GLOBAL DYNAMICS OF A MATHEMATICAL MODEL OF TUBERCULOSIS
|
|
- Rose Flowers
- 5 years ago
- Views:
Transcription
1 CANADIAN APPIED MATHEMATICS QUARTERY Volume 13, Number 4, Winter 2005 GOBA DYNAMICS OF A MATHEMATICA MODE OF TUBERCUOSIS HONGBIN GUO ABSTRACT. Mathematical analysis is carried out for a mathematical model of Tuberculosis (TB that incorporates direct progression and latent reactivation. Our analysis establishes that the global dynamics of the model are completely determined by a basic reproduction number R 0. If R 0 1, the TB always dies out. If R 0 > 1, the TB becomes endemic, and a unique endemic equilibrium is globally asymptotically stable in the interior of the feasible region. 1 Introduction Tuberculosis (TB is an ancient disease caused by the infection of bacterium Mycobacterium tuberculosis. Once thought under control using antibiotic therapies, TB made a dramatic come back in the late eighties and early nineties, largely due to the emergence of antibiotic resistant strains and to co-infection with the HIV. Currently, the global per capita incidence rate of TB is growing at approximately 1.1% per year, and the number of cases at 2.4% per year. According to the 2004 WHO report Global Tuberculosis Control [1], there were 8.8 million new cases of TB worldwide in 2002, with close to 2 million TB-related deaths, more than any other infectious diseases. TB remains as one of the most serious health problems facing the world today. Mathematical models have been used to improve our understanding of the basic transmission dynamics of the TB and to evaluate the effectiveness of various control and prevention strategies [2 11]. The TB bacteria can spread in the air from a person with active TB disease to others when they are in close contact. When first infected with TB bacteria, a person typically goes through a latent, asymptomatic and noninfectious period during which the body s immune system fights the TB bacteria. We assume that there are two distinct pathogenic mechanisms of the TB infection. One is direct progression or primary progressive Keywords: Tuberculosis(TB, basic reproduction number, endemic equilibrium, global stability, yapunov functions. Copyright c Applied Mathematics Institute, University of Alberta. 313
2 314 HONGBIN GUO TB that the disease develops soon after infection. Another is endogenous reactivation or slow TB that the disease can develop many years after infection. Using a compartmental approach, the total host population can be partitioned into three compartments: susceptible individuals (, latently infected individuals ( and individuals with active TB disease (T. Only individuals in compartment T are infectious, and new infections result from contacts between a susceptible and an infectious individual, with an incidence rate β(tt (t. Here (t, (t, and T (t denote the number of individuals in the three corresponding compartments at time t. Once infected, a fraction p, 0 p 1, of the newly infected individuals develop tuberculosis directly and the remaining 1 p fraction of the newly infected progresses to the latent class. Once there, the rate of progression to active disease is at a lower rate ν. Recruitment to the susceptible population occurs at a constant rate π and removal rates for the three compartments are µ, µ and µ T, respectively. Here removal rate may include natural death, death due to TB. The dynamical transfer among the three compartments is depicted in the following transfer diagram. Here all parameters are assumed to be positive. FIGURE 1: The transfer diagram for model (1. Based on our assumptions and the transfer diagram, the model can be described by three ordinary differential equations as follows: (1 = π βt µ, = (1 pβt (ν + µ, T = pβt + ν µ T T. The model contains an earlier model proposed by Blower et al. [3] to discuss effectiveness of treating TB patients at the early infection stage. A basic reproduction number R 0 is derived in [3], (2 R 0 = β(pµ + νπ µ T (µ + νµ,
3 A MATHEMATICA MODE OF TUBERCUOSIS 315 based on which quantitative analysis was carried out. The parameter R 0 measures the average number of infections caused by one infectious individual throughout the infectious period when introduced into an entirely susceptible population. It is expected that if R 0 < 1, then no TB epidemic can develop in the population, and if R 0 > 1, a TB epidemic can develop and become endemic in the population. In the present paper, we give a rigorous mathematical analysis of model (1, and prove that the global dynamics of the model are completely determined by the parameter R 0 in (2. More specifically, we prove that if R 0 1, then the disease-free equilibrium P 0 = (π/µ, 0, 0 is globally stable in the feasible region; if R 0 > 1, P 0 is unstable, and a unique endemic equilibrium P = (,, T with,, T > 0 exists and is asymptotically stable. Furthermore, all solutions in the interior of the feasible region converge to P. In particular, our results establish that the R 0 in (2 as derived in [3] is a sharp threshold parameter for the global dynamics of (1. In the next section, we discuss the feasible region of the model and its equilibria. The global dynamics when R 0 1 are established in Section 3, and the global results when R 0 > 1 are given in Section 4. 2 Feasible region and equilibria of the system From (1 we have π µ, and thus lim sup t (t π/µ along each solution to (1. et N(t = (t + (t + T (t. Then using (1 we have N = π µ µ µ T T π µn, where µ = min{µ, µ, µ T }. This implies that lim sup t N(t π/µ. Therefore the model can be studied in the feasible region (3 Γ = { (,, T R 3 + : 0 π, T π }, µ µ where R 3 + denotes the non-negative cone of R3 including its lower dimensional faces. It can be verified that Γ is positively invariant with respect to (1. We denote by Γ and Int Γ the closure and the interior of Γ in R 3 +, respectively.
4 316 HONGBIN GUO An equilibrium (,, T of (1 satisfies the following equations (4 π = βt + µ, (1 pβt = (ν + µ, pβt + ν = µ T T. Simplifying these equations we obtain Therefore, [ ] ν(1 pβ µ + ν + pβ µ T T = 0. either T = 0 or = µ T (µ + ν β(pµ + ν. Correspondingly, system (1 has two possible equilibria: the diseasefree equilibrium P 0 = (π/µ, 0, 0 and the endemic equilibrium P = (,, T where (5 = µ T (µ + ν β(pµ + ν. From (2, the basic reproduction number R 0 satisfies R 0 = π µ. This relation can be used to estimate the value of R 0 from data on the susceptible fraction /(π/µ. Using R 0 we have the following expressions for the coordinates of P : (6 = π µ R 0, = (1 pπ (1 1R0, T = µ µ + ν β (R 0 1. It follows from (6 that P exists in IntΓ only when R 0 > 1. The following result is immediate. Proposition 1. System (1 has two possible equilibria. When R 0 1, the disease-free P 0 = (π/µ, 0, 0 is the only equilibrium in Γ; when R 0 > 1, both P 0 and the unique endemic equilibrium P = (,, T exist in Γ, where, and T are given in (6.
5 A MATHEMATICA MODE OF TUBERCUOSIS 317 For epidemic models of this type, it is generally expected that the global dynamics are determined by the basic reproduction number R 0 : if R 0 1, then all solutions converge to the disease-free equilibrium P 0, and the TB dies out from the population irrespective of the initial incidence; while if R 0 > 1, all solutions with positive initial conditions will be persistent and converge to the unique endemic equilibrium P, and any initial TB epidemics will become endemic in the population. In the next two sections, we rigorously establish this threshold behaviour. 3 Stability of the disease-free equilibrium P 0 In this section, we show that the disease-free equilibrium P 0 is globally asymptotically stable with respect to Γ if R 0 1, and P 0 is unstable if R 0 > 1. Theorem 2. If R 0 1, then the disease-free equilibrium P 0 is locally asymptotically stable and all solutions in Γ converge to P 0. If R 0 > 1, then P 0 is unstable. Proof. Consider a yapunov function Direct calculation leads to W = ν + (ν + µ T W = ν + (ν + µ T. = ν(1 pβt + (ν + µ pβt (ν + µ µ T T ( = β(pµ + ν π T. µ R 0 Therefore W 0 in Γ if R 0 1. Furthermore, W = 0 T = 0 or = π µ R 0. Therefore, the largest compact invariant set in G = {(,, T Γ : W = 0}, when R 0 1, is the singleton {P 0 }. asalle s Invariance Principle ([12], Chapter 2, Theorem 6.4 implies that all solutions in Γ converge to P 0. This global convergence also implies that P 0 is locally stable, since otherwise P 0 will have a homoclinic orbit that has to belong entirely to the set G Γ where W = 0, and thus contradicting the fact that the largest compact invariant set in G is the singleton {P 0 }.
6 318 HONGBIN GUO If R 0 > 1, then W > 0 at (,, T Int Γ if is sufficiently close to π/µ, except when T = 0. Solutions in Γ starting sufficiently close to P 0 leave a neighborhood of P 0 except those on the invariant -axis, on which (1 reduces to = π µ and thus (t π/µ, as t. This establishes the theorem. By Theorem 2, the disease-free equilibrium point P 0 is unstable when R 0 > 1. Moreover, the local dynamics near P 0 imply that system (1 is uniformly persistent with respect to R 3 + if R 0 > 1. Namely, there exists constant c > 0 such that lim inf t (t > c, lim inf t (t > c, lim inf t T (t > c, provided ((0, (0, T (0 R 3 +. Here the constant c is independent of initial data in R 3 +. We thus have the following corollary, whose proof is similar to that of Proposition 3.3 of [13]. Corollary 3. System (1 is uniformly persistent if and only if R 0 > 1. 4 Stability of the endemic equilibrium P when R 0 > 1 We have shown in the previous section that system (1 is uniformly persistent if and only if R 0 > 1. In this section, we further establish that all solutions in the interior of the feasible region Γ converge to the unique endemic equilibrium P if R 0 > 1. Therefore, the TB will persist at the endemic equilibrium level. The proof is accomplished by constructing a global yapunov function. yapunov functions of similar type have been used in the literature, see [14 16]. Theorem 4. Assume R 0 > 1. Then the endemic equilibrium P = (,, T is asymptotically stable. Furthermore, all solutions in the interior of Γ converge to P. Proof. Set z = (,, T Γ R 3 +. Consider a yapunov function V = V (z = ( ln + b ( ln + c (T T T ln TT, where z = P = (,, T is the endemic equilibrium and (7 b = ν pµ + ν = βν µ T (µ + ν, c = µ + ν pµ + ν = β µ T.
7 A MATHEMATICA MODE OF TUBERCUOSIS 319 We note that V (z 0, for z Int Γ, the interior of Γ, and V (z = 0 z = z. So the function V is positive definite with respect to the endemic equilibrium z = P. Computing the derivative of V along the solution of system (1, we obtain (8 dv dt = (1 + b (1 + c (1 T Using (1 and π = µ + β T from (4, we have T T. (9 (1 = π βt µ π + β T + µ = 2µ + β T βt µ (10 µ 2 β 2 T = β T + µ ( 2 + β T βt β 2 T + β T. Similarly, b (1 = b(1 pβt b(ν + µ Define b(1 pβt + b(µ + ν, c (1 T T = cpβt + cν cµ T T T (11 q = cpβt cνt T (1 pν pµ + ν, r = p(µ + ν pµ + ν. + cµ T T. Then q + r = 1, q > 0, r > 0. It follows from b(1 p + cp = q + r = 1 and (7 (12 cµ T = β, b(µ + ν = cν.
8 320 HONGBIN GUO Using (9 (12 we can simplify (8 as (13 dv dt = β T + µ (2 From (7, (4 and (5 we have β 2 T cνt T qβt rβt + b(µ + ν + cµ T T. (14 b(µ + ν = qβ T, cµ T T = β T. Using (14, we obtain (15 and qβt = q 2 b(µ + ν βt β T (16 rβt = rβ cµ T β T, cνt T = ν µ T T β T. Substitute (14, (15 and (16 into (13, then we get (17 dv dt = (2 + q β T + µ (2 β 2 T q 2 βt b(µ + ν β T rβ cµ T β T ν µ T T β T ( = µ 2 + β T ((2 + q q 2 βt b(µ + ν rβ ν. cµ T µ T T Since 2 + q = 2(q + r + q = 3q + 2r,
9 A MATHEMATICA MODE OF TUBERCUOSIS 321 we can rewrite (17 as (18 dv dt = µ (2 + β T (2r r ( + β T 3q q q 2 βt b(µ + ν. = I 1 + I 2 + I 3. Applying the inequality rβ cµ T ν µ T T a 1 + a a n n n a 1 a 2 a n, for a i 0, i = 1,, n, we obtain ( (19 I 1 = µ 2 0. Moreover, (20 ( I 2 = β T 2r r rβ cµ T β T (2r 2 r rβ cµ T = β T (2r β 2r = 0, cµ T by the definition of c in (7. Similarly (21 ( I 3 = β T 3q q q 2 βt b(µ + ν ( β T 3q 3 3 q q 2 βt b(µ + ν ( = β T βν 3q 3q 3 = 0, bµ T (µ + ν ν µ T T ν µ T T
10 322 HONGBIN GUO by the definition of b in (7. Using (19 (21 we obtain (22 dv dt = I 1 + I 2 + I 3 0, z Int Γ. Furthermore, dv /dt = 0 if and only if equalities hold in (19 (21, if and only if z = z = P. Therefore dv /dt is negative definite in Int Γ with respect to the endemic equilibrium P. This implies that the basin of attraction of P contains Int Γ. The positive definiteness of V (z with respect to P implies that P is also locally stable. This completes the proof. 5 Summary In this paper, mathematical analysis is carried out for a TB model. Global dynamics of the model are shown to be completely determined by a basic reproduction number R 0, first derived in [3]. More specifically, we have proved that if R 0 1, then the disease-free equilibrium P 0 is asymptotically stable and all solutions in the feasible region converge to P 0. If R 0 > 1, then P 0 becomes unstable, and a unique endemic equilibrium P exists and is asymptotically stable. In this case, all solutions in the interior of the feasible region converge to P. The proof of global convergence uses the method of yapunov functions. 6 Acknowledgement I wish to thank Professor Michael Y. i for direction on this research. REFERENCES 1. Global Tuberculosis Control, WHO Report, R. M. Anderson and R. M. May, Infectious Diseases of Humans, Dynamics and Control, Oxford University Press, Oxford, S. M. Blower, A. R. Mclean, T. C. Porco, P. M. Small, P. C. Hopewell, M. A. Sanchez, and A. R. Moss, The intrinsic transmission dynamics of tuberculosis epidemics, Nature Medicine 1 (1995, S. M. Blower, P. M. Small, P. C. Hopewell, Control strategies for tuberculosis epidemics: new models for old problems, Science 273 (1996, C. Castillo-Chavez and Z. Feng, To treat or not to treat: the case of tuberculosis, J. Math. Biol. 35 (1997, C. Castillo-Chavez and Z. Feng, Global stability of an age-structure model for TB and its applications to optimal vaccination strategies, Math. Biosci. 151 (1998, Z. Feng, C. Castillo-Chavez, and A. F. Capurro, A model for tuberculosis with exogenous reinfection, Theor. Popul. Biol. 57 (2000,
11 A MATHEMATICA MODE OF TUBERCUOSIS C. J.. Murray and J. A. Salomon, Modelling the impact of global tuberculosis control strategies, Proc. Natl. Acad. Sci. 95 (1998, T. C. Porco and S. M. Blower, Quantifying the intrinsic transmission dynamics of tuberculosis, Theor. Popul, Biol. 54 (1998, H. Waaler, A. Geser, and S. Andersen, The use of mathematical models in the study of the epidemiology of tuberculosis, Am. J. Public Health 52 (1962, Elad Ziv, Charles. Daley, and Sally M. Blower, Early therapy for latent tuberculosis infection, Amer. J. Epidemiol. 153 (2001, J. P. asalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, M. Y. i, J. R. Graef,. Wang, and J. Karsai, Global dynamics of a SEIR model with a varying total population size, Math. Biosci. 160 (1999, H. I. Freedman and J. W. -H. So, Global stability and persistence of simple food chains, Math. Biosci. 76 (1985, Z. Ma, J. iu, and J. i, Stability analysis for differential infectivity epidemic models, Nonlinear Anal. 4 (2003, A. Korobeinikov and P. K. Maini, A yapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng. 1 (2004, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 address: hguo@math.ualberta.ca
12
GLOBAL STABILITY OF THE ENDEMIC EQUILIBRIUM OF A TUBERCULOSIS MODEL WITH IMMIGRATION AND TREATMENT
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 19, Number 1, Spring 2011 GLOBAL STABILITY OF THE ENDEMIC EQUILIBRIUM OF A TUBERCULOSIS MODEL WITH IMMIGRATION AND TREATMENT HONGBIN GUO AND MICHAEL Y. LI
More informationMathematical Model of Tuberculosis Spread within Two Groups of Infected Population
Applied Mathematical Sciences, Vol. 10, 2016, no. 43, 2131-2140 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.63130 Mathematical Model of Tuberculosis Spread within Two Groups of Infected
More informationIntroduction to SEIR Models
Department of Epidemiology and Public Health Health Systems Research and Dynamical Modelling Unit Introduction to SEIR Models Nakul Chitnis Workshop on Mathematical Models of Climate Variability, Environmental
More informationGlobal Stability of SEIRS Models in Epidemiology
Global Stability of SRS Models in pidemiology M. Y. Li, J. S. Muldowney, and P. van den Driessche Department of Mathematics and Statistics Mississippi State University, Mississippi State, MS 39762 Department
More informationThursday. Threshold and Sensitivity Analysis
Thursday Threshold and Sensitivity Analysis SIR Model without Demography ds dt di dt dr dt = βsi (2.1) = βsi γi (2.2) = γi (2.3) With initial conditions S(0) > 0, I(0) > 0, and R(0) = 0. This model can
More informationA GRAPH-THEORETIC APPROACH TO THE METHOD OF GLOBAL LYAPUNOV FUNCTIONS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 8, August 2008, Pages 2793 2802 S 0002-993908)09341-6 Article electronically published on March 27, 2008 A GRAPH-THEORETIC APPROACH TO
More informationMathematical Analysis of Epidemiological Models: Introduction
Mathematical Analysis of Epidemiological Models: Introduction Jan Medlock Clemson University Department of Mathematical Sciences 8 February 2010 1. Introduction. The effectiveness of improved sanitation,
More informationGlobal stability for a four dimensional epidemic model
Note di Matematica SSN 1123-2536, e-ssn 1590-0932 Note Mat. 30 (2010) no. 2, 83 95. doi:10.1285/i15900932v30n2p83 Global stability for a four dimensional epidemic model Bruno Buonomo Department of Mathematics
More informationGLOBAL STABILITY OF SIR MODELS WITH NONLINEAR INCIDENCE AND DISCONTINUOUS TREATMENT
Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 304, pp. 1 8. SSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu GLOBAL STABLTY
More informationQualitative Analysis of a Discrete SIR Epidemic Model
ISSN (e): 2250 3005 Volume, 05 Issue, 03 March 2015 International Journal of Computational Engineering Research (IJCER) Qualitative Analysis of a Discrete SIR Epidemic Model A. George Maria Selvam 1, D.
More informationThe dynamics of disease transmission in a Prey Predator System with harvesting of prey
ISSN: 78 Volume, Issue, April The dynamics of disease transmission in a Prey Predator System with harvesting of prey, Kul Bhushan Agnihotri* Department of Applied Sciences and Humanties Shaheed Bhagat
More informationBehavior Stability in two SIR-Style. Models for HIV
Int. Journal of Math. Analysis, Vol. 4, 2010, no. 9, 427-434 Behavior Stability in two SIR-Style Models for HIV S. Seddighi Chaharborj 2,1, M. R. Abu Bakar 2, I. Fudziah 2 I. Noor Akma 2, A. H. Malik 2,
More informationGlobal Analysis of an SEIRS Model with Saturating Contact Rate 1
Applied Mathematical Sciences, Vol. 6, 2012, no. 80, 3991-4003 Global Analysis of an SEIRS Model with Saturating Contact Rate 1 Shulin Sun a, Cuihua Guo b, and Chengmin Li a a School of Mathematics and
More informationGlobal stability of an SEIS epidemic model with recruitment and a varying total population size
Mathematical Biosciences 170 2001) 199±208 www.elsevier.com/locate/mbs Global stability of an SEIS epidemic model with recruitment and a varying total population size Meng Fan a, Michael Y. Li b, *, Ke
More informationA Modeling Approach for Assessing the Spread of Tuberculosis and Human Immunodeficiency Virus Co-Infections in Thailand
Kasetsart J. (at. Sci.) 49 : 99 - (5) A Modeling Approach for Assessing the Spread of Tuberculosis and uman Immunodeficiency Virus Co-Infections in Thailand Kornkanok Bunwong,3, Wichuta Sae-jie,3,* and
More informationGLOBAL STABILITY OF A VACCINATION MODEL WITH IMMIGRATION
Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 92, pp. 1 10. SSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu GLOBAL STABLTY
More informationarxiv: v2 [q-bio.pe] 3 Oct 2018
Journal of Mathematical Biology manuscript No. (will be inserted by the editor Global stability properties of renewal epidemic models Michael T. Meehan Daniel G. Cocks Johannes Müller Emma S. McBryde arxiv:177.3489v2
More informationHETEROGENEOUS MIXING IN EPIDEMIC MODELS
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 2, Number 1, Spring 212 HETEROGENEOUS MIXING IN EPIDEMIC MODELS FRED BRAUER ABSTRACT. We extend the relation between the basic reproduction number and the
More informationStability Analysis of a SIS Epidemic Model with Standard Incidence
tability Analysis of a I Epidemic Model with tandard Incidence Cruz Vargas-De-León Received 19 April 2011; Accepted 19 Octuber 2011 leoncruz82@yahoo.com.mx Abstract In this paper, we study the global properties
More informationSI j RS E-Epidemic Model With Multiple Groups of Infection In Computer Network. 1 Introduction. Bimal Kumar Mishra 1, Aditya Kumar Singh 2
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.13(2012) No.3,pp.357-362 SI j RS E-Epidemic Model With Multiple Groups of Infection In Computer Network Bimal Kumar
More informationIntroduction: What one must do to analyze any model Prove the positivity and boundedness of the solutions Determine the disease free equilibrium
Introduction: What one must do to analyze any model Prove the positivity and boundedness of the solutions Determine the disease free equilibrium point and the model reproduction number Prove the stability
More informationThe Existence and Stability Analysis of the Equilibria in Dengue Disease Infection Model
Journal of Physics: Conference Series PAPER OPEN ACCESS The Existence and Stability Analysis of the Equilibria in Dengue Disease Infection Model Related content - Anomalous ion conduction from toroidal
More informationStability of SEIR Model of Infectious Diseases with Human Immunity
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1811 1819 Research India Publications http://www.ripublication.com/gjpam.htm Stability of SEIR Model of Infectious
More informationDynamics of Disease Spread. in a Predator-Prey System
Advanced Studies in Biology, vol. 6, 2014, no. 4, 169-179 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/asb.2014.4845 Dynamics of Disease Spread in a Predator-Prey System Asrul Sani 1, Edi Cahyono
More informationSTABILITY ANALYSIS OF A GENERAL SIR EPIDEMIC MODEL
VFAST Transactions on Mathematics http://vfast.org/index.php/vtm@ 2013 ISSN: 2309-0022 Volume 1, Number 1, May-June, 2013 pp. 16 20 STABILITY ANALYSIS OF A GENERAL SIR EPIDEMIC MODEL Roman Ullah 1, Gul
More informationMathematical models on Malaria with multiple strains of pathogens
Mathematical models on Malaria with multiple strains of pathogens Yanyu Xiao Department of Mathematics University of Miami CTW: From Within Host Dynamics to the Epidemiology of Infectious Disease MBI,
More informationReproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission
Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission P. van den Driessche a,1 and James Watmough b,2, a Department of Mathematics and Statistics, University
More informationGlobal Analysis of an Epidemic Model with Nonmonotone Incidence Rate
Global Analysis of an Epidemic Model with Nonmonotone Incidence Rate Dongmei Xiao Department of Mathematics, Shanghai Jiaotong University, Shanghai 00030, China E-mail: xiaodm@sjtu.edu.cn and Shigui Ruan
More informationSIR Epidemic Model with total Population size
Advances in Applied Mathematical Biosciences. ISSN 2248-9983 Volume 7, Number 1 (2016), pp. 33-39 International Research Publication House http://www.irphouse.com SIR Epidemic Model with total Population
More informationBifurcations in an SEIQR Model for Childhood Diseases
Bifurcations in an SEIQR Model for Childhood Diseases David J. Gerberry Purdue University, West Lafayette, IN, USA, 47907 Conference on Computational and Mathematical Population Dynamics Campinas, Brazil
More informationEpidemic Model of HIV/AIDS Transmission Dynamics with Different Latent Stages Based on Treatment
American Journal of Applied Mathematics 6; 4(5): -4 http://www.sciencepublishinggroup.com/j/ajam doi:.648/j.ajam.645.4 ISSN: -4 (Print); ISSN: -6X (Online) Epidemic Model of HIV/AIDS Transmission Dynamics
More informationModels of Infectious Disease Formal Demography Stanford Summer Short Course James Holland Jones, Instructor. August 15, 2005
Models of Infectious Disease Formal Demography Stanford Summer Short Course James Holland Jones, Instructor August 15, 2005 1 Outline 1. Compartmental Thinking 2. Simple Epidemic (a) Epidemic Curve 1:
More informationGlobal Properties for Virus Dynamics Model with Beddington-DeAngelis Functional Response
Global Properties for Virus Dynamics Model with Beddington-DeAngelis Functional Response Gang Huang 1,2, Wanbiao Ma 2, Yasuhiro Takeuchi 1 1,Graduate School of Science and Technology, Shizuoka University,
More informationGlobal Stability of a Computer Virus Model with Cure and Vertical Transmission
International Journal of Research Studies in Computer Science and Engineering (IJRSCSE) Volume 3, Issue 1, January 016, PP 16-4 ISSN 349-4840 (Print) & ISSN 349-4859 (Online) www.arcjournals.org Global
More informationA comparison of delayed SIR and SEIR epidemic models
Nonlinear Analysis: Modelling and Control, 2011, Vol. 16, No. 2, 181 190 181 A comparison of delayed SIR and SEIR epidemic models Abdelilah Kaddar a, Abdelhadi Abta b, Hamad Talibi Alaoui b a Université
More informationGLOBAL STABILITY OF A 9-DIMENSIONAL HSV-2 EPIDEMIC MODEL
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 9 Number 4 Winter 0 GLOBAL STABILITY OF A 9-DIMENSIONAL HSV- EPIDEMIC MODEL Dedicated to Professor Freedman on the Occasion of his 70th Birthday ZHILAN FENG
More informationResearch Article A Delayed Epidemic Model with Pulse Vaccination
Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2008, Article ID 746951, 12 pages doi:10.1155/2008/746951 Research Article A Delayed Epidemic Model with Pulse Vaccination
More informationA Mathematical Analysis on the Transmission Dynamics of Neisseria gonorrhoeae. Yk j N k j
North Carolina Journal of Mathematics and Statistics Volume 3, Pages 7 20 (Accepted June 23, 2017, published June 30, 2017 ISSN 2380-7539 A Mathematical Analysis on the Transmission Dynamics of Neisseria
More informationModelling of the Hand-Foot-Mouth-Disease with the Carrier Population
Modelling of the Hand-Foot-Mouth-Disease with the Carrier Population Ruzhang Zhao, Lijun Yang Department of Mathematical Science, Tsinghua University, China. Corresponding author. Email: lyang@math.tsinghua.edu.cn,
More informationStability Analysis of an SVIR Epidemic Model with Non-linear Saturated Incidence Rate
Applied Mathematical Sciences, Vol. 9, 215, no. 23, 1145-1158 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.215.41164 Stability Analysis of an SVIR Epidemic Model with Non-linear Saturated
More informationPARAMETER ESTIMATION IN EPIDEMIC MODELS: SIMPLIFIED FORMULAS
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 19, Number 4, Winter 211 PARAMETER ESTIMATION IN EPIDEMIC MODELS: SIMPLIFIED FORMULAS Dedicated to Herb Freedman on the occasion of his seventieth birthday
More informationModelling HIV/AIDS and Tuberculosis Coinfection
Modelling HIV/AIDS and Tuberculosis Coinfection C. P. Bhunu 1, W. Garira 1, Z. Mukandavire 1 Department of Applied Mathematics, National University of Science and Technology, Bulawayo, Zimbabwe 1 Abstract
More informationA Delayed HIV Infection Model with Specific Nonlinear Incidence Rate and Cure of Infected Cells in Eclipse Stage
Applied Mathematical Sciences, Vol. 1, 216, no. 43, 2121-213 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.216.63128 A Delayed HIV Infection Model with Specific Nonlinear Incidence Rate and
More informationMathematical Analysis of Epidemiological Models III
Intro Computing R Complex models Mathematical Analysis of Epidemiological Models III Jan Medlock Clemson University Department of Mathematical Sciences 27 July 29 Intro Computing R Complex models What
More informationStochastic Model for the Spread of the Hepatitis C Virus with Different Types of Virus Genome
Australian Journal of Basic and Applied Sciences, 3(): 53-65, 009 ISSN 99-878 Stochastic Model for the Spread of the Hepatitis C Virus with Different Types of Virus Genome I.A. Moneim and G.A. Mosa, Department
More informationDelay SIR Model with Nonlinear Incident Rate and Varying Total Population
Delay SIR Model with Nonlinear Incident Rate Varying Total Population Rujira Ouncharoen, Salinthip Daengkongkho, Thongchai Dumrongpokaphan, Yongwimon Lenbury Abstract Recently, models describing the behavior
More informationOptimal Treatment Strategies for Tuberculosis with Exogenous Reinfection
Optimal Treatment Strategies for Tuberculosis with Exogenous Reinfection Sunhwa Choi, Eunok Jung, Carlos Castillo-Chavez 2 Department of Mathematics, Konkuk University, Seoul, Korea 43-7 2 Department of
More informationOn CTL Response against Mycobacterium tuberculosis
Applied Mathematical Sciences, Vol. 8, 2014, no. 48, 2383-2389 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.43150 On CTL Response against Mycobacterium tuberculosis Eduardo Ibargüen-Mondragón
More informationMATHEMATICAL MODELS Vol. III - Mathematical Models in Epidemiology - M. G. Roberts, J. A. P. Heesterbeek
MATHEMATICAL MODELS I EPIDEMIOLOGY M. G. Roberts Institute of Information and Mathematical Sciences, Massey University, Auckland, ew Zealand J. A. P. Heesterbeek Faculty of Veterinary Medicine, Utrecht
More informationA Note on the Spread of Infectious Diseases. in a Large Susceptible Population
International Mathematical Forum, Vol. 7, 2012, no. 50, 2481-2492 A Note on the Spread of Infectious Diseases in a Large Susceptible Population B. Barnes Department of Mathematics Kwame Nkrumah University
More informationMathematical Modeling and Analysis of Infectious Disease Dynamics
Mathematical Modeling and Analysis of Infectious Disease Dynamics V. A. Bokil Department of Mathematics Oregon State University Corvallis, OR MTH 323: Mathematical Modeling May 22, 2017 V. A. Bokil (OSU-Math)
More informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
This article appeared in a ournal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution
More informationResearch Article Two Quarantine Models on the Attack of Malicious Objects in Computer Network
Mathematical Problems in Engineering Volume 2012, Article ID 407064, 13 pages doi:10.1155/2012/407064 Research Article Two Quarantine Models on the Attack of Malicious Objects in Computer Network Bimal
More informationStability and bifurcation analysis of epidemic models with saturated incidence rates: an application to a nonmonotone incidence rate
Published in Mathematical Biosciences and Engineering 4 785-85 DOI:.3934/mbe.4..785 Stability and bifurcation analysis of epidemic models with saturated incidence rates: an application to a nonmonotone
More informationMathematical Epidemiology Lecture 1. Matylda Jabłońska-Sabuka
Lecture 1 Lappeenranta University of Technology Wrocław, Fall 2013 What is? Basic terminology Epidemiology is the subject that studies the spread of diseases in populations, and primarily the human populations.
More information(mathematical epidemiology)
1. 30 (mathematical epidemiology) 2. 1927 10) * Anderson and May 1), Diekmann and Heesterbeek 3) 7) 14) NO. 538, APRIL 2008 1 S(t), I(t), R(t) (susceptibles ) (infectives ) (recovered/removed = βs(t)i(t)
More informationDENSITY DEPENDENCE IN DISEASE INCIDENCE AND ITS IMPACTS ON TRANSMISSION DYNAMICS
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 19, Number 3, Fall 2011 DENSITY DEPENDENCE IN DISEASE INCIDENCE AND ITS IMPACTS ON TRANSMISSION DYNAMICS REBECCA DE BOER AND MICHAEL Y. LI ABSTRACT. Incidence
More informationSmoking as Epidemic: Modeling and Simulation Study
American Journal of Applied Mathematics 2017; 5(1): 31-38 http://www.sciencepublishinggroup.com/j/ajam doi: 10.11648/j.ajam.20170501.14 ISSN: 2330-0043 (Print); ISSN: 2330-006X (Online) Smoking as Epidemic:
More informationSimple Mathematical Model for Malaria Transmission
Journal of Advances in Mathematics and Computer Science 25(6): 1-24, 217; Article no.jamcs.37843 ISSN: 2456-9968 (Past name: British Journal of Mathematics & Computer Science, Past ISSN: 2231-851) Simple
More informationThe E ect of Occasional Smokers on the Dynamics of a Smoking Model
International Mathematical Forum, Vol. 9, 2014, no. 25, 1207-1222 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.46120 The E ect of Occasional Smokers on the Dynamics of a Smoking Model
More informationAnalysis of a model for hepatitis C virus transmission that includes the effects of vaccination and waning immunity
Analysis of a model for hepatitis C virus transmission that includes the effects of vaccination and waning immunity Daniah Tahir Uppsala University Department of Mathematics 7516 Uppsala Sweden daniahtahir@gmailcom
More informationSUBTHRESHOLD AND SUPERTHRESHOLD COEXISTENCE OF PATHOGEN VARIANTS: THE IMPACT OF HOST AGE-STRUCTURE
SUBTHRESHOLD AND SUPERTHRESHOLD COEXISTENCE OF PATHOGEN VARIANTS: THE IMPACT OF HOST AGE-STRUCTURE MAIA MARTCHEVA, SERGEI S. PILYUGIN, AND ROBERT D. HOLT Abstract. It is well known that in the most general
More informationA Model on the Impact of Treating Typhoid with Anti-malarial: Dynamics of Malaria Concurrent and Co-infection with Typhoid
International Journal of Mathematical Analysis Vol. 9, 2015, no. 11, 541-551 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.412403 A Model on the Impact of Treating Typhoid with Anti-malarial:
More informationGlobal Dynamics of an SEIRS Epidemic Model with Constant Immigration and Immunity
Global Dynamics of an SIRS pidemic Model with Constant Immigration and Immunity Li juan Zhang Institute of disaster prevention Basic Course Department Sanhe, Hebei 065201 P. R. CHIA Lijuan262658@126.com
More informationAustralian Journal of Basic and Applied Sciences
AENSI Journals Australian Journal of Basic and Applied Sciences ISSN:1991-8178 Journal home page: www.ajbasweb.com A SIR Transmission Model of Political Figure Fever 1 Benny Yong and 2 Nor Azah Samat 1
More informationSensitivity and Stability Analysis of Hepatitis B Virus Model with Non-Cytolytic Cure Process and Logistic Hepatocyte Growth
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 3 2016), pp. 2297 2312 Research India Publications http://www.ripublication.com/gjpam.htm Sensitivity and Stability Analysis
More informationGlobal Stability Results for a Tuberculosis Epidemic Model
Research Journal of Mathematics and Statistics 4(): 4-20, 202 ISSN:2040-7505 Maxwell Scientific Organization, 202 Submitted: December 09, 20 Accepted: February 5, 202 Published: February 25, 202 Global
More informationModelling the spread of bacterial infectious disease with environmental effect in a logistically growing human population
Nonlinear Analysis: Real World Applications 7 2006) 341 363 www.elsevier.com/locate/na Modelling the spread of bacterial infectious disease with environmental effect in a logistically growing human population
More informationInfectious Disease Modeling with Interpersonal Contact Patterns as a Heterogeneous Network
Infectious Disease Modeling with Interpersonal Contact Patterns as a Heterogeneous Network by Joanna Boneng A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for
More informationApplications in Biology
11 Applications in Biology In this chapter we make use of the techniques developed in the previous few chapters to examine some nonlinear systems that have been used as mathematical models for a variety
More informationModels of Infectious Disease Formal Demography Stanford Spring Workshop in Formal Demography May 2008
Models of Infectious Disease Formal Demography Stanford Spring Workshop in Formal Demography May 2008 James Holland Jones Department of Anthropology Stanford University May 3, 2008 1 Outline 1. Compartmental
More informationThe Effect of Stochastic Migration on an SIR Model for the Transmission of HIV. Jan P. Medlock
The Effect of Stochastic Migration on an SIR Model for the Transmission of HIV A Thesis Presented to The Faculty of the Division of Graduate Studies by Jan P. Medlock In Partial Fulfillment of the Requirements
More informationAnalysis of Numerical and Exact solutions of certain SIR and SIS Epidemic models
Journal of Mathematical Modelling and Application 2011, Vol. 1, No. 4, 51-56 ISSN: 2178-2423 Analysis of Numerical and Exact solutions of certain SIR and SIS Epidemic models S O Maliki Department of Industrial
More informationMathematical Analysis of HIV/AIDS Prophylaxis Treatment Model
Applied Mathematical Sciences, Vol. 12, 2018, no. 18, 893-902 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.8689 Mathematical Analysis of HIV/AIDS Prophylaxis Treatment Model F. K. Tireito,
More informationTransmission Dynamics of an Influenza Model with Vaccination and Antiviral Treatment
Bulletin of Mathematical Biology (2010) 72: 1 33 DOI 10.1007/s11538-009-9435-5 ORIGINAL ARTICLE Transmission Dynamics of an Influenza Model with Vaccination and Antiviral Treatment Zhipeng Qiu a,, Zhilan
More informationApparent paradoxes in disease models with horizontal and vertical transmission
Apparent paradoxes in disease models with horizontal and vertical transmission Thanate Dhirasakdanon, Stanley H.Faeth, Karl P. Hadeler*, Horst R. Thieme School of Life Sciences School of Mathematical and
More informationDynamical models of HIV-AIDS e ect on population growth
Dynamical models of HV-ADS e ect on population growth David Gurarie May 11, 2005 Abstract We review some known dynamical models of epidemics, given by coupled systems of di erential equations, and propose
More informationApplied Mathematics Letters
Applied athematics Letters 25 (212) 156 16 Contents lists available at SciVerse ScienceDirect Applied athematics Letters journal homepage: www.elsevier.com/locate/aml Globally stable endemicity for infectious
More informationA sharp threshold for disease persistence in host metapopulations
A sharp threshold for disease persistence in host metapopulations Thanate Dhirasakdanon, Horst R. Thieme, and P. van den Driessche Department of Mathematics and Statistics, Arizona State University, Tempe,
More informationPermanence Implies the Existence of Interior Periodic Solutions for FDEs
International Journal of Qualitative Theory of Differential Equations and Applications Vol. 2, No. 1 (2008), pp. 125 137 Permanence Implies the Existence of Interior Periodic Solutions for FDEs Xiao-Qiang
More informationAssessing the effects of multiple infections and long latency in the dynamics of tuberculosis
RESEARCH Open Access Assessing the effects of multiple infections and long latency in the dynamics of tuberculosis Hyun M Yang, Silvia M Raimundo Correspondence: hyunyang@ime. unicamp.br UNICAMP-IMECC.
More informationGLOBAL DYNAMICS OF A TWO-STRAIN DISEASE MODEL WITH LATENCY AND SATURATING INCIDENCE RATE
CANADIAN APPLIED MATHEMATIC QUARTERLY Volume 2, Number 1, pring 212 GLOBAL DYNAMIC OF A TWO-TRAIN DIEAE MODEL WITH LATENCY AND ATURATING INCIDENCE RATE Dedicated to Professor H.I. Freedman s 7th birthday.
More informationDYNAMICAL MODELS OF TUBERCULOSIS AND THEIR APPLICATIONS. Carlos Castillo-Chavez. Baojun Song. (Communicated by Yang Kuang)
MATHEMATICAL BIOSCIENCES http://math.asu.edu/ mbe/ AND ENGINEERING Volume 1, Number 2, September 2004 pp. 361 404 DYNAMICAL MODELS OF TUBERCULOSIS AND THEIR APPLICATIONS Carlos Castillo-Chavez Department
More informationA Time Since Recovery Model with Varying Rates of Loss of Immunity
Bull Math Biol (212) 74:281 2819 DOI 1.17/s11538-12-978-7 ORIGINAL ARTICLE A Time Since Recovery Model with Varying Rates of Loss of Immunity Subhra Bhattacharya Frederick R. Adler Received: 7 May 212
More informationDemographic impact and controllability of malaria in an SIS model with proportional fatality
Demographic impact and controllability of malaria in an SIS model with proportional fatality Muntaser Safan 1 Ahmed Ghazi Mathematics Department, Faculty of Science, Mansoura University, 35516 Mansoura,
More informationMODELING MAJOR FACTORS THAT CONTROL TUBERCULOSIS (TB) SPREAD IN CHINA
MODELING MAJOR FACTORS THAT CONTROL TUBERCULOSIS (TB) SPREAD IN CHINA XUE-ZHI LI +, SOUVIK BHATTACHARYA, JUN-YUAN YANG, AND MAIA MARTCHEVA Abstract. This article introduces a novel model that studies the
More informationResearch Article Modelling the Role of Diagnosis, Treatment, and Health Education on Multidrug-Resistant Tuberculosis Dynamics
International Scholarly Research Network ISRN Biomathematics Volume, Article ID 5989, pages doi:.5//5989 Research Article Modelling the Role of Diagnosis, Treatment, and Health Education on Multidrug-Resistant
More informationMATHEMATICAL ANALYSIS OF THE TRANSMISSION DYNAMICS OF HIV/TB COINFECTION IN THE PRESENCE OF TREATMENT
MATHEMATICAL BIOSCIENCES http://www.mbejournal.org/ AND ENGINEERING Volume 5 Number 1 January 28 pp. 145 174 MATHEMATICAL ANALYSIS OF THE TRANSMISSION DYNAMICS OF HIV/TB COINFECTION IN THE PRESENCE OF
More informationMathematical modelling and controlling the dynamics of infectious diseases
Mathematical modelling and controlling the dynamics of infectious diseases Musa Mammadov Centre for Informatics and Applied Optimisation Federation University Australia 25 August 2017, School of Science,
More informationBifurcation Analysis in Simple SIS Epidemic Model Involving Immigrations with Treatment
Appl. Math. Inf. Sci. Lett. 3, No. 3, 97-10 015) 97 Applied Mathematics & Information Sciences Letters An International Journal http://dx.doi.org/10.1785/amisl/03030 Bifurcation Analysis in Simple SIS
More informationThreshold Conditions in SIR STD Models
Applied Mathematical Sciences, Vol. 3, 2009, no. 7, 333-349 Threshold Conditions in SIR STD Models S. Seddighi Chaharborj 1,, M. R. Abu Bakar 1, V. Alli 2 and A. H. Malik 1 1 Department of Mathematics,
More informationSupplemental Information Population Dynamics of Epidemic and Endemic States of Drug-Resistance Emergence in Infectious Diseases
1 2 3 4 Supplemental Information Population Dynamics of Epidemic and Endemic States of Drug-Resistance Emergence in Infectious Diseases 5 Diána Knipl, 1 Gergely Röst, 2 Seyed M. Moghadas 3, 6 7 8 9 1 1
More informationStability analysis of delayed SIR epidemic models with a class of nonlinear incidence rates
Published in Applied Mathematics and Computation 218 (2012 5327-5336 DOI: 10.1016/j.amc.2011.11.016 Stability analysis of delayed SIR epidemic models with a class of nonlinear incidence rates Yoichi Enatsu
More informationBifurcation Analysis of a Vaccination Model of Tuberculosis Infection
merican Journal of pplied and Industrial Cemistry 2017; 1(1): 5-9 ttp://www.sciencepublisinggroup.com/j/ajaic doi: 10.11648/j.ajaic.20170101.12 Bifurcation nalysis of a Vaccination Model of Tuberculosis
More informationAustralian Journal of Basic and Applied Sciences. Effect of Personal Hygiene Campaign on the Transmission Model of Hepatitis A
Australian Journal of Basic and Applied Sciences, 9(13) Special 15, Pages: 67-73 ISSN:1991-8178 Australian Journal of Basic and Applied Sciences Journal home page: wwwajbaswebcom Effect of Personal Hygiene
More informationStability of a Numerical Discretisation Scheme for the SIS Epidemic Model with a Delay
Stability of a Numerical Discretisation Scheme for the SIS Epidemic Model with a Delay Ekkachai Kunnawuttipreechachan Abstract This paper deals with stability properties of the discrete numerical scheme
More informationMathematical Analysis of Visceral Leishmaniasis Model
vol. 1 (2017), Article I 101263, 16 pages doi:10.11131/2017/101263 AgiAl Publishing House http://www.agialpress.com/ Research Article Mathematical Analysis of Visceral Leishmaniasis Model F. Boukhalfa,
More informationSTUDY OF THE DYNAMICAL MODEL OF HIV
STUDY OF THE DYNAMICAL MODEL OF HIV M.A. Lapshova, E.A. Shchepakina Samara National Research University, Samara, Russia Abstract. The paper is devoted to the study of the dynamical model of HIV. An application
More informationNew results on the existences of solutions of the Dirichlet problem with respect to the Schrödinger-prey operator and their applications
Chen Zhang Journal of Inequalities Applications 2017 2017:143 DOI 10.1186/s13660-017-1417-9 R E S E A R C H Open Access New results on the existences of solutions of the Dirichlet problem with respect
More informationThe death of an epidemic
LECTURE 2 Equilibrium Stability Analysis & Next Generation Method The death of an epidemic In SIR equations, let s divide equation for dx/dt by dz/ dt:!! dx/dz = - (β X Y/N)/(γY)!!! = - R 0 X/N Integrate
More information