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)

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.

Statistical physics of vaccination

https://www.math.miami.edu/%7Eruan/MyPapers/RuanWang-jde03.pdf

Dynamical behavior of an epidemic model with a nonlinear incidence rate

A simple two-dimensional SIS model with vaccination exhibits a backward bifurcation for some parameter values. A two-population version of the model leads to the consideration of vaccination policies in paired border towns. The results of our mathematical analysis indicate that a vaccination campaign φ meant to reduce a disease's reproduction number R(φ) below one may fail to control the disease. If the aim is to prevent an epidemic outbreak, a large initial number of infective persons can cause a high endemicity level to arise rather suddenly even if the vaccine-reduced reproduction number is below threshold. If the aim is to eradicate an already established disease, bringing the vaccine-reduced reproduction number below one may not be sufficient to do so. The complete bifurcation analysis of the model in terms of the vaccine-reduced reproduction number is given, and some extensions are considered.

/pdf/a-simple-vaccination-model-with-multiple-endemic-states-3ub5v8zzql.pdf

A simple vaccination model with multiple endemic states.

Models for sexually transmitted diseases generally assume that the size of the core group is fixed. Publicly available information on disease prevalence may influence the recruitment of new susceptibles into highly sexually active populations. It is assumed that the recruitment rate into the core population is low while disease prevalence is high, core group members mix only with each other, disease levels outside the core are negligible, and some core group members reduce their risk through the use of a partially effective vaccine or prophylactics. A demographic-epidemic model is formulated in which the combined size of the core and non-core population is constant. A simpler version models the epidemic in an isolated core population of constant size under the influence of educational programs and measures that reduce susceptibility. The threshold condition for an endemic infection is determined. Backward bifurcations, multiple infective stationary states, and hysteresis phenomena can be observed even in the simplified version. Abrupt changes in disease prevalence levels may result from small changes in the disease management parameters and do not occur in the absence of such a program. The general conclusion is that partially effective vaccination or education programs may increase the total number of cases while decreasing the relative frequency of cases in the core group. The study throws some new light on the role of the reproduction number in connection with elimination attempts. It shows that although the reproduction number defines the threshold for the spread of the disease in a susceptible population, it is of limited value when elimination of an existing epidemic is planned.

/pdf/a-core-group-model-for-disease-transmission-2fkoitqghs.pdf

A core group model for disease transmission

For a class of epidemiological SIRS models that include public health policies, the stability at the uninfected state and the prevalence at the infected state are investigated. Backward bifurcation from the uninfected state and hysteresis effects are shown to occur for some range of parameters. In such cases, the reproduction number does not describe the necessary elimination effort; rather the effort is described by the value of the critical parameter at the turning point. An explicit expression is given for this quantity. The phenomenon of subcritical bifurcation in epidemic modeling is also discussed in terms of group models, pair formation, and macroparasite infection.

Backward bifurcation in epidemic control

On copositive matrices

Starting from a recent paper of Pollicott, Wang and Weiss we try to obtain improved representation formulas for the estimation of the time-dependent transmission rate of an epidemic in terms of either incidence or prevalence data. Although the formulas are (trivially) mathematically equivalent to previous formulas, the new representations need no additional estimates and they should be more stable numerically. We review the discrete time and the stochastic continuous time approach. We replace the assumption that recovery follows an exponential distribution and get estimates for the transmission rate for constant duration of the infectious phase.

Parameter identification in epidemic models.

Two optimization problems are considered: Harvesting from a structured population with maximal gain subject to the condition of non-extinction, and vaccinating a population with prescribed reduction of the reproduction number of the disease at minimal costs. It is shown that these problems have a similar structure and can be treated by the same mathematical approach. The optimal solutions have a 'two-window' structure: Optimal harvesting and vaccination strategies or policies are concentrated on one or two preferred age classes. The results are first shown for a linear age structure problem and for an epidemic situation at the uninfected state (minimize costs for a given reduction of the reproduction number) and then extended to populations structured by size, to harvesting at Gurtin-MacCamy equilibria and to vaccination at infected equilibria.

K.P. Hadeler

Papers

A core group model for disease transmission

Backward bifurcation in epidemic control

On copositive matrices

Parameter identification in epidemic models.

Optimal harvesting and optimal vaccination.