Mathematical models on Malaria with multiple strains of pathogens
|
|
- Tracy Reeves
- 6 years ago
- Views:
Transcription
1 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, Columbus, Ohio
2 Outline Background Within-host Level Between-host Level Discussion and Future Work
3 Geographic Distribution of Malaria WHO, World Malaria Report 2010, December 2010.
4 The Pathogen of Malaria Malaria is a mosquito-borne infectious disease caused by Malaria parasites. Malaria parasites are members of eukaryotic protists of the genus Plasmodium. In general, there are five kinds of plasmodiums associated with human malaria infections.
5 Multiple Strains
6 Different Characters of Multiple Strains Some comparative characters of the five human malaria parasites: P. falciparum P.vivax P.ovale P.malaria P. knowlesi Duration of primary exoerythrocytlc cycle (days) Number of exoerythrocytlc merozoites Duration of erythrocytic cycle (hours) Duration of mosquito cycle at 27 C (days) (Source:
7 Multiple Strains
8 The Facts Newly transmitted P. falciparum infections were suppressing patient infections (either new or latent) with P. vivax. - K. Maitland, et al. (Parastitol Today 1997) On the Thai-Burma border, pregnant women whose first attack of malaria during pregnancy was caused by P. vivax had a significantly lower risk of developing P. falciparum later in the pregnancy. - M. Mayxay, et al. (Trend Parasitol 2004) Authors have detected..., including the co-occurrence of all 4 species in populations in Madagascar and New Guinea. - F. E. McKenzie and W. H. Bossert (J Parasitol 1997)
9 The Facts Another fact: There is no obvious cross-immunity between two species. - S.L. Hoffman (J Infect Dis 2002), K. Jangpatarapongsa (PLoS One 2012) This work answers the question by using mathematical model. To this end, we need model at within-host level, and at population level.
10 Parasites Life Cycle
11 Single Strain Within-host Level T l k T p V I ec (1- e)c V M d d m( p ) d V M d Ṫ = λ dt kv M T, Ṫ = kv M T µ(p)t, V I = pt d 1 V I cv I, V M = ɛcv I d 1 V M, V M = (1 ɛ)cv I.
12 Single Strain Within-host Level Basic reproduction number (the number of secondary cases one case generates on average over the course of its infectious period, in an otherwise uninfected population): R 0 = λkɛc d(d 1 + c)d 1 N, where N = 0 pe µ(p)a da If R 0 < 1, the parasites will be cleaned up in the host cells; if R 0 > 1, the parasites will establish a stable steady state inside of the host cells globally.
13 Double Strains Within-host Level T l d k k T m( p ) p V d I d e c V M T V I V p e c M m( p ) d d Ṫ = λ dt k 1 V M1 T k 2 V M2 T, Ṫ 1 = k 1 V M1 T µ(p 1 )T 1, Ṫ2 = k 2 V M2 T µ(p 2 )T2, V I1 = p 1 T1 d 1V I1 c 1 V I1, V I2 = p 2 T2 d 2V I2 c 2 V I2, V M1 = ɛ 1 c 1 V I1 d 1 V M1, V M2 = ɛ 2 c 2 V I2 d 2 V M2.
14 Double Strains Within-host Level The basic reproduction number R 0 = max i (R 1, R 2 ): Theorem R i = λk i ɛc i p i dµ(p i )(d i + c i )d i. If R 0 < 1, the infection free equilibrium E 0 is G-A-S. If R 0 > 1, (i) If R 1 > 1, and R 2 < R 1, E 1 exists and is G-A-S. (ii) If R 2 > 1, and R 1 < R 2, E 2 exists and is G-A-S. (iii) If R 1 = R 2 > 1, there are infinitely many co-infection equilibria. where E 1 and E 2 are boundary equilibrium for species 1 and 2, respectively. Principle of Competitive Exclusion, Hardin science 1960; Iggidr et. al. SIAP 2006;
15 Double Strains Within-host Level
16 Double Strains Within-host Level
17 Double Strains Within-host Level
18 Single Strain Between-host Level S H = b H N H d H S H ac 1 S H N H I M + βr H, I H = ac 1 S H N H I M d H I H γi H, R H = γi H d H R H βr H, S M = b M N M d M S M ac 2 S M I H N H, I M = ac 2 S M I H N H d M I M.
19 Single Strain Between-host Level Set n = N M N H and nondimensionalize the system, we have the basic reproduction number: a R 0 = 2 c 1 c 2 n d M (d H + γ) The stability of disease free equilibrium (DFE) E 0 = (1, 0, 0, 1, 0) is fully determined by R 0 : Theorem (Stability) If R 0 < 1, E 0 is G-A-S; if R 0 > 1, it is unstable.
20 Single Strain Between-host Level When R 0 > 1, there is a unique endemic equilibrium (EE) E = (SH, I H, R H, S M, I M ), and Theorem (Stability) Assume R 0 > 1, the EE E is G-A-S, provided that d H + d M max { β, β γ} > 0.
21 Double Strains Between-host Level bh b b 1 SH 2 dm dh dm ae11 ae12 IM1 IM2 ae21 ae22 RH1 g 1 IH1 dm IH2 g 2 RH2 dh dh dh dh SM IM2 IM1 bm aer2 aer1
22 Double Strains Between-host Level S H = b H N H d H S H ae 11 S H N H I M1 ae 12 S H N H I M2 + β 1 R H1 + β 2 R H2, I H1 = ae 11 S H N H I M1 d H I H1 γ 1 I H1 +ae R1 R H2 N H I M1, R H1 = γ 1 I H1 ae R2 R H1 N H I M2 d H R H1 β 1 R H1, I H2 = ae 12 S H N H I M2 d H I H2 γ 2 I H2 +ae R2 R H1 N H I M2, R H2 = γ 2 I H2 ae R1 R H2 N H I M1 d H R H2 β 2 R H2, S M I = b M N M d M S M ae 21 S H1 I M N H ae 22 S H2 M N H, I M1 = ae 21 S M I H1 N H d M I M1, I M2 = ae 22 S M I H2 N H d M I M2.
23 Double Strains Between-host Level Rescale the system S H = d H d H S H ae 11 ns H I M1 ae 12 ns H I M2 + β 1 R H1 + β 2 R H2, I H1 = ae 11 ns H I M1 d H I H1 γ 1 I H1 + ae R1 nr H2 I M1, R H1 = γ 1 I H1 ae R2 nr H1 I M2 d H R H1 β 1 R H1, I H2 = ae 12 ns H I M2 d H I H2 γ 2 I H2 + ae R2 nr H1 I M2, R H2 = γ 2 I H2 ae R1 nr H2 I M1 d H R H2 β 2 R H2, S M = d M d M S M ae 21 S M I H1 ae 22 S M I H2, I M1 = ae 21 S M I H1 d M I M1, I M2 = ae 22 S M I H2 d M I M2. where n = N M NH is the number of mosquitoes per person.
24 Double Strains Between-host Level The basic reproduction number for species i in the absence of species j, j i is: R i = a2 e 1i e 2i n, i = 1, 2. d M (d H + γ i ) Further, R 0 = max { R1, R 2 }, The system has a DFE Ē0 = (1, 0, 0, 0, 0, 1, 0, 0). Theorem (Stability) If R 0 < 1, Ē0 is G-A-S; if R 0 > 1, Ē0 becomes unstable.
25 Double Strains Between-host Level When R i > 1, i = 1, 2, there are two boundary equilibria: If R 1 > 1, Ē1 = (S H, I H1, R H1, 0, 0, S M, I M1, 0). If R 2 > 1, Ē2 = (SH, 0, 0, I H2, R H2, S M, 0, I M2 ). The stabilities of Ē1 and Ē2 are not simply decided by R i, i = 1, 2.
26 Double Strains Between-host Level Define R ji as the species i-mediated reproduction number for species j by R 21 = a2 e 12 e 22 ns H S M +a2 e 22 e R2 ns M R H1 d M (d H +γ 2 ), R 12 = a2 e 11 e 21 ns H S M +a2 e 21 e R1 nsm R H2 d M (d H +γ 1 ). Rij measures the number of secondary infections caused by an individual infected by species i, assuming the species j has been settled at Ēj. Rji can be considered as the threshold parameter for invasion of species j to residence species i.
27 Double Strains Between-host Level R ji can be considered as the threshold parameter for invasion of species j to residence species i. Theorem (Stability) (i) If R 1 > 1, R 21 < 1 and (a) d H + d M max ( β 1, β 1 γ 1 ) > 0, then Ē1 is L-A-S; (ii) If R 2 > 1, R 12 < 1 and (b) d H + d M max ( β 2, β 2 γ 2 ) > 0, then Ē2 is L-A-S.
28 Double Strains Between-host Level Theorem (Persistence) Species 1 is uniformly persistent if (C1) R 1 > 1 and R 2 < 1; or (C2) R 2 > 1, R 12 > 1 and (b) exists. Species 2 is uniformly persistent if (C3) R 2 > 1 and R 1 < 1; or (C4) R 1 > 1, R 21 > 1 and (a) exists.
29 Double Strains Between-host Level Theorem (Persistence) If one of the three holds, (i) R 1 > 1, R 2 < 1, R 21 > 1 and (b); (ii) R 2 > 1, R 1 < 1, R 12 > 1 and (a); or (iii) R 1 > 1, R 2 > 1, R 12 > 1, R 21 > 1 and (a), (b) hold; both species are uniformly persistent.
30 Double Strains Between-host Level
31 Double Strains Between-host Level
32 Conclusions We modeled the transmission of Malaria in both within- and between- host level. At within-host level: co-infection (super-infection) is generically impossible (unless R 1 = R 2 > 1). Parasites will compete with each other until only one species survives. At population level: co-existence of two species in a region is possible, as they not only compete but also benefit each other! Remark: Both within- and between- host level models can be extended to scenarios with more than two strains, but conditions are more compicated at between-host level.
33 Future Work Explore the special case, R 1 = R 2, for the within-host model (super-infection); More strains of pathogens involved; Disease latency within host and vector; Spatial impacts.
34 Some References C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity, I. Theory of Epidemics, O. Arino, D. E. Axelrod, and M. Kimmel, eds., Wuerz, Winnepeg, Canada, 1995, pp P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), pp A. Iggidr, J.C. Kamgang, G. Sallet, and J.J. Tewa, Global analysis of new malaria intrahost models with a competitive exclusion principle, SIAM J. Appl. Math., 67 (2006), pp M. Y. Li and J. S. Muldowney, A geometric approach to global-stability problems, SIAM J. Math. Anal., 27 (1996), pp
35 Thank you!
Can multiple species of Malaria co-persist in a region? Dynamics of multiple malaria species
Can multiple species of Malaria co-persist in a region? Dynamics of multiple malaria species Xingfu Zou Department of Applied Mathematics University of Western Ontario London, Ontario, Canada (Joint work
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 analysis of multi-strains SIS, SIR and MSIR epidemic models
Global analysis of multi-strains I, IR and MIR epidemic models Derdei Bichara, Abderrahman Iggidr, Gauthier allet To cite this version: Derdei Bichara, Abderrahman Iggidr, Gauthier allet. Global analysis
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 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 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 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 informationGLOBAL DYNAMICS OF A MATHEMATICAL MODEL OF TUBERCULOSIS
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
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 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 informationA mathematical model for malaria involving differential susceptibility, exposedness and infectivity of human host
A mathematical model for malaria involving differential susceptibility exposedness and infectivity of human host A. DUCROT 1 B. SOME 2 S. B. SIRIMA 3 and P. ZONGO 12 May 23 2008 1 INRIA-Anubis Sud-Ouest
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 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 informationCompetitive exclusion principle for SIS and SIR models with n strains
Competitive exclusion principle for SIS and SIR models with n strains Derdei Bichara, Abderrahman Iggidr, Gauthier Sallet To cite this version: Derdei Bichara, Abderrahman Iggidr, Gauthier Sallet. Competitive
More informationImpact of Enhanced Malaria Control on the Competition between Plasmodium falciparum and Plasmodium vivax in India
Impact of Enhanced Malaria Control on the Competition between Plasmodium falciparum and Plasmodium vivax in India Olivia Prosper 1, Maia Martcheva 1 1 Department of Mathematics, University of Florida,
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 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 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 informationMathematical modelling of the impact of vaccination on malaria epidemiology
International International Jr, of Qualitative Journal Theory of Qualitative Differential Equations Theory of and Differential ApplicationsEquations and Applications Vol. 1 No. 1 (June, Vol. 1, 2015) No.
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 informationAccepted Manuscript. Backward Bifurcations in Dengue Transmission Dynamics. S.M. Garba, A.B. Gumel, M.R. Abu Bakar
Accepted Manuscript Backward Bifurcations in Dengue Transmission Dynamics S.M. Garba, A.B. Gumel, M.R. Abu Bakar PII: S0025-5564(08)00073-4 DOI: 10.1016/j.mbs.2008.05.002 Reference: MBS 6860 To appear
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 informationSIS and SIR Epidemic Models Under Virtual Dispersal
Bull Math Biol DOI 10.1007/s11538-015-0113-5 ORIGINAL ARTICLE SIS and SIR Epidemic Models Under Virtual Dispersal Derdei Bichara 1 Yun Kang 2 Carlos Castillo-Chavez 1 Richard Horan 3 Charles Perrings 4
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 informationSlow and fast dynamics model of a Malaria with Sickle-Cell genetic disease with multi-stage infections of the mosquitoes population
Journal of Physics: Conference Series PAPER OPEN ACCESS Slow and fast dynamics model of a Malaria with Sickle-Cell genetic disease with multi-stage infections of the mosquitoes population To cite this
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 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 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 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 informationGLOBAL 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 informationA Mathematical Model for the Spatial Spread of HIV in a Heterogeneous Population
A Mathematical Model for the Spatial Spread of HIV in a Heterogeneous Population Titus G. Kassem * Department of Mathematics, University of Jos, Nigeria. Abstract Emmanuel J.D. Garba Department of Mathematics,
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 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 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 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 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 informationA mathematical model for malaria involving differential susceptibility, exposedness and infectivity of human host
Journal of Biological Dynamics ISSN: 75-3758 (Print) 75-3766 (Online) Journal homepage: http://www.tandfonline.com/loi/tjbd20 A mathematical model for malaria involving differential susceptibility exposedness
More informationIN mathematical epidemiology, deterministic models are
tability and ensitivity Analysis of a Deterministic Epidemiological Model with Pseudo-recovery amson Olaniyi, Maruf A. Lawal, Olawale. Obabiyi Abstract A deterministic epidemiological model describing
More informationMulti-strain persistence induced by host age structure
Multi-strain persistence induced by host age structure Zhipeng Qiu 1 Xuezhi Li Maia Martcheva 3 1 Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, 194, P R China
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 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 informationMODELING THE SPREAD OF DENGUE FEVER BY USING SIR MODEL. Hor Ming An, PM. Dr. Yudariah Mohammad Yusof
MODELING THE SPREAD OF DENGUE FEVER BY USING SIR MODEL Hor Ming An, PM. Dr. Yudariah Mohammad Yusof Abstract The establishment and spread of dengue fever is a complex phenomenon with many factors that
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 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 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 informationMODELLING AND ANALYSIS OF THE SPREAD OF MALARIA: ENVIRONMENTAL AND ECOLOGICAL EFFECTS
Journal of Biological Systems, Vol. 13, No. 1 (2005) 1 11 c World Scientific Publishing Company MODELLING AND ANALYSIS OF THE SPREAD OF MALARIA: ENVIRONMENTAL AND ECOLOGICAL EFFECTS SHIKHA SINGH,J.B.SHUKLA
More informationGLOBAL DYNAMICS OF A TICK IXODES SCAPULARIS MODEL
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 19, Number 1, Spring 2011 GLOBAL DYNAMICS OF A TICK IXODES SCAPULARIS MODEL YIJUN LOU AND JIANHONG WU ABSTRACT. Lyme disease remains the world s most frequently
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 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 informationA multi-species epidemic model with spatial dynamics
A multi-species epidemic model with spatial dynamics Julien Arino Jonathan R. Davis David Hartley Richard Jordan Joy M. Miller P. van den Driessche Version of January 6, 25 Abstract A model is formulated
More informationThe effect of population dispersal on the spread of a disease
J. Math. Anal. Appl. 308 (2005) 343 364 www.elsevier.com/locate/jmaa The effect of population dispersal on the spread of a disease Yu Jin, Wendi Wang Department of Mathematics, Southwest China Normal University,
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 informationStability Analysis of an HIV/AIDS Epidemic Model with Screening
International Mathematical Forum, Vol. 6, 11, no. 66, 351-373 Stability Analysis of an HIV/AIDS Epidemic Model with Screening Sarah Al-Sheikh Department of Mathematics King Abdulaziz University Jeddah,
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 informationResearch Article Modeling Computer Virus and Its Dynamics
Mathematical Problems in Engineering Volume 213, Article ID 842614, 5 pages http://dx.doi.org/1.1155/213/842614 Research Article Modeling Computer Virus and Its Dynamics Mei Peng, 1 Xing He, 2 Junjian
More informationTransmission Dynamics of Some Epidemiological Patch Models
University of Miami Scholarly Repository Open Access Dissertations Electronic Theses and Dissertations 2012-05-08 Transmission Dynamics of Some Epidemiological Patch Models Daozhou Gao University of Miami,
More informationMathematical Model of Dengue Disease Transmission Dynamics with Control Measures
Journal of Advances in Mathematics and Computer Science 23(3): 1-12, 2017; Article no.jamcs.33955 Previously known as British Journal of Mathematics & Computer Science ISSN: 2231-0851 Mathematical Model
More informationGlobal dynamics of SEIRS epidemic model with non-linear generalized incidences and preventive vaccination
Khan et al Advances in Difference Equations (2015) 2015:88 DOI 101186/s13662-015-0429-3 R E S E A R C H Open Access Global dynamics of SEIRS epidemic model with non-linear generalized incidences and preventive
More informationGLOBAL DYNAMICS OF A TIME-DELAYED DENGUE TRANSMISSION MODEL
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 2, Number 1, Spring 212 GLOBAL DYNAMICS OF A TIME-DELAYED DENGUE TRANSMISSION MODEL Dedicated to Herb Freedman on the occasion of his 7th birthday ZHEN WANG
More informationStability analysis of an SEIR epidemic model with non-linear saturated incidence and temporary immunity
Int. J. Adv. Appl. Math. and Mech. 2(3) (215) 1-14 (ISSN: 2347-2529) Journal homepage: www.ijaamm.com International Journal of Advances in Applied Mathematics and Mechanics Stability analysis of an SEIR
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 informationHIV/AIDS Treatment Model with the Incorporation of Diffusion Equations
Applied Mathematical Sciences, Vol. 12, 2018, no. 12, 603-615 HIKARI Ltd www.m-hikari.com https://doi.org/10.12988/ams.2018.8467 HIV/AIDS Treatment Model with the Incorporation of Diffusion Equations K.
More informationResilience and stability of harvested predator-prey systems to infectious diseases in the predator
Resilience and stability of harvested predator-prey systems to infectious diseases in the predator Morgane Chevé Ronan Congar Papa A. Diop November 1, 2010 Abstract In the context of global change, emerging
More informationModeling Co-Dynamics of Cervical Cancer and HIV Diseases
Global ournal of Pure Applied Mathematics. SSN 093-8 Volume 3 Number (0) pp. 05-08 Research ndia Publications http://www.riblication.com Modeling o-dynamics of ervical ancer V Diseases Geomira G. Sanga
More informationThree Disguises of 1 x = e λx
Three Disguises of 1 x = e λx Chathuri Karunarathna Mudiyanselage Rabi K.C. Winfried Just Department of Mathematics, Ohio University Mathematical Biology and Dynamical Systems Seminar Ohio University November
More informationOn the Spread of Epidemics in a Closed Heterogeneous Population
On the Spread of Epidemics in a Closed Heterogeneous Population Artem Novozhilov Applied Mathematics 1 Moscow State University of Railway Engineering (MIIT) the 3d Workshop on Mathematical Models and Numerical
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 informationAN ABSTRACT OF THE THESIS OF. Margaret-Rose W. Leung for the degree of Honors Baccalaureate of Science in Mathematics
AN ABSTRACT OF THE THESIS OF Margaret-Rose W. Leung for the degree of Honors Baccalaureate of Science in Mathematics presented on June 5, 2012. Title: A Vector Host Model for Coinfection by Barley/Cereal
More informationResearch Article Stability of a Mathematical Model of Malaria Transmission with Relapse
Abstract and Applied Analysis, Article ID 289349, 9 pages http://dx.doi.org/10.1155/2014/289349 Research Article Stability of a Mathematical Model of Malaria Transmission with Relapse Hai-Feng Huo and
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 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 informationMULTI-SCALE MODELING OF MALARIA: FROM ENDEMICITY TO ELIMINATION
MULTI-SCALE MODELING OF MALARIA: FROM ENDEMICITY TO ELIMINATION or the DEATH of SEIR models Juan B. Gutierrez UGA s Department of Mathematics & Institute of Bioinformatics December 14, 2012 1 / 19 Collaborators
More informationSTUDY OF THE BRUCELLOSIS TRANSMISSION WITH MULTI-STAGE KE MENG, XAMXINUR ABDURAHMAN
Available online at http://scik.org Commun. Math. Biol. Neurosci. 208, 208:20 https://doi.org/0.2899/cmbn/3796 ISSN: 2052-254 STUDY OF THE BRUCELLOSIS TRANSMISSION WITH MULTI-STAGE KE MENG, XAMXINUR ABDURAHMAN
More informationUNIFORM WEAK IMPLIES UNIFORM STRONG PERSISTENCE FOR NON-AUTONOMOUS SEMIFLOWS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 8, Pages 2395 2403 S 0002-9939(99)05034-0 Article electronically published on April 15, 1999 UNIFORM WEAK IMPLIES UNIFORM STRONG PERSISTENCE
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 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 informationDetermining Important Parameters in the Spread of Malaria Through the Sensitivity Analysis of a Mathematical Model
Bulletin of Mathematical Biology (2008) 70: 1272 1296 DOI 10.1007/s11538-008-9299-0 ORIGINAL ARTICLE Determining Important Parameters in the Spread of Malaria Through the Sensitivity Analysis of a Mathematical
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 informationA simple two-patch epidemiological model with Allee effects and disease-modified fitness
Contemporary Mathematics A simple two-patch epidemiological model with Allee effects and disease-modified fitness Yun Kang and Carlos Castillo-Chavez This paper is dedicated to Professor Ronald Mickens
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 informationRevisiting a two-patch SIS model with infection during transport
Mathematical Medicine and Biology 2016) 33, 29 55 doi:10.1093/imammb/dqv001 Advance Access publication on February 11, 2015 Revisiting a two-patch SIS model with infection during transport Julien Arino
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 informationVector Hazard Report: Malaria in Ghana Part 1: Climate, Demographics and Disease Risk Maps
Vector Hazard Report: Malaria in Ghana Part 1: Climate, Demographics and Disease Risk Maps Information gathered from products of The Walter Reed Biosystematics Unit (WRBU) VectorMap Systematic Catalogue
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 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 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 informationAnalysis of SIR Mathematical Model for Malaria disease with the inclusion of Infected Immigrants
IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 14, Issue 5 Ver. I (Sep - Oct 218), PP 1-21 www.iosrjournals.org Analysis of SIR Mathematical Model for Malaria disease
More informationTransmission Dynamics of Malaria in Ghana
Journal of Mathematics Research; Vol. 4, No. 6; 2012 ISSN 1916-9795 E-ISSN 1916-9809 Published by Canadian Center of Science and Education Transmission Dynamics of Malaria in Ghana Francis T. Oduro 1,
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 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 informationUnderstanding the contribution of space on the spread of Influenza using an Individual-based model approach
Understanding the contribution of space on the spread of Influenza using an Individual-based model approach Shrupa Shah Joint PhD Candidate School of Mathematics and Statistics School of Population and
More informationEpidemics in Networks Part 2 Compartmental Disease Models
Epidemics in Networks Part 2 Compartmental Disease Models Joel C. Miller & Tom Hladish 18 20 July 2018 1 / 35 Introduction to Compartmental Models Dynamics R 0 Epidemic Probability Epidemic size Review
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 informationAsynchronous oscillations due to antigenic variation in Malaria Pf
Asynchronous oscillations due to antigenic variation in Malaria Pf Jonathan L. Mitchell and Thomas W. Carr Department of Mathematics Southern Methodist University Dallas, TX SIAM LS, Pittsburgh, 21 Outline
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 informationDisease Spread in Metapopulations
Fields Institute Communications Volume 00, 0000 Disease Spread in Metapopulations Julien Arino Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada R3T 22, arinoj@cc.umanitoba.ca P.
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 informationTransmission in finite populations
Transmission in finite populations Juliet Pulliam, PhD Department of Biology and Emerging Pathogens Institute University of Florida and RAPIDD Program, DIEPS Fogarty International Center US National Institutes
More informationSupplementary Information
Supplementary Information This document shows the supplementary figures referred to in the main article. The contents are as follows: a. Malaria maps b. Dengue maps c. Yellow fever maps d. Chikungunya
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 information