The Final Size of a Serious Epidemic.

In an epidemic of a serious disease, there is likely to be behavioral response that decreases the epidemic size considerably, and taking this into account may lead to estimates of the final epidemic size that are much smaller and more realistic than estimates that do not take this into account.

Can treatment increase the epidemic size?

Antiviral treatment is one of the key pharmacological interventions against many infectious diseases. This is particularly important in the absence of preventive measures such as vaccination. However, the evolution of drug-resistance in treated patients and its subsequent spread to the population pose significant impediments to the containment of disease epidemics using treatment. Previous models of population dynamics of influenza infection have shown that in the presence of drug-resistance, the epidemic final size (i.e., the total number of infections throughout the epidemic) is affected by the treatment rate. These models, through simulation experiments, illustrate the existence of an optimal treatment rate, not necessarily the highest possible rate, for minimizing the epidemic final size. However, the conditions for the existence of such an optimal treatment rate have never been found. Here, we provide these conditions for a class of models covered in the literature previously, and investigate the combination effect of treatment and transmissibility of the drug-resistant pathogen strain on the epidemic final size. For the first time, we obtain the final size relations for an epidemic model with two strains of a pathogen (i.e., drug-sensitive and drug-resistant). We also discuss this model with specific functional forms of de novo resistance emergence, and illustrate the theoretical findings with numerical simulations.

Some simple nosocomial disease transmission models.

The SARS epidemic of 2002-2003 drew attention to nosocomial disease transmission as many of the disease cases were transmitted through hospital staff and visitors. Various types of model have been proposed to describe this, including metapopulation models. We formulate and analyze a simple compartmental model with heterogeneous mixing to describe nosocomial transmission and determine the reproduction number and final size relation.

Viral dynamics model with CTL immune response incorporating antiretroviral therapy.

We present two HIV models that include the CTL immune response, antiretroviral therapy and a full logistic growth term for uninfected CD4+ T-cells. The difference between the two models lies in the inclusion or omission of a loss term in the free virus equation. We obtain critical conditions for the existence of one, two or three steady states, and analyze the stability of these steady states. Through numerical simulation we find substantial differences in the reproduction numbers and the behaviour at the infected steady state between the two models, for certain parameter sets. We explore the effect of varying the combination drug efficacy on model behaviour, and the possibility of reconstituting the CTL immune response through antiretroviral therapy. Furthermore, we employ Latin hypercube sampling to investigate the existence of multiple infected equilibria.

A simple model for behaviour change in epidemics.

BACKGROUND: People change their behaviour during an epidemic. Infectious members of a population may reduce the number of contacts they make with other people because of the physical effects of their illness and possibly because of public health announcements asking them to do so in order to decrease the number of new infections, while susceptible members of the population may reduce the number of contacts they make in order to try to avoid becoming infected. METHODS: We consider a simple epidemic model in which susceptible and infectious members respond to a disease outbreak by reducing contacts by different fractions and analyze the effect of such contact reductions on the size of the epidemic. We assume constant fractional reductions, without attempting to consider the way in which susceptible members might respond to information about the epidemic. RESULTS: We are able to derive upper and lower bounds for the final size of an epidemic, both for simple and staged progression models. CONCLUSIONS: The responses of uninfected and infected individuals in a disease outbreak are different, and this difference affects estimates of epidemic size.

Vaccination against 2009 pandemic H1N1 in a population dynamical model of Vancouver, Canada: timing is everything.

BACKGROUND: Much remains unknown about the effect of timing and prioritization of vaccination against pandemic (pH1N1) 2009 virus on health outcomes. We adapted a city-level contact network model to study different campaigns on influenza morbidity and mortality. METHODS: We modeled different distribution strategies initiated between July and November 2009 using a compartmental epidemic model that includes age structure and transmission network dynamics. The model represents the Greater Vancouver Regional District, a major North American city and surrounding suburbs with a population of 2 million, and is parameterized using data from the British Columbia Ministry of Health, published studies, and expert opinion. Outcomes are expressed as the number of infections and deaths averted due to vaccination. RESULTS: The model output was consistent with provincial surveillance data. Assuming a basic reproduction number = 1.4, an 8-week vaccination campaign initiated 2 weeks before the epidemic onset reduced morbidity and mortality by 79-91% and 80-87%, respectively, compared to no vaccination. Prioritizing children and parents for vaccination may have reduced transmission compared to actual practice, but the mortality benefit of this strategy appears highly sensitive to campaign timing. Modeling the actual late October start date resulted in modest reductions in morbidity and mortality (13-25% and 16-20%, respectively) with little variation by prioritization scheme. CONCLUSION: Delays in vaccine production due to technological or logistical barriers may reduce potential benefits of vaccination for pandemic influenza, and these temporal effects can outweigh any additional theoretical benefits from population targeting. Careful modeling may provide decision makers with estimates of these effects before the epidemic peak to guide production goals and inform policy. Integration of real-time surveillance data with mathematical models holds the promise of enabling public health planners to optimize the community benefits from proposed interventions before the pandemic peak.

The Effect of Face Mask Use on COVID-19 Models.

We begin with a simple model for the COVID-19 epidemic and add face mask usages and testing and quarantine of infectives. We estimate the effect on the reproduction number and discuss the question of whether the epidemic can be controlled by increased use of face masks.

Modelling and simulating Chikungunya spread with an unstructured triangular cellular automata.

In this work we propose a mathematical model to simulate Chikungunya spread; the spread model is implemented in a C++ cellular automata code defined on unstructured triangular grids and space visualizations are performed with Python. In order to simulate the time space spread of the Chikungunya diseases we include assumptions such as: heterogeneous human and vector densities, population mobility, geographically localized points of infection using geographical information systems, changes in the probabilities of infection, extrinsic incubation and mosquito death rate due to environmental variables. Numerical experiments reproduce the qualitative behavior of diseases spread and provide an insight to develop strategies to prevent the diseases spread.

A co-interaction model of HIV and syphilis infection among gay, bisexual and other men who have sex with men.

We developed a mathematical model to study the co-interaction of HIV and syphilis infection among gay, bisexual and other men who have sex with men (gbMSM). We qualitatively analysed the model and established necessary conditions under which disease-free and endemic equilibria are asymptotically stable. We gave analytical expressions for the reproduction number, and showed that whenever the reproduction numbers of sub-models and co-interaction model are less than unity, the epidemics die out, while epidemics persist when they are greater than unity. We presented numerical simulations of the full model and showed qualitative changes of the dynamics of the full model to changes in the transmission rates. Our numerical simulations using a set of reasonable parameter values showed that: (a) both diseases die out or co-exist whenever their reproduction number is less than or exceed unity. (b) HIV infection impacts syphilis prevalence negatively and vice versa. (c) one possibility of lowering the co-infection of HIV and syphilis among gbMSM is to increase both testing and treatment rates for syphilis and HIV infection, and decrease the rate at which HIV infected individuals go off treatment.

A novel approach to modelling the spatial spread of airborne diseases: an epidemic model with indirect transmission.

We formulated and analyzed a class of coupled partial and ordinary differential equation (PDE-ODE) model to study the spread of airborne diseases. Our model describes human populations with patches and the movement of pathogens in the air with linear diffusion. The diffusing pathogens are coupled to the SIR dynamics of each population patch using an integro-differential equation. Susceptible individuals become infected at some rate whenever they are in contact with pathogens (indirect transmission), and the spread of infection in each patch depends on the density of pathogens around the patch. In the limit where the pathogens are diffusing fast, a matched asymptotic analysis is used to reduce the coupled PDE-ODE model into a nonlinear system of ODEs, which is then used to compute the basic reproduction number and final size relation for different scenarios. Numerical simulations of the reduced system of ODEs and the full PDE-ODE model are consistent, and they predict a decrease in the spread of infection as the diffusion rate of pathogens increases. Furthermore, we studied the effect of patch location on the spread of infections for the case of two population patches. Our model predicts higher infections when the patches are closer to each other.

Mathematical epidemiology is not an oxymoron.

A brief description of the importance of communicable diseases in history and the development of mathematical modelling of disease transmission is given. This includes reasons for mathematical modelling, the history of mathematical modelling from the foundations laid in the late nineteenth century to the present, some of the accomplishments of mathematical modelling, and some challenges for the future. Our purpose is to demonstrate the importance of mathematical modelling for the understanding and management of infectious disease transmission.

A singular perturbation approach to epidemics of vector-transmitted diseases.

In vector-borne epidemic models there is often a substantial difference between the vector and host time scales. This makes it possible to use the quasi-steady-state to obtain final size relations.

Early estimates of epidemic final sizes.

Early in a disease outbreak, it is important to be able to estimate the final size of the epidemic in order to assess needs for treatment and to be able to compare the effects of different treatment approaches. However, it is common for epidemics, especially of diseases considered dangerous, to grow much more slowly than expected. We suggest that by assuming behavioural changes in the face of an epidemic and heterogeneity of mixing in the population it is possible to obtain reasonable early estimates.

A model for influenza with vaccination and antiviral treatment.

Compartmental models for influenza that include control by vaccination and antiviral treatment are formulated. Analytic expressions for the basic reproduction number, control reproduction number and the final size of the epidemic are derived for this general class of disease transmission models. Sensitivity and uncertainty analyses of the dependence of the control reproduction number on the parameters of the model give a comparison of the various intervention strategies. Numerical computations of the deterministic models are compared with those of recent stochastic simulation influenza models. Predictions of the deterministic compartmental models are in general agreement with those of the stochastic simulation models.

Epidemic models with heterogeneous mixing and treatment.

We consider a two-group epidemic model with treatment and establish a final size relation that gives the extent of the epidemic. This relation can be established with arbitrary mixing between the groups even though it may not be feasible to determine the reproduction number for the model. If the mixing of the two groups is proportionate, there is an explicit expression for the reproductive number and the final size relation is expressible in terms of the components of the reproduction number. We also extend the results to a two-group influenza model with proportionate mixing. Some numerical simulations suggest that (i) the assumption of no disease deaths is a good approximation if the disease death rate is small and (ii) a one-group model is a close approximation to a two-group model but a two-group model is necessary for comparing targeted treatment strategies.

Oscillations in a patchy environment disease model.

For a single patch SIRS model with a period of immunity of fixed length, recruitment-death demographics, disease related deaths and mass action incidence, the basic reproduction number R(0) is identified. It is shown that the disease-free equilibrium is globally asymptotically stable if R(0)<1. For R(0)>1, local stability of the endemic equilibrium and Hopf bifurcation analysis about this equilibrium are carried out. Moreover, a practical numerical approach to locate the bifurcation values for a characteristic equation with delay-dependent coefficients is provided. For a two patch SIRS model with travel, it is shown that there are several threshold quantities determining its dynamic behavior and that travel can reduce oscillations in both patches; travel may enhance oscillations in both patches; or travel can switch oscillations from one patch to another.

A new epidemic model with indirect transmission.

We consider an epidemic model in which all disease transmission is through shedding of virus by infectives and acquisition by susceptibles, rather than by direct contact. This leads to an susceptible-infectious-virus-removed (SIVR) model for which we can determine the basic reproduction number and the final size relation. We extend the model to an age of infection model with virus shedding a function of the age of infection.

A final size relation for epidemic models of vector-transmitted diseases.

We formulate and analyze an age of infection model for epidemics of diseases transmitted by a vector, including the possibility of direct transmission as well. We show how to determine a basic reproduction number. While there is no explicit final size relation as for diseases transmitted directly, we are able to obtain estimates for the final size of the epidemic.

Mathematical epidemiology: Past, present, and future.

We give a brief outline of some of the important aspects of the development of mathematical epidemiology.

Some models for epidemics of vector-transmitted diseases.

Vector-transmitted diseases such as dengue fever and chikungunya have been spreading rapidly in many parts of the world. The Zika virus has been known since 1947 and invaded South America in 2013. It can be transmitted not only by (mosquito) vectors but also directly through sexual contact. Zika has developed into a serious global health problem because, while most cases are asymptomatic or very light, babies born to Zika - infected mothers may develop microcephaly and other very serious birth defects. We formulate and analyze two epidemic models for vector-transmitted diseases, one appropriate for dengue and chikungunya fever outbreaks and one that includes direct transmission appropriate for Zika virus outbreaks. This is especially important because the Zika virus is the first example of a disease that can be spread both indirectly through a vector and directly (through sexual contact). In both cases, we obtain expressions for the basic reproduction number and show how to use the initial exponential growth rate to estimate the basic reproduction number. However, for the model that includes direct transmission some additional data would be needed to identify the fraction of cases transmitted directly. Data for the 2015 Zika virus outbreak in Barranquilla, Colombia has been used to fit parameters to the model developed here and to estimate the basic reproduction number.