For infectious diseases such as pertussis, susceptibility is determined by immunity, which is chronological age-dependent. We consider an age-structured epidemiological model that accounts for both passively acquired maternal antibodies that decay and active immunity that wanes, permitting reinfection. The model is a 6-dimensional system of partial differential equations (PDE). By assuming constant rates within each age-group, the PDE system can be reduced to an ordinary differential equation (ODE) system with aging from one age-group to the next. We derive formulae for the effective reproduction number ℛ and provide their biological interpretation in some special cases. We show that the disease-free equilibrium is stable when ℛ < 1 and unstable if ℛ > 1.

When modeling infectious diseases such as pertussis, age-dependent immunity and susceptibility to disease are important to consider. One of the main reasons is that infants may receive maternal antibodies that do not confer permanent immunity, permitting re-infection after immunity wanes. In [

The model considered in this paper is a system of PDEs with age-dependent force of infection based on proportionate or preferential mixing between age-groups. Although there are methods for deriving the reproduction number ℛ for PDEs, they usually require the force of infection to be in a separable form (see, for example, [

Similar age-structured models for infectious diseases including pertussis have been discussed in previous studies including [

We derive the reproduction number ℛ using the next generation matrix, which provides a clear biological interpretation of the elements in ℛ. The inclusion of aging terms makes the computation of ℛ more challenging, particularly when the number of age groups

Because the next generation matrix

Let _{V}

In ^{∞} (0,∞);

The function ^{∞}(0

The specific form of

Alternatively, the PDF can take the form

Because of the types of biological questions we intend to study and the complexities of the model due to factors related to preferential mixing, maternal immunity and multiple infections, our model ignore the latent period and disease-induced mortality.

We will first derive the NGM (

The density of the total population ^{qt}A^{qt}

The PDE model described in _{i−}_{1}_{i}_{0} < _{1} < ⋯ < _{n−1} < _{n} = _{i−1}, _{i}), and assume that the parameters _{i}_{−1}, _{i}_{i}_{i+1}), which has the same meaning as in Hethcote [^{qt}_{i−}_{1}_{i}_{i}

Because these densities are all changing exponentially by ^{qt}_{i}

Note that the force of infection _{j}_{i−}_{1}_{i}_{n}_{i}_{i}_{i}_{j}

For ease of presentation, introduce the following vector notation:
^{0} = (^{0}^{0}^{0},
^{0}^{0}), where

As in many epidemiological models, the reproduction number represents the average number of secondary infections produced by a typical infected individual during the entire period of infection when introduced into a completely susceptible population [

The product _{i}a_{i}_{i}_{i}a_{i}_{F}_{G}_{ε}_{F}_{G}_{1}_{2},⋯_{n}_{1}_{2}, ⋯_{n}

The inverse of _{2} has a similar form given by

The matrices _{1} and _{2} in

Consider the case of proportionate mixing in the contacts. That is, _{i}_{i}_{i}_{i}_{1} and _{2} in

Using the results in [

Let ℛ be given in ^{0} is locally asymptotically stable if ℛ < 1 and unstable if ℛ > 1.

The proof of Theorem 3.1 is provided in the

Besides the assumption of proportionate mixing (i.e., _{i}_{i}_{i} = μ, γ_{i} = γ, ω_{i}_{i} = c_{i}_{i}

Then the matrix

Thus, the matrices

Noticing that

We illustrate this using the formula for ℛ given in ^{−1}

Similarly, because

The age-structured epidemiological models considered in this paper incorporate several features not present in existing models. This includes preferential mixing, temporary immunity from maternal antibodies and prior infections, and the possibility of multiple infections at reduced rates due to immunity. The model is formulated as a system of partial differential equations (PDEs), but the analysis is carried out by reducing the system of PDEs to a system of ordinary differential equations (ODEs). The ODE system includes the aging rate _{i}

When preferential mixing is considered, two typical forms have been used: the Kronecker delta function and a Gaussian kernel (see Glasser

Explicit formulae for ℛ can be helpful for applications of the models to study specific questions that interest epidemiologists and public health policymakers. We remark that while age-structured ODE models with the aging terms have been used for various infectious diseases (see, for example, Dye and Williams [_{i}

In _{i}_{1}_{2}_{3}_{4}_{5}_{6}_{7}_{8}_{9}_{10}_{11}_{12}_{13}_{14}_{15}_{i}_{1}_{2}_{3}_{4}_{5}_{6}_{7}_{8}_{9}_{10}_{11}_{12}_{13}_{14}_{15}

In the case of ^{+}

We consider a system of partial differential equations with age-dependent force of infection based on proportionate or preferential mixing between age-groups. Following Hethcote [

The research is supported in part by the NSF Grant DMS-1022758, and Zhipeng Qiu is supported by the NSFC Grant 11271190. We thank the Reviewers for helpful comments which improved the presentation.

The findings and conclusions in this report are those of the authors and do not necessarily represent the official position of the Centers for Disease Control and Prevention or other institutions with which they are affiliated.

2010

For ease of notation, let
^{0} has the following form:

Let

Then

Because

From the above discussion, we have that

Note that all leading principal minors of

We then conclude from Theorem 2.5.3 in [_{0} is locally asymptotically stable, and if ℛ > 1, then _{0} is unstable. This completes the proof of Theorem 3.1. □

Plots of ℛ as a function of immunity period 1/