Some mathematical methods for formulation and numerical simulation of stochastic epidemic models are presented. Specifically, models are formulated for continuous-time Markov chains and stochastic ...differential equations. Some well-known examples are used for illustration such as an SIR epidemic model and a host-vector malaria model. Analytical methods for approximating the probability of a disease outbreak are also discussed.
Seasonal variation affects the dynamics of many infectious diseases including influenza, cholera and malaria. The time when infectious individuals are first introduced into a population is crucial in ...predicting whether a major disease outbreak occurs. In this investigation, we apply a time-nonhomogeneous stochastic process for a cholera epidemic with seasonal periodicity and a multitype branching process approximation to obtain an analytical estimate for the probability of an outbreak. In particular, an analytic estimate of the probability of disease extinction is shown to satisfy a system of ordinary differential equations which follows from the backward Kolmogorov differential equation. An explicit expression for the mean (resp. variance) of the first extinction time given an extinction occurs is derived based on the analytic estimate for the extinction probability. Our results indicate that the probability of a disease outbreak, and mean and standard derivation of the first time to disease extinction are periodic in time and depend on the time when the infectious individuals or free-living pathogens are introduced. Numerical simulations are then carried out to validate the analytical predictions using two examples of the general cholera model. At the end, the developed theoretical results are extended to more general models of infectious diseases.
► New stochastic virus-cell models distinguish between two viral release strategies. ► In the absence of an immune response, bursting is more successful than budding. ► With an immune response, in a ...model for HIV-1, budding is more successful.
New stochastic models are developed for the dynamics of a viral infection and an immune response during the early stages of infection. The stochastic models are derived based on the dynamics of deterministic models. The simplest deterministic model is a well-known system of ordinary differential equations which consists of three populations: uninfected cells, actively infected cells, and virus particles. This basic model is extended to include some factors of the immune response related to Human Immunodeficiency Virus-1 (HIV-1) infection. For the deterministic models, the basic reproduction number,
R
0
, is calculated and it is shown that if
R
0
<
1
, the disease-free equilibrium is locally asymptotically stable and is globally asymptotically stable in some special cases. The new stochastic models are systems of stochastic differential equations (SDEs) and continuous-time Markov chain (CTMC) models that account for the variability in cellular reproduction and death, the infection process, the immune system activation, and viral reproduction. Two viral release strategies are considered: budding and bursting. The CTMC model is used to estimate the probability of virus extinction during the early stages of infection. Numerical simulations are carried out using parameter values applicable to HIV-1 dynamics. The stochastic models provide new insights, distinct from the basic deterministic models. For the case
R
0
>
1
, the deterministic models predict the viral infection persists in the host. But for the stochastic models, there is a positive probability of viral extinction. It is shown that the probability of a successful invasion depends on the initial viral dose, whether the immune system is activated, and whether the release strategy is bursting or budding.
•The ODE model consists of three steady-states: Disease free, Asymptomatic, and AIDS.•The model illustrates the importance of the CTL immune response during the asymptomatic stage of HIV ...infection.•The stimulation of CTLs occurs in the presence of both CD4 T cells and infectious virus particles.•Strong stimulation rate promotes the long asymptomatic stage.•When the stimulation rate is low, the solutions converge to the AIDS stage very fast.
CD4 T cells play a fundamental role in the adaptive immune response including the stimulation of cytotoxic lymphocytes (CTLs). Human immunodeficiency virus (HIV) which infects and kills CD4 T cells causes progressive failure of the immune system. However, HIV particles are also reproduced by the infected CD4 T cells. Therefore, during HIV infection, infected and healthy CD4 T cells act in opposition to each other, reproducing virus particles and activating and stimulating cellular immune responses, respectively. In this investigation, we develop and analyze a simple system of four ordinary differential equations that accounts for these two opposing roles of CD4 T cells. The model illustrates the importance of the CTL immune response during the asymptomatic stage of HIV infection. In addition, the solution behavior exhibits the two stages of infection, asymptomatic and final AIDS stages. In the model, a weak immune response results in a short asymptomatic stage and faster development of AIDS, whereas a strong immune response illustrates the long asymptomatic stage. A model with a latent stage for infected CD4 T cells is also investigated and compared numerically with the original model. The model shows that strong stimulation of CTLs by CD4 T cells is necessary to prevent progression to the AIDS stage.
Factors such as seasonality and spatial connectivity affect the spread of an infectious disease. Accounting for these factors in infectious disease models provides useful information on the times and ...locations of greatest risk for disease outbreaks. In this investigation, stochastic multi-patch epidemic models are formulated with seasonal and demographic variability. The stochastic models are used to investigate the probability of a disease outbreak when infected individuals are introduced into one or more of the patches. Seasonal variation is included through periodic transmission and dispersal rates. Multi-type branching process approximation and application of the backward Kolmogorov differential equation lead to an estimate for the probability of a disease outbreak. This estimate is also periodic and depends on the time, the location, and the number of initial infected individuals introduced into the patch system as well as the magnitude of the transmission and dispersal rates and the connectivity between patches. Examples are given for seasonal transmission and dispersal in two and three patches.
The U.S. has incorporated environmental policies into its all free trade agreements since it negotiated the North American Free Trade Agreement (NAFTA) in the early 1990s. The inclusion of ...environmental policies represented a major shift in trade policy but the environmental policies have not drastically changed in subsequent trade agreements over the past 25 years despite the continued involvement of environmental constituencies and policymakers. The punctuated equilibrium model provides the analytical framework for understanding the factors that gave rise to the drastic policy shift under NAFTA as well as the subsequent policy stasis, in order to inform future policymaking efforts. Based on this analysis, it appears that environmentalists and policymakers will likely be able to maintain the environmental policy status quo within the trade policy domain but should consider another policy arena for advancing their new environmental policy priorities.
Seasonal changes in temperature, humidity, and rainfall affect vector survival and emergence of mosquitoes and thus impact the dynamics of vector-borne disease outbreaks. Recent studies of ...deterministic and stochastic epidemic models with periodic environments have shown that the average basic reproduction number is not sufficient to predict an outbreak. We extend these studies to time-nonhomogeneous stochastic dengue models with demographic variability wherein the adult vectors emerge from the larval stage vary periodically. The combined effects of variability and periodicity provide a better understanding of the risk of dengue outbreaks. A multitype branching process approximation of the stochastic dengue model near the disease-free periodic solution is used to calculate the probability of a disease outbreak. The approximation follows from the solution of a system of differential equations derived from the backward Kolmogorov differential equation. This approximation shows that the risk of a disease outbreak is also periodic and depends on the particular time and the number of the initial infected individuals. Numerical examples are explored to demonstrate that the estimates of the probability of an outbreak from that of branching process approximations agree well with that of the continuous-time Markov chain. In addition, we propose a simple stochastic model to account for the effects of environmental variability on the emergence of adult vectors from the larval stage.
•Single-serotype dengue models with seasonal forcing in the vector population.•Seasonality, demographic and environmental variability drive dengue dynamics.•Markov chain and stochastic differential equation models account for variability.•Branching process estimates obtained for periodic probability of a dengue outbreak.•Outbreaks depend on number of infected hosts and vectors and time of introduction.
Lyme disease is the most common vector-borne disease in the United States impacting the Northeast and Midwest at the highest rates. Recently, it has become established in southeastern and ...south-central regions of Canada. In these regions, Lyme disease is caused by
Borrelia burgdorferi
, which is transmitted to humans by an infected
Ixodes scapularis
tick. Understanding the parasite-host interaction is critical as the white-footed mouse is one of the most competent reservoir for
B. burgdorferi
. The cycle of infection is driven by tick larvae feeding on infected mice that molt into infected nymphs and then transmit the disease to another susceptible host such as mice or humans. Lyme disease in humans is generally caused by the bite of an infected nymph. The main aim of this investigation is to study how diapause delays and demographic and seasonal variability in tick births, deaths, and feedings impact the infection dynamics of the tick-mouse cycle. We model tick-mouse dynamics with fixed diapause delays and more realistic Erlang distributed delays through delay and ordinary differential equations (ODEs). To account for demographic and seasonal variability, the ODEs are generalized to a continuous-time Markov chain (CTMC). The basic reproduction number and parameter sensitivity analysis are computed for the ODEs. The CTMC is used to investigate the probability of Lyme disease emergence when ticks and mice are introduced, a few of which are infected. The probability of disease emergence is highly dependent on the time and the infected species introduced. Infected mice introduced during the summer season result in the highest probability of disease emergence.
The dynamic nature of the COVID-19 pandemic has demanded a public health response that is constantly evolving due to the novelty of the virus. Many jurisdictions in the USA, Canada, and across the ...world have adopted social distancing and recommended the use of face masks. Considering these measures, it is prudent to understand the contributions of subpopulations—such as “silent spreaders”—to disease transmission dynamics in order to inform public health strategies in a jurisdiction-dependent manner. Additionally, we and others have shown that demographic and environmental stochasticity in transmission rates can play an important role in shaping disease dynamics. Here, we create a model for the COVID-19 pandemic by including two classes of individuals: silent spreaders, who either never experience a symptomatic phase or remain undetected throughout their disease course; and symptomatic spreaders, who experience symptoms and are detected. We fit the model to real-time COVID-19 confirmed cases and deaths to derive the transmission rates, death rates, and other relevant parameters for multiple phases of outbreaks in British Columbia (BC), Canada. We determine the extent to which SilS contributed to BC’s early wave of disease transmission as well as the impact of public health interventions on reducing transmission from both SilS and SymS. To do this, we validate our model against an existing COVID-19 parameterized framework and then fit our model to clinical data to estimate key parameter values for different stages of BC’s disease dynamics. We then use these parameters to construct a hybrid stochastic model that leverages the strengths of both a time-nonhomogeneous discrete process and a stochastic differential equation model. By combining these previously established approaches, we explore the impact of demographic and environmental variability on disease dynamics by simulating various scenarios in which a COVID-19 outbreak is initiated. Our results demonstrate that variability in disease transmission rate impacts the probability and severity of COVID-19 outbreaks differently in high- versus low-transmission scenarios.
If pathogen species, strains, or clones do not interact, intuition suggests the proportion of coinfected hosts should be the product of the individual prevalences. Independence consequently underpins ...the wide range of methods for detecting pathogen interactions from cross-sectional survey data. However, the very simplest of epidemiological models challenge the underlying assumption of statistical independence. Even if pathogens do not interact, death of coinfected hosts causes net prevalences of individual pathogens to decrease simultaneously. The induced positive correlation between prevalences means the proportion of coinfected hosts is expected to be higher than multiplication would suggest. By modelling the dynamics of multiple noninteracting pathogens causing chronic infections, we develop a pair of novel tests of interaction that properly account for nonindependence between pathogens causing lifelong infection. Our tests allow us to reinterpret data from previous studies including pathogens of humans, plants, and animals. Our work demonstrates how methods to identify interactions between pathogens can be updated using simple epidemic models.
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