Many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. Threshold theorems involving the basic reproduction number $R_{0}$, the contact number $\sigma$, and the replacement number $R$ are reviewed for the classic SIR epidemic and endemic models. Similar results with new expressions for $R_{0}$ are obtained for MSEIR and SEIR endemic models with either continuous age or age groups. Values of $R_{0}$ and $\sigma$ are estimated for various diseases including measles in Niger and pertussis in the United States. Previous models with age structure, heterogeneity, and spatial structure are surveyed.

https://epubs.siam.org/doi/pdf/10.1137/S0036144500371907

The Mathematics of Infectious Diseases

Between the 1960s and 1980s, most life scientists focused their attention on studies of nucleic acids and the translation of the coded information. Protein degradation was a neglected area, conside...

The Ubiquitin-Proteasome Proteolytic Pathway: Destruction for the Sake of Construction

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Networks and the epidemiology of directly transmitted infectious diseases are fundamentally linked. The foundations of epidemiology and early epidemiological models were based on population wide random-mixing, but in practice each individual has a finite set of contacts to whom they can pass infection; the ensemble of all such contacts forms a ‘mixing network’. Knowledge of the structure of the network allows models to compute the epidemic dynamics at the population scale from the individual-level behaviour of infections. Therefore, characteristics of mixing networks—and how these deviate from the random-mixing norm—have become important applied concerns that may enhance the understanding and prediction of epidemic patterns and intervention measures. Here, we review the basis of epidemiological theory (based on random-mixing models) and network theory (based on work from the social sciences and graph theory). We then describe a variety of methods that allow the mixing network, or an approximation to the network, to be ascertained. It is often the case that time and resources limit our ability to accurately find all connections within a network, and hence a generic understanding of the relationship between network structure and disease dynamics is needed. Therefore, we review some of the variety of idealized network types and approximation techniques that have been utilized to elucidate this link. Finally, we look to the future to suggest how the two fields of network theory and epidemiological modelling can deliver an improved understanding of disease dynamics and better public health through effective disease control.

/pdf/networks-and-epidemic-models-1qt352hzh3.pdf

Networks and epidemic models.

The reemergence of tuberculosis (TB) from the 1980s to the early
1990s instigated extensive researches on the mechanisms behind the
transmission dynamics of TB epidemics. This article provides a
detailed review of the work on the dynamics and control of TB. The
earliest mathematical models describing the TB dynamics appeared in
the 1960s and focused on the prediction and control strategies using
simulation approaches. Most recently developed models not only pay
attention to simulations but also take care of dynamical analysis
using modern knowledge of dynamical systems. Questions addressed by
these models mainly concentrate on TB control strategies, optimal
vaccination policies, approaches toward the elimination of TB in the
U.S.A., TB co-infection with HIV/AIDS, drug-resistant TB, responses
of the immune system, impacts of demography, the role of public
transportation systems, and the impact of contact patterns. Model
formulations involve a variety of mathematical areas, such as ODEs
(Ordinary Differential Equations) (both autonomous and
non-autonomous systems), PDEs (Partial Differential Equations),
system of difference equations, system of integro-differential
equations, Markov chain model, and simulation models.

Dynamical models of tuberculosis and their applications.

A predator-prey model, where both species are subjected to parasitism, is developed and analyzed. For the case where there is coexistence of the predator with the uninfected prey, an epidemic threshold theorem is proved. It is shown that in the case where the uninfected predator cannot survive only on uninfected prey, the parasitization could lead to persistence of the predator provided a certain threshold of transmission is surpassed.

Predator-prey populations with parasitic infection.

In Fisher's model for the migration of advantageous genes, in epidemic models and in the theory of combustion similar existence problems for travelling fronts and waves occur. For a general two-dimensional system of ordinary differential equations depending on a parameter the existence of trajectories connecting stationary points is established. For systems derived from diffusion problems these trajectories describe the shape of a travelling front, the corresponding value of the parameter is the propagation speed. The method allows to determine the exact value of the minimal speed in Fisher's model for all interesting choices of selection parameters, i.e. for intermediate heterozygotes and for inferior heterozygotes.

/pdf/travelling-fronts-in-nonlinear-diffusion-equations-5gc8ohnikj.pdf

Travelling fronts in nonlinear diffusion equations

The classical models for sexually transmitted infections assume homogeneous mixing either between all males and females or between certain subgroups of males and females with heterogeneous contact rates. This implies that everybody is all the time at risk of acquiring an infection. These models ignore the fact that the formation of a pair of two susceptibles renders them in a sense temporarily immune to infection as long as the partners do not separate and have no contacts with other partners. The present paper takes into account the phenomenon of pair formation by introducing explicitly a pairing rate and a separation rate. The infection transmission dynamics depends on the contact rate within a pair and the duration of a partnership. It turns out that endemic equilibria can only exist if the separation rate is sufficiently large in order to ensure the necessary number of sexual partners. The classical models are recovered if one lets the separation rate tend to infinity.

Epidemiological models for sexually transmitted diseases

An algorithm for the prediction of proteasomal cleavages.

Birth, death, pair formation, and separation are described by a system of three nonlinear homogeneous ordinary differential equations. The qualitative properties of the system are investigated, in particular the conditions for existence and global stability of the bisexual state.

K. P. Hadeler

Papers

Predator-prey populations with parasitic infection.

Travelling fronts in nonlinear diffusion equations

Epidemiological models for sexually transmitted diseases

An algorithm for the prediction of proteasomal cleavages.

Models for pair formation in bisexual populations.