtisi, Herieth Rwezaura, Jean Michel Tchuenche. A mathematical analysis of malaria and tuberculosis co-dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 827-864. doi: 10.3934/dcdsb.2009.12.827

[16]

Carlota Rebelo, Alessandro Margheri, Nicolas Bacaër. Persistence in some periodic epidemic models with infection age or constant periods of infection. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1155-1170. doi: 10.3934/dcdsb.2014.19.1155

[17]

Fred Brauer. Age-of-infection and the final size relation. Mathematical Biosciences & Engineering, 2008, 5 (4) : 681-690. doi: 10.3934/mbe.2008.5.681

[18]

Fred Brauer, Zhisheng Shuai, P. van den Driessche. Dynamics of an age-of-infection cholera model. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1335-1349. doi: 10.3934/mbe.2013.10.1335

[19]

Jinliang Wang, Ran Zhang, Toshikazu Kuniya. A note on dynamics of an age-of-infection cholera model. Mathematical Biosciences & Engineering, 2016, 13 (1) : 227-247. doi: 10.3934/mbe.2016.13.227

[20]

Robert G. McLeod, John F. Brewster, Abba B. Gumel, Dean A. Slonowsky. Sensitivity and uncertainty analyses for a SARS model with time-varying inputs and outputs. Mathematical Biosciences & Engineering, 2006, 3 (3) : 527-544. doi: 10.3934/mbe.2006.3.527

2020 Impact Factor: 1.327

Article outline

Figures and Tables

[Back to Top]