An introdution to the basic reproductive number in mathematical epidemiology

This article introduces the notion of basic reproduction number R0 in mathematical epi-demiology. After an historic reminder describing the steps leading to the statement of its mathematical deﬁnition, we explain the next-generation matrix method allowing its calculation in the case of epidemic models described by ordinary diﬀerential equations (ODEs). The article then focuses, through four ODEs examples and an infection load structured PDE model, on the usefulness of the R0 to address biological as well mathematical issues.

R 0 calculation What to do with a R 0 ? Difficulties with PDEs contents 1 A brief history of R 0 2 A recipee for R 0 calculation 3 What to do with a R 0 ?
4 Main difficulties arising with structured PDE models Question : Can we extract a tool to measure the disease risk ? Pest epidemic in Mumbai [Kermack & McKendrick (1927)] Link with demographic concept [MacDonald (1952)] Epidemiological concept R0 : number of secondary infections resulting from a single primary infection into an otherwise susceptible population.
Why is R0 a threshold marker of epidemic ? → introduction of p infected individuals ⇒ (R0) k p infected individuals after step k.
R 0 calculation What to do with a R 0 ? Difficulties with PDEs ... to a mathematical definition of R0 Mathematical translation through dynamical systems [Diekmann & Heersterbeck (1990)] Mathematical translation R0 : bifurcation threshold that ensures (R0 < 1) or not (R0 > 1) the stability of a specific equilibrium point, the disease-free equilibrium (DFE) with Fi flux of newly infected V + i (resp. V − i ) other entering fluxes (resp. leaving fluxes) What to do with a R 0 ? Difficulties with PDEs

The next generation matrix
With DFE x * = (x * 1 , . . . , x * p , 0, . . . , 0), The R0 value related to the epidemic systemẋ(t) = f (x(t)) is given by Sketch of proof : Example 2 : Control strategy 1-Malaria and Ross' "Mosquitoe theorem" Ross model What to do with a R 0 ? Difficulties with PDEs Example 2 : Control strategy Vaccination of a proportion of new borns :     1 existence of DFE (z * , x * 1 , x * 2 , 0, 0, 0) 2 Next generation matrix : 3 Basic reproductive number : Figure: Impact of prey availability on R0, with Γ1 = Γ2 (left) and Γ1 < Γ2 (right) R 0 History R 0 calculation What to do with a R 0 ? Difficulties with PDEs Example 3 : Impact of biodiversity on the disease dynamics Eco-epidemiological question : How variability in host competence impacts the parasite dynamics ? → Density-dependant dilution of the parasite !
Can we say more than "locally" when R0 < 1 ?
Persistence of the disease when R0 > 1 ? → the instability of DFE is not enough ! And what about R0 = 1 ?

Definition (uniform persistence)
The disease is uniformly persistent if R 0 History R 0 calculation What to do with a R 0 ? Difficulties with PDEs
Can we say more than "locally" when R0 < 1 ?
Persistence of the disease when R0 > 1 ? → the instability of DFE is not enough ! And what about R0 = 1 ?