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Calvetti D, Somersalo E. Post-pandemic modeling of COVID-19: Waning immunity determines recurrence frequency. Math Biosci 2023; 365:109067. [PMID: 37708989 DOI: 10.1016/j.mbs.2023.109067] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/09/2023] [Accepted: 08/16/2023] [Indexed: 09/16/2023]
Abstract
There are many factors in the current phase of the COVID-19 pandemic that signal the need for new modeling ideas. In fact, most traditional infectious disease models do not address adequately the waning immunity, in particular as new emerging variants have been able to break the immune shield acquired either by previous infection by a different strain of the virus, or by inoculation of vaccines not effective for the current variant. Furthermore, in a post-pandemic landscape in which reporting is no longer a default, it is impossible to have reliable quantitative data at the population level. Our contribution to COVID-19 post-pandemic modeling is a simple mathematical predictive model along the age-distributed population framework, that can take into account the waning immunity in a transparent and easily controllable manner. Numerical simulations show that under static conditions, the model produces periodic solutions that are qualitatively similar to the reported data, with the period determined by the immunity waning profile. Evidence from the mathematical model indicates that the immunity dynamics is the main factor in the recurrence of infection spikes, however, irregular perturbation of the transmission rate, due to either mutations of the pathogen or human behavior, may result in suppression of recurrent spikes, and irregular time intervals between consecutive peaks. The spike amplitudes are sensitive to the transmission rate and vaccination strategies, but also to the skewness of the profile describing the waning immunity, suggesting that these factors should be taken into consideration when making predictions about future outbreaks.
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Affiliation(s)
- D Calvetti
- Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, 30100 Euclid Avenue, Cleveland, OH 44106, United States of America
| | - E Somersalo
- Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, 30100 Euclid Avenue, Cleveland, OH 44106, United States of America.
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Opoku-Sarkodie R, Bartha FA, Polner M, Röst G. Dynamics of an SIRWS model with waning of immunity and varying immune boosting period. JOURNAL OF BIOLOGICAL DYNAMICS 2022; 16:596-618. [PMID: 35943129 DOI: 10.1080/17513758.2022.2109766] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/18/2022] [Accepted: 07/29/2022] [Indexed: 06/15/2023]
Abstract
SIRS models capture transmission dynamics of infectious diseases for which immunity is not lifelong. Extending these models by a W compartment for individuals with waning immunity, the boosting of the immune system upon repeated exposure may be incorporated. Previous analyses assumed identical waning rates from R to W and from W to S. This implicitly assumes equal length for the period of full immunity and of waned immunity. We relax this restriction, and allow an asymmetric partitioning of the total immune period. Stability switches of the endemic equilibrium are investigated with a combination of analytic and numerical tools. Then, continuation methods are applied to track bifurcations along the equilibrium branch. We find rich dynamics: Hopf bifurcations, endemic double bubbles, and regions of bistability. Our results highlight that the length of the period in which waning immunity can be boosted is a crucial parameter significantly influencing long term epidemiological dynamics.
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Affiliation(s)
| | | | - Mónika Polner
- Bolyai Institute, University of Szeged, Szeged, Hungary
| | - Gergely Röst
- Bolyai Institute, University of Szeged, Szeged, Hungary
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Okuwa K, Inaba H, Kuniya T. An age-structured epidemic model with boosting and waning of immune status. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2021; 18:5707-5736. [PMID: 34517509 DOI: 10.3934/mbe.2021289] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
In this paper, we developed an age-structured epidemic model that takes into account boosting and waning of immune status of host individuals. For many infectious diseases, the immunity of recovered individuals may be waning as time evolves, so reinfection could occur, but also their immune status could be boosted if they have contact with infective agent. According to the idea of the Aron's malaria model, we incorporate a boosting mechanism expressed by reset of recovery-age (immunity clock) into the SIRS epidemic model. We established the mathematical well-posedness of our formulation and showed that the initial invasion condition and the endemicity can be characterized by the basic reproduction number $ R_0 $. Our focus is to investigate the condition to determine the direction of bifurcation of endemic steady states bifurcated from the disease-free steady state, because it is a crucial point for disease prevention strategy whether there exist subcritical endemic steady states. Based on a recent result by Martcheva and Inaba [1], we have determined the direction of bifurcation that endemic steady states bifurcate from the disease-free steady state when the basic reproduction number passes through the unity. Finally, we have given a necessary and sufficient condition for backward bifurcation to occur.
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Affiliation(s)
- Kento Okuwa
- Manulife Life Insurance Company, 3-20-2 Nishishinjuku Shinjuku-ku Tokyo 163-1430, Japan
| | - Hisashi Inaba
- Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba Meguro-ku Tokyo 153-8914, Japan
| | - Toshikazu Kuniya
- Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho Nada-ku Kobe 657-8501, Japan
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STRUBE LAURAF, WALTON MAYA, CHILDS LAURENM. ROLE OF REPEAT INFECTION IN THE DYNAMICS OF A SIMPLE MODEL OF WANING AND BOOSTING IMMUNITY. J BIOL SYST 2021. [DOI: 10.1142/s021833902140012x] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Some infectious diseases produce lifelong immunity while others only produce temporary immunity. In the case of short-lived immunity, the level of protection wanes over time and may be boosted upon re-exposure, via infection or vaccination. Previous work developed a simple model capturing waning and boosting immunity, known as the Susceptible-Infectious-Recovered-Waned-Susceptible (SIRWS) model, which exhibits rich dynamical behavior including supercritical and subcritical Hopf bifurcations among other structures. Here, we extend the bifurcation analyses of the SIRWS model to examine the influence of all parameters on these bifurcation structures. We show that the bistable region, involving both a stable fixed point and a stable limit cycle, exists only for a small region of biologically realistic parameter space. Furthermore, we contrast the SIRWS model with a modified version, where immune boosting may involve the occurrence of a secondary infection. Analysis of this extended model shows that oscillations and bistability, as found in the SIRWS model, depend on strong assumptions about infectivity and recovery rate from secondary infection. Understanding the dynamics of models of waning and boosting immunity is important for accurately assessing epidemiological data.
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Affiliation(s)
- LAURA F. STRUBE
- Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061, USA
| | - MAYA WALTON
- Division of Systems Biology, Virginia Tech, Blacksburg, Virginia 24061, USA
| | - LAUREN M. CHILDS
- Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061, USA
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Leung T, Campbell PT, Hughes BD, Frascoli F, McCaw JM. Infection-acquired versus vaccine-acquired immunity in an SIRWS model. Infect Dis Model 2018; 3:118-135. [PMID: 30839933 PMCID: PMC6326260 DOI: 10.1016/j.idm.2018.06.002] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/15/2018] [Revised: 05/18/2018] [Accepted: 06/05/2018] [Indexed: 12/02/2022] Open
Abstract
In some disease systems, the process of waning immunity can be subtle, involving a complex relationship between the duration of immunity-acquired either through natural infection or vaccination-and subsequent boosting of immunity through asymptomatic re-exposure. We present and analyse a model of infectious disease transmission where primary and secondary infections are distinguished to examine the interplay between infection and immunity. Additionally we allow the duration of infection-acquired immunity to differ from that of vaccine-acquired immunity to explore the impact on long-term disease patterns and prevalence of infection in the presence of immune boosting. Our model demonstrates that vaccination may induce cyclic behaviour, and the ability of vaccinations to reduce primary infections may not lead to decreased transmission. Where the boosting of vaccine-acquired immunity delays a primary infection, the driver of transmission largely remains primary infections. In contrast, if the immune boosting bypasses a primary infection, secondary infections become the main driver of transmission under a sufficiently long duration of immunity. Our results show that the epidemiological patterns of an infectious disease may change considerably when the duration of vaccine-acquired immunity differs from that of infection-acquired immunity. Our study highlights that for any particular disease and associated vaccine, a detailed understanding of the waning and boosting of immunity and how the duration of protection is influenced by infection prevalence are important as we seek to optimise vaccination strategies.
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Affiliation(s)
- Tiffany Leung
- School of Mathematics and Statistics, University of Melbourne, Parkville, Victoria 3010, Australia
| | - Patricia T. Campbell
- Victorian Infectious Diseases Reference Laboratory Epidemiology Unit, Peter Doherty Institute for Infection and Immunity, University of Melbourne and Royal Melbourne Hospital, Parkville, Victoria 3010, Australia
- Infection and Immunity Research Theme, Murdoch Children's Research Institute, Royal Children's Hospital, Parkville, Victoria 3052, Australia
| | - Barry D. Hughes
- School of Mathematics and Statistics, University of Melbourne, Parkville, Victoria 3010, Australia
| | - Federico Frascoli
- Department of Mathematics, Faculty of Science, Engineering and Technology, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia
| | - James M. McCaw
- School of Mathematics and Statistics, University of Melbourne, Parkville, Victoria 3010, Australia
- Victorian Infectious Diseases Reference Laboratory Epidemiology Unit, Peter Doherty Institute for Infection and Immunity, University of Melbourne and Royal Melbourne Hospital, Parkville, Victoria 3010, Australia
- Infection and Immunity Research Theme, Murdoch Children's Research Institute, Royal Children's Hospital, Parkville, Victoria 3052, Australia
- Melbourne School of Population and Global Health, University of Melbourne, Parkville, Victoria 3010, Australia
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Chen C, Li C, Kang Y. Modelling the effects of cutting off infected branches and replanting on fire-blight transmission using Filippov systems. J Theor Biol 2017; 439:127-140. [PMID: 29175425 DOI: 10.1016/j.jtbi.2017.11.017] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/08/2017] [Revised: 11/18/2017] [Accepted: 11/21/2017] [Indexed: 11/26/2022]
Abstract
Fire blight is one of the most devastating plant diseases in the world. This paper proposes a Filippov fire-blight model incorporating cutting off infected branches and replanting susceptible trees. The Filippov-type model is formulated by considering that no control strategy is taken if the number of infected trees is less than an infected threshold level Ic; further, we cut off infected branches once the number of infected trees exceeds Ic; meanwhile, we replant trees if the number of susceptible trees is less than a susceptible threshold level Sc. The global dynamical behaviour of the Filippov system is investigated. It is shown that model solutions ultimately converge to the positive equilibrium that lies in the region above Ic, or below Ic, or on I=Ic, as we vary the susceptible and infected threshold values Sc and Ic. Our results indicate that proper combinations of the susceptible and infected threshold values based on the threshold policy can lead the number of infected trees to an acceptable level, when complete eradication is not economically desirable.
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Affiliation(s)
- Can Chen
- Department of Mathematics and Physics, College of Science, Zhengzhou University of Aeronautics, Zhengzhou 450015, PR China; Department of Applied Mathematics, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, PR China.
| | - Chentong Li
- Department of Applied Mathematics, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, PR China.
| | - Yanmei Kang
- Department of Applied Mathematics, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, PR China.
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