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Mathematical Biosciences & Engineering

2009 , Volume 6 , Issue 2

A special issue on
A tribute to the mathematical epidemiology work of Fred Brauer and Karl Hadeler

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From the Guest Editors
Carlos Castillo-Chávez , Christopher Kribs Zaleta , Yang Kuang and  Baojun Song
2009, 6(2): i-ii doi: 10.3934/mbe.2009.6.2i +[Abstract](60) +[PDF](48.5KB)
Those of us who met the field of mathematical biology as a well-developed, flourishing, and rewarding discipline owe much to those who made it so. This special issue of Mathematical Biosciences and Engineering is dedicated to two such pioneers: Fred Brauer and Karl Hadeler. Since retrospectives of both men have been published in other venues [1, 2], we will only summarize their contributions briefly here.
Fred Brauer obtained his Ph.D. from MIT in 1956 under Norman Levinson, and during a long tenure at the University of Wisconsin he co-wrote several texts on ordinary differential equations that have become classics. His research entered mathematical biology first through early studies in predator-prey systems and harvesting, both with and without delays. He then moved into mathematical epidemiology, and the text he co-authored with Carlos Castillo-Chavez in both these areas earlier this decade is already in wide use.

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Mathematical modelling of tuberculosis epidemics
Juan Pablo Aparicio and  Carlos Castillo-Chávez
2009, 6(2): 209-237 doi: 10.3934/mbe.2009.6.209 +[Abstract](93) +[PDF](378.3KB)
The strengths and limitations of using homogeneous mixing and heterogeneous mixing epidemic models are explored in the context of the transmission dynamics of tuberculosis. The focus is on three types of models: a standard incidence homogeneous mixing model, a non-homogeneous mixing model that incorporates 'household' contacts, and an age-structured model. The models are parameterized using demographic and epidemiological data and the patterns generated from these models are compared. Furthermore, the effects of population growth, stochasticity, clustering of contacts, and age structure on disease dynamics are explored. This framework is used to asses the possible causes for the observed historical decline of tuberculosis notifications.
The reproduction number $R_t$ in structured and nonstructured populations
Tom Burr and  Gerardo Chowell
2009, 6(2): 239-259 doi: 10.3934/mbe.2009.6.239 +[Abstract](58) +[PDF](226.7KB)
Using daily counts of newly infected individuals, Wallinga and Teunis (WT) introduced a conceptually simple method to estimate the number of secondary cases per primary case ($R_t$) for a given day. The method requires an estimate of the generation interval probability density function (pdf), which specifies the probabilities for the times between symptom onset in a primary case and symptom onset in a corresponding secondary case. Other methods to estimate $R_t$ are based on explicit models such as the SIR model; therefore, one might expect the WT method to be more robust to departures from SIR-type behavior. This paper uses simulated data to compare the quality of daily $R_t$ estimates based on a SIR model to those using the WT method for both structured (classical SIR assumptions are violated) and nonstructured (classical SIR assumptions hold) populations. By using detailed simulations that record the infection day of each new infection and the donor-recipient identities, the true $R_t$ and the generation interval pdf is known with negligible error. We find that the generation interval pdf is time dependent in all cases, which agrees with recent results reported elsewhere. We also find that the WT method performs essentially the same in the structured populations (except for a spatial network) as it does in the nonstructured population. And, the WT method does as well or better than a SIR-model based method in three of the four structured populations. Therefore, even if the contact patterns are heterogeneous as in the structured populations evaluated here, the WT method provides reasonable estimates of $R_t$, as does the SIR method.
The estimation of the effective reproductive number from disease outbreak data
Ariel Cintrón-Arias , Carlos Castillo-Chávez , Luís M. A. Bettencourt , Alun L. Lloyd and  H. T. Banks
2009, 6(2): 261-282 doi: 10.3934/mbe.2009.6.261 +[Abstract](82) +[PDF](1573.6KB)
We consider a single outbreak susceptible-infected-recovered (SIR) model and corresponding estimation procedures for the effective reproductive number $\mathcal{R}(t)$. We discuss the estimation of the underlying SIR parameters with a generalized least squares (GLS) estimation technique. We do this in the context of appropriate statistical models for the measurement process. We use asymptotic statistical theories to derive the mean and variance of the limiting (Gaussian) sampling distribution and to perform post statistical analysis of the inverse problems. We illustrate the ideas and pitfalls (e.g., large condition numbers on the corresponding Fisher information matrix) with both synthetic and influenza incidence data sets.
The dynamics of a delay model of hepatitis B virus infection with logistic hepatocyte growth
Steffen Eikenberry , Sarah Hews , John D. Nagy and  Yang Kuang
2009, 6(2): 283-299 doi: 10.3934/mbe.2009.6.283 +[Abstract](65) +[PDF](667.9KB)
Chronic HBV affects 350 million people and can lead to death through cirrhosis-induced liver failure or hepatocellular carcinoma. We analyze the dynamics of a model considering logistic hepatocyte growth and a standard incidence function governing viral infection. This model also considers an explicit time delay in virus production. With this model formulation all model parameters can be estimated from biological data; we also simulate a course of lamivudine therapy and find that the model gives good agreement with clinical data. Previous models considering constant hepatocyte growth have permitted only two dynamical possibilities: convergence to a virus free or a chronic steady state. Our model admits a third possibility of sustained oscillations. We show that when the basic reproductive number is greater than 1 there exists a biologically meaningful chronic steady state, and the stability of this steady state is dependent upon both the rate of hepatocyte regeneration and the virulence of the disease. When the chronic steady state is unstable, simulations show the existence of an attracting periodic orbit. Minimum hepatocyte populations are very small in the periodic orbit, and such a state likely represents acute liver failure. Therefore, the often sudden onset of liver failure in chronic HBV patients can be explained as a switch in stability caused by the gradual evolution of parameters representing the disease state.
Culling structured hosts to eradicate vector-borne diseases
Xinli Hu , Yansheng Liu and  Jianhong Wu
2009, 6(2): 301-319 doi: 10.3934/mbe.2009.6.301 +[Abstract](40) +[PDF](845.1KB)
A compartmental model is developed, in the form of a nonautonomous system of delay differential equations subject to impulses at specific times, for mosquito-born disease control involving larvicides and insecticide sprays. Sufficient conditions in terms of the frequencies and rates of larvicides and insecticide sprays are derived, and numerical simulations are provided to illustrate the sharpness of these disease eradication conditions.
Epidemic models with differential susceptibility and staged progression and their dynamics
James M. Hyman and  Jia Li
2009, 6(2): 321-332 doi: 10.3934/mbe.2009.6.321 +[Abstract](39) +[PDF](158.9KB)
We formulate and study epidemic models with differential susceptibilities and staged-progressions, based on systems of ordinary differential equations, for disease transmission where the susceptibility of susceptible individuals vary and the infective individuals progress the disease gradually through stages with different infectiousness in each stage. We consider the contact rates to be proportional to the total population or constant such that the infection rates have a bilinear or standard form, respectively. We derive explicit formulas for the reproductive number $R_0$, and show that the infection-free equilibrium is globally asymptotically stable if $R_0<1$ when the infection rate has a bilinear form. We investigate existence of the endemic equilibrium for the two cases and show that there exists a unique endemic equilibrium for the bilinear incidence, and at least one endemic equilibrium for the standard incidence when $R_0>1$.
Mathematical analysis of a model for HIV-malaria co-infection
Zindoga Mukandavire , Abba B. Gumel , Winston Garira and  Jean Michel Tchuenche
2009, 6(2): 333-362 doi: 10.3934/mbe.2009.6.333 +[Abstract](56) +[PDF](1834.4KB)
A deterministic model for the co-interaction of HIV and malaria in a community is presented and rigorously analyzed. Two sub-models, namely the HIV-only and malaria-only sub-models, are considered first of all. Unlike the HIV-only sub-model, which has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction number is less than unity, the malaria-only sub-model undergoes the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium, for a certain range of the associated reproduction number less than unity. Thus, for malaria, the classical requirement of having the associated reproduction number to be less than unity, although necessary, is not sufficient for its elimination. It is also shown, using centre manifold theory, that the full HIV-malaria co-infection model undergoes backward bifurcation. Simulations of the full HIV-malaria model show that the two diseases co-exist whenever their reproduction numbers exceed unity (with no competitive exclusion occurring). Further, the reduction in sexual activity of individuals with malaria symptoms decreases the number of new cases of HIV and the mixed HIV-malaria infection while increasing the number of malaria cases. Finally, these simulations show that the HIV-induced increase in susceptibility to malaria infection has marginal effect on the new cases of HIV and malaria but increases the number of new cases of the dual HIV-malaria infection.
Examination of a simple model of condom usage and individual withdrawal for the HIV epidemic
Jeff Musgrave and  James Watmough
2009, 6(2): 363-376 doi: 10.3934/mbe.2009.6.363 +[Abstract](45) +[PDF](212.5KB)
Since the discovery of HIV/AIDS there have been numerous mathematical models proposed to explain the epidemic of the disease and to evaluate possible control measures. In particular, several recent studies have looked at the potential impact of condom usage on the epidemic [1, 2, 3, 4]. We develop a simple model for HIV/AIDS, and investigate the effectiveness of condoms as a possible control strategy. We show that condoms can greatly reduce the number of outbreaks and the size of the epidemic. However, the necessary condom usage levels are much higher than the current estimates. We conclude that condoms alone will not be sufficient to halt the epidemic in most populations unless current estimates of the transmission probabilities are high. Our model has only five independent parameters, which allows for a complete analysis. We show that the assumptions of mass action and standard incidence provide similar results, which implies that the results of the simpler mass action model can be used as a good first approximation to the peak of the epidemic.
The discounted reproductive number for epidemiology
Timothy C. Reluga , Jan Medlock and  Alison Galvani
2009, 6(2): 377-393 doi: 10.3934/mbe.2009.6.377 +[Abstract](71) +[PDF](326.1KB)
The basic reproductive number, $\Ro$, and the effective reproductive number, $R$, are commonly used in mathematical epidemiology as summary statistics for the size and controllability of epidemics. However, these commonly used reproductive numbers can be misleading when applied to predict pathogen evolution because they do not incorporate the impact of the timing of events in the life-history cycle of the pathogen. To study evolution problems where the host population size is changing, measures like the ultimate proliferation rate must be used. A third measure of reproductive success, which combines properties of both the basic reproductive number and the ultimate proliferation rate, is the discounted reproductive number $\mathcal{R}_d$. The discounted reproductive number is a measure of reproductive success that is an individual's expected lifetime offspring production discounted by the background population growth rate. Here, we draw attention to the discounted reproductive number by providing an explicit definition and a systematic application framework. We describe how the discounted reproductive number overcomes the limitations of both the standard reproductive numbers and proliferation rates, and show that $\mathcal{R}_d$ is closely connected to Fisher's reproductive values for different life-history stages.
On the eradicability of infections with partially protective vaccination in models with backward bifurcation
Muntaser Safan and  Klaus Dietz
2009, 6(2): 395-407 doi: 10.3934/mbe.2009.6.395 +[Abstract](44) +[PDF](208.0KB)
The SIS model of Hadeler and Castillo-Chavez [9] with a constant transfer rate of susceptibles into a partially protected state has been modified to take into account vaccination at birth. The model shows backward bifurcation (existence of multiple endemic stationary states) for certain values of parameters. Parameter values ensuring the existence and nonexistence of endemic equilibria have been discussed. Local and global stability of equilibria have been investigated. The minimum effort required to eradicate the infection has been determined.
Global stability of a class of discrete age-structured SIS models with immigration
Yicang Zhou and  Zhien Ma
2009, 6(2): 409-425 doi: 10.3934/mbe.2009.6.409 +[Abstract](47) +[PDF](200.9KB)
Immigration has an important influence on the growth of population and the transmission dynamics of infectious diseases. A discrete age-structured epidemic SIS model with immigration is formulated and its dynamical behavior is studied in this paper. It is found that population growth will be determined by the reproductive number and the immigration rate. In the simple case without infected immigration, the basic reproductive number is defined, and the global stability of equilibria is investigated. In the case with infected immigration, there is no disease-free equilibrium, and there always exists an endemic equilibrium, and the global stability conditions of the unique endemic equilibrium is obtained.

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