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Journal ArticleDOI

Human-vector malaria transmission model structured by age, time since infection and waning immunity

TL;DR: An age-structured model accounting for the chronological age of humans and mosquito population, the time since humans and mosquitoes are infected and humans waning immunity is formulated, which highlights the effect of above structural variables on key important epidemiological traits of the human-vector association.
Abstract: Malaria is one of the most common mosquito-borne diseases widespread in tropical and subtropical regions, causing thousands of deaths every year in the world. Few models considering a multiple structure model formulation including (i) the chronological age of human and mosquito populations, (ii) the time since they are infected, and (iii) humans waning immunity (i.e. the progressive loss of protective antibodies after recovery) have been developed. In this paper we formulate an age-structured model containing three structural variables. Using the integrated semigroups theory, we first handle the well-posedness of the model proposed. We also investigate the existence of steady-states. A disease-free equilibrium always exists while the existence of endemic equilibria is discussed. We derive the basic reproduction number R 0 which expression highlights the effect of the above structural variables on key important epidemiological traits of the human-vector association such as vectorial capacity (i.e., vector daily reproduction rate), humans transmission probability, and survival rate. The expression of R 0 obtained here generalizes the classical formula of the basic reproduction number. Next, we derive a necessary and sufficient condition that implies the bifurcation of an endemic equilibrium. In the specific case where the age-structure of the human population is neglected, we show that a bifurcation, either backward of forward, may occur at R 0 = 1 leading to the existence, or not, of multiple endemic equilibrium when 0 ≪ R 0 1 . Finally, the latter theoretical results are enlightened by numerical simulations.

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Citations
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Journal ArticleDOI
TL;DR: This study shows the flexibility and robustness of PDE formalism to capture national COVID-19 dynamics and opens perspectives to study medium or long-term scenarios involving immune waning or virus evolution.
Abstract: The Covid-19 outbreak was followed by a huge amount of modelling studies in order to rapidly gain insights to implement the best public health policies. Most of these compartmental models involved ordinary differential equations (ODEs) systems. Such a formalism implicitly assumes that the time spent in each compartment does not depend on the time already spent in it, which is unrealistic. To overcome this “memoryless” issue, a widely used solution is to chain the number of compartments of a unique reality (e.g. have infected individual move between several compartments). This allows for greater heterogeneity, but also tends to make the whole model more difficult to apprehend and parameterize. We develop a non-Markovian alternative formalism based on partial differential equations (PDEs) instead of ODEs, which, by construction, provides a memory structure for each compartment. We apply our model to the French 2021 SARS-CoV-2 epidemic and we determine the major components that contributed to the Covid-19 hospital admissions. A global sensitivity analysis highlights a huge uncertainty attributable to the age-structured contact matrix. Our study shows the flexibility and robustness of PDE formalism to capture national COVID-19 dynamics and opens perspectives to study medium or long-term scenarios involving immune waning or virus evolution.

9 citations

Journal ArticleDOI
TL;DR: In this paper , the authors proposed a two-group malaria model structured by age with the SEIS dynamic in individuals aged below 5 years old, and SEIRS dynamic in those aged above 5 years.

6 citations

References
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01 Jan 2005

5,145 citations

Book
29 Oct 1999
TL;DR: In this paper, Spectral Theory for Semigroups and Generators is used to describe the exponential function of a semigroup and its relation to generators and resolvents.
Abstract: Linear Dynamical Systems.- Semigroups, Generators, and Resolvents.- Perturbation and Approximation of Semigroups.- Spectral Theory for Semigroups and Generators.- Asymptotics of Semigroups.- Semigroups Everywhere.- A Brief History of the Exponential Function.

4,348 citations

Journal ArticleDOI
TL;DR: It is shown that in certain special cases one can easily compute or estimate the expected number of secondary cases produced by a typical infected individual during its entire period of infectiousness in a completely susceptible population.
Abstract: The expected number of secondary cases produced by a typical infected individual during its entire period of infectiousness in a completely susceptible population is mathematically defined as the dominant eigenvalue of a positive linear operator. It is shown that in certain special cases one can easily compute or estimate this eigenvalue. Several examples involving various structuring variables like age, sexual disposition and activity are presented.

3,885 citations


"Human-vector malaria transmission m..." refers background in this paper

  • ...The number of new infections in humans that one human causes through his/her infectious period is given by R(2)0, where R0 is the basic reproduction number characterized as the spectral radius based on the next generation operator approach [13, 28]....

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  • ...Step 1: we begin by computing the next generation operator (see [13, 28])....

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  • ...It then follows (see [13, 28]), that the next generation operator is given by:...

    [...]

01 Jan 1989
TL;DR: In this paper, the expected number of secondary cases produced by a typical infected individual during its entire period of infectiousness in a completely susceptible population is defined as the dominant eigenvalue of a positive linear operator.
Abstract: The expected number of secondary cases produced by a typical infected individual during its entire period of infectiousness in a completely susceptible population is mathematically defined as the dominant eigenvalue of a positive linear operator. It is shown that in certain special cases one can easily compute or estimate this eigenvalue. Several examples involving various structuring variables like age, sexual disposition and activity are presented.

3,037 citations