Waning and boosting: on the dynamics of immune status

The aim is to describe the distribution of immune status (as captured by antibody level) on the basis of a within-host submodel for continuous waning and occasional boosting. Inspired by Feller’s fundamental work and the more recent delay equation formulation of models for the dynamics of physiologically structured populations, we derive, for given force of infection, a linear renewal equation. The solution is obtained by generation expansion, with the generation number corresponding to the number of times the individual became infected. Our main result provides a precise characterization of the stable distribution of immune status.

[1]  H.J.A.M. Heijmans,et al.  Markov Semigroups and Structured Population Dynamics , 1986 .

[2]  Mark H. Davis.  Markov Models and Optimization , 1995 .

[3]  T. Szarek,et al.  Existence of a unique invariant measure for a class of equicontinuous Markov operators with application to a stochastic model for an autoregulated gene , 2016 .

[4]  M. Kretzschmar,et al.  Incidence and Reproduction Numbers of Pertussis: Estimates from Serological and Social Contact Data in Five European Countries , 2010, PLoS medicine.

[5]  R. Rudnicki,et al.  Piecewise Deterministic Markov Processes in Biological Models , 2015, 1512.02093.

[6]  On a Convergence Theorem for Semigroups of Positive Integral Operators , 2017, 1706.01146.

[7]  C Jessica E Metcalf,et al.  Use of serological surveys to generate key insights into the changing global landscape of infectious disease , 2016, The Lancet.

[8]  O. Diekmann,et al.  On the formulation and analysis of general deterministic structured population models II. Nonlinear theory , 2000 .

[9]  Upendra Dave,et al.  Applied Probability and Queues , 1987 .

[10]  S. Lunel,et al.  Delay Equations. Functional-, Complex-, and Nonlinear Analysis , 1995 .

[11]  A. W. Kemp,et al.  Applied Probability and Queues , 1989 .

[12]  Odo Diekmann,et al.  On the formulation and analysis of general deterministic structured population models I. Linear Theory , 1998, Journal of mathematical biology.

[13]  Asymptotic decomposition of substochastic operators and semigroups , 2016 .

[15]  Ryszard Rudnicki,et al.  Continuous Markov Semigroups and Stability of Transport Equations , 2000 .

[16]  M. Kretzschmar,et al.  A two-phase within-host model for immune response and its application to serological profiles of pertussis. , 2014, Epidemics.

[17]  M. Mackey,et al.  Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics , 1998 .

[18]  R. Rudnicki,et al.  Asynchronous Exponential Growth of a General Structured Population Model , 2012 .

[19]  Sarah E. Wilson,et al.  The role of seroepidemiology in the comprehensive surveillance of vaccine-preventable diseases , 2012, Canadian Medical Association Journal.

[20]  Günther Greiner.  Spektrum und Asymptotik stark stetiger Halbgruppen positiver Operatoren , 1982 .

[21]  Michael C. Mackey,et al.  Dynamic Behavior of Stochastic Gene Expression Models in the Presence of Bursting , 2013, SIAM J. Appl. Math..

[22]  Y T H P van Duynhoven,et al.  Biomarker dynamics: estimating infection rates from serological data , 2012, Statistics in medicine.

[23]  Michael C. Mackey,et al.  Chaos, Fractals, and Noise , 1994 .

[24]  Michael C. Mackey,et al.  The statistical dynamics of recurrent biological events , 1992 .

[25]  M. Kretzschmar,et al.  Linking the seroresponse to infection to within-host heterogeneity in antibody production. , 2016, Epidemics.

[26]  J. Schellekens,et al.  Age-specific long-term course of IgG antibodies to pertussis toxin after symptomatic infection with Bordetella pertussis , 2005, Epidemiology and Infection.

[27]  Mirjam Kretzschmar,et al.  The incidence of Bordetella pertussis infections estimated in the population from a combination of serological surveys. , 2006, The Journal of infection.

[28]  H. J. A. M. Heijmans,et al.  Structured populations, linear semigroups and positivity , 1984 .

[29]  M Gyllenberg,et al.  Steady-state analysis of structured population models. , 2003, Theoretical population biology.