Qualitative analysis of a stochastic SEITR epidemic model with multiple stages of infection and treatment

We present a mathematical analysis of the transmission of certain diseases using a stochastic susceptible-exposed-infectious-treated-recovered (SEITR) model with multiple stages of infection and treatment and explore the effects of treatments and external fluctuations in the transmission, treatment and recovery rates. We assume external fluctuations are caused by variability in the number of contacts between infected and susceptible individuals. It is shown that the expected number of secondary infections produced (in the absence of noise) reduces as treatment is introduced into the population. By defining RT,n and RT,n as the basic deterministic and stochastic reproduction numbers, respectively, in stage n of infection and treatment, we show mathematically that as the intensity of the noise in the transmission, treatment and recovery rates increases, the number of secondary cases of infection increases. The global stability of the disease-free and endemic equilibrium for the deterministic and stochastic SEITR models is also presented. The work presented is demonstrated using parameter values relevant to the transmission dynamics of Influenza in the United States from October 1, 2018 through May 4, 2019 influenza seasons.

[1]  J. Caylà,et al.  Influenza vaccine effectiveness in reducing severe outcomes over six influenza seasons, a case-case analysis, Spain, 2010/11 to 2015/16 , 2018, Euro surveillance : bulletin Europeen sur les maladies transmissibles = European communicable disease bulletin.

[2]  Anna Mummert,et al.  Parameter identification for a stochastic SEIRS epidemic model: case study influenza , 2019, Journal of mathematical biology.

[3]  J. Watmough,et al.  Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. , 2002, Mathematical biosciences.

[4]  E. Arias,et al.  Mortality in the United States, 2017. , 2018, NCHS data brief.

[5]  A. Gumel,et al.  Global asymptotic properties of an SEIRS model with multiple infectious stages , 2010 .

[6]  O. Diekmann,et al.  On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations , 1990, Journal of mathematical biology.

[7]  Zhilan Feng,et al.  Modeling the Effects of Vaccination and Treatment on Pandemic Influenza , 2011, The AAPS Journal.

[8]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[9]  Yanni Xiao,et al.  An algebraic approach to proving the global stability of a class of epidemic models , 2012 .

[10]  Tailei Zhang,et al.  Global stability for a tuberculosis model , 2011, Math. Comput. Model..

[11]  Karima R. Nigmatulina,et al.  The scaling of contact rates with population density for the infectious disease models. , 2013, Mathematical biosciences.

[12]  D. J. H. Garling,et al.  The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities by J. Michael Steele , 2005, Am. Math. Mon..

[13]  R. Khasminskii.  Stochastic Stability of Differential Equations , 1980 .

[14]  L. Balluz,et al.  Self-reported influenza-like illness and receipt of influenza antiviral drugs during the 2009 pandemic, United States, 2009-2010. , 2012, American journal of public health.

[15]  Paul J. Birrell,et al.  Prospects of elimination of HIV with test-and-treat strategy , 2013, Proceedings of the National Academy of Sciences.

[16]  Adi R. Bulsara,et al.  Stochastic processes with non-additive fluctuations , 1979 .

[17]  O. M. Otunuga.  Global stability for a 2n+1 dimensional HIV/AIDS epidemic model with treatments. , 2018, Mathematical biosciences.

[18]  James F. Blowey,et al.  Theory and Numerics of Differential Equations , 2001 .

[19]  Andrei Korobeinikov,et al.  Global Properties of SIR and SEIR Epidemic Models with Multiple Parallel Infectious Stages , 2009, Bulletin of mathematical biology.

[20]  Zhipeng Qiu,et al.  Transmission Dynamics of an Influenza Model with Vaccination and Antiviral Treatment , 2010, Bulletin of mathematical biology.

[21]  L. Arnold.  Stochastic Differential Equations: Theory and Applications , 1992 .

[22]  N. Grassly,et al.  Seasonal infectious disease epidemiology , 2006, Proceedings of the Royal Society B: Biological Sciences.

[23]  Carrie Reed,et al.  Seasonal Incidence of Symptomatic Influenza in the United States , 2018, Clinical infectious diseases : an official publication of the Infectious Diseases Society of America.

[24]  Hai-Feng Huo,et al.  Modelling and stability of HIV/AIDS epidemic model with treatment , 2016 .

[25]  O. M. Otunuga.  Global Stability of Nonlinear Stochastic SEI Epidemic Model with Fluctuations in Transmission Rate of Disease , 2017 .

[26]  Christophe Fraser,et al.  HIV-1 transmission, by stage of infection. , 2008, The Journal of infectious diseases.

[27]  Z. Poljak,et al.  An SEIR model of influenza A virus infection and reinfection within a farrow-to-finish swine farm , 2018, PloS one.

[28]  Xuerong Mao,et al.  Extinction and recurrence of multi-group SEIR epidemic , 2013 .

[29]  E. Wong,et al.  On the Convergence of Ordinary Integrals to Stochastic Integrals , 1965 .

[30]  Gerardo Chowell,et al.  Assessing parameter identifiability in compartmental dynamic models using a computational approach: application to infectious disease transmission models , 2019, Theoretical Biology and Medical Modelling.

[31]  Vicenç Méndez,et al.  Stochastic fluctuations of the transmission rate in the susceptible-infected-susceptible epidemic model. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  Pasquale Vetro,et al.  Stability of a stochastic SIR system , 2005 .

[33]  R. Plemmons.  M-matrix characterizations.I—nonsingular M-matrices , 1977 .

[34]  Steve Leach,et al.  Potential impact of antiviral drug use during influenza pandemic. , 2005 .