A model for meningitis is developed by adding a class of carriers to the basic SIR model. This model is used to analyze the impact a vaccination program can have on the health of the population of epidemic prone countries. Analysis of the model shows the local stability of the disease free equilibrium, the existence of an endemic equilibrium and computation of the reproduction number, ℜ0 . Using a MATLAB program we simulate a time course of the model using parameters gathered from the World Health Organization. The numerical solution demonstrates that our reproduction number was correct. We thenconcluded that a high infection transmission rate requires a high vaccine rate.

abstract: The purpose of information source detection problem (or called rumor source detection) is to identify the source of information diffusion in networks based on available observations like the states of the nodes and the timestamps at which nodes adopted the information (or called infected). The solution of the problem can be used to answer a wide range of important questions in epidemiology, computer network security, etc. This dissertation studies the fundamental theory and the design of efficient and robust algorithms for the information source detection problem. For tree networks, the maximum a posterior (MAP) estimator of the information source is derived under the independent cascades (IC) model with a complete snapshot and a Short-Fat Tree (SFT) algorithm is proposed for general networks based on the MAP estimator. Furthermore, the following possibility and impossibility results are established on the Erdos-Renyi (ER) random graph: $(i)$ when the infection duration $<\frac{2}{3}t_u,$ SFT identifies the source with probability one asymptotically, where $t_u=\left\lceil\frac{\log n}{\log \mu}\right\rceil+2$ and $\mu$ is the average node degree, $(ii)$ when the infection duration $>t_u,$ the probability of identifying the source approaches zero asymptotically under any algorithm; and $(iii)$ when infection duration $<t_u,$ the breadth-first search (BFS) tree starting from the source is a fat tree. Numerical experiments on tree networks, the ER random graphs and real world networks show that the SFT algorithm outperforms existing algorithms. In practice, other than the nodes' states, side information like partial timestamps may also be available. Such information provides important insights of the diffusion process. To utilize the partial timestamps, the information source detection problem is formulated as a ranking problem on graphs and two ranking algorithms, cost-based ranking (CR) and tree-based ranking (TR), are proposed. Extensive experimental evaluations of synthetic data of different diffusion models and real world data demonstrate the effectiveness and robustness of CR and TR compared with existing algorithms. / Dissertation/Thesis / Doctoral Dissertation Electrical Engineering 2015

This thesis applies the theory of uncertainty quantification and sensitivity analysis on the SIR-model and SEIR-model for the spread of diseases. We attempt to determine if we can apply this theory to estimate the model parameters to an acceptable degree of accuracy.  Using sensitivity analysis we determine which parameters of the models are the most significant for some quantity of interest. We apply forward uncertainty quantification to determine how the uncertainty of the model parameters propagates to the quantities of interests. And lastly, we apply uncertainty quantification based on the maximum likelihood method to estimate the model parameters. To easily verify the results, we use synthetic data when estimating the parameters. After applying these methods we see that the importance of the model parameters heavily depend on the choice of quantity of interest. We also note that the uncertainty method reduces the uncertainty in the quantities of interests, although there are a lot of sources of errors that still needs to be considered.

本篇論文討論的是SIR模型的反應擴散方程 　　　　　　　　　s_t = d_1 s_xx − βsi/(s + i), 　　　　　　　　　i_t = d_2 i_xx + βsi/(s + i) − γi, 　　　　　　　　　r_t = d_3 r_xx + γi, 之行進波的存在性，其中模型描述的是在一個封閉區域裡流行疾病爆發的狀態。這裡的 β 是傳播係數，γ 是治癒或移除(即死亡)速率，s 是未被傳染個體數，i 是傳染源個體數，d_1、d_2、d_3分別為其擴散之係數。 　　我們將使用Schauder不動點定理(Schauder fixed point theorem)、Arzela-Ascoli定理和最大值原理(maximum principle)來證明：該系統存在最小速度為c=c*:=2√(d2( β - γ ))之行進波解。我們的結果回答了[11]裡所提出的開放式問題。 / In this thesis, we study the existence of traveling waves of a reaction-diffusion equation for a diffusive epidemic SIR model 　　　　　　　　　s_t = d_1 s_xx − βsi/(s + i), 　　　　　　　　　i_t = d_2 i_xx + βsi/(s + i) − γi, 　　　　　　　　　r_t = d_3 r_xx + γi, which describes an infectious disease outbreak in a closed population. Here β is the transmission coefficient, γ is the recovery or remove rate, and s, i, and r rep-resent numbers of susceptible individuals, infected individuals, and removed individuals, respectively, and d_1, d_2, and d_3 are their diffusion rates. We use the Schauder fixed point theorem, the Arzela-Ascoli theorem, and the maximum principle to show that this system has a traveling wave solution with minimum speed c=c*:=2√(d2( β - γ )). Our result answers an open problem proposed in [11].

We introduce random fluctuations on contact and recovery rates in three basic deterministic models in mathematical epidemiology and obtain stochastic counterparts. This paper addresses qualitative and quantitative analysis of stochastic SIS model with disease deaths and demographic effects, and stochastic SIR models with/without disease deaths and demographic effects. We prove the global existence of a unique strong solution and discuss stochastic asymptotic stability of disease free and endemic equilibria. We also investigate numerical properties of these models and prove the convergence of the Balanced Implicit Method approximation to the analytic solution. We simulate the models with fairly realistic parameters to visualize our conclusions.

For decades, dengue virus has been a cause of major public health concern in Costa Rica, due to its landscape and climatic conditions that favor the circumstances in which the vector, Aedes aegypti, thrives. The emergence and introduction throughout tropical and subtropical countries of the chikungunya virus, as of 2014, challenged Costa Rican health authorities to provide a correct diagnosis since it is also transmitted by the same vector and infected hosts may share similar symptoms. We study the 2015–2016 dengue and chikungunya outbreaks in Costa Rica while establishing how point estimates of epidemic parameters for both diseases compare to one another. Longitudinal weekly incidence reports of these outbreaks signal likely misdiagnosis of infected individuals: underreporting of chikungunya cases, while overreporting cases of dengue. Our comparative analysis is formulated with a single-outbreak deterministic model that features an undiagnosed class. Additionally, we also used a genetic algorithm in the context of weighted least squares to calculate point estimates of key model parameters and initial conditions, while formally quantifying misdiagnosis.

The SIR can be expressed either as a system of nonlinear ordinary differential equations or as a nonlinear Volterra integral equation. In general, neither of these can be solved in closed form. In this thesis, it is shown that if we assume S(t) is a finite multi-exponential, i.e. function of the form S(t) = a+ ∑nk=1 rke-σkt or a logistic function which is an infinite-multi-exponential, i.e. function of the form S(t) = c + a/b+ewt, then we can have closed form solution. Also we will formulate a method to determine R0 the basic reproductive rate of an infection.

No description available.

<p>Mathematical modelling has become a powerful tool used to predict the spread of infectious diseases in populations. Successful analysis and modeling of historical infectious disease data can explain changes in the pattern of past epidemics and lead to a better understanding of epidemiological processes. The lessons learned can be used to predict future epidemics and help to improve public healthstrategies for control and eradication.</p> <p>This thesis is focused on the analysis and modelling of smallpox dynamics based on the weekly smallpox mortality records in London, England, 1664-1930. Statistical analysis of these records is presented. A timeline of significant historical events related to changes in variolation and vaccination uptake levels and demographics was established. These events were correlated with transitions observed in smallpox dynamics. Seasonality of the smallpox time series was investigated and the contact rate between susceptible and infectious individuals was found to be seasonally forced. Seasonal variations in smallpox transmission and changes in their seasonality over long time scale were estimated. The method of transition analysis, which is used to predict qualitative changes in epidemiological patterns, was used to explain the transitions observed in the smallpox time series. We found that the standard SIR model exhibits dynamics similar to the more realistic Gamma distributed SEIR model if the mean serial interval is chosen the same, so we used the standard SIR model for our analysis. We conclude that transitions observed in the temporal pattern of smallpox dynamics can be explained by the changes in birth, immigration and intervention uptake levels.</p> / Doctor of Philosophy (PhD)

<p>Infectious diseases spreading in a human population can occasionally exhibit sudden transitions in their qualitative dynamics. Previous work has been very successful in predicting such transitions in New York City's measles incidence rates using the standard SIR model (susceptible, infected, recovered). This work relied on a dataset spanning 45 years, which we have extended to 93 years (1891-1984). We continue previous research in transition analysis on this larger dataset, and compare resonant and transient periods predicted to exist in NYC's measles incidence rates with those observed through a continuous wavelet transform of the data. We find good agreement between SIR predictions and observation, and in particular note the likely existence of previously unobserved hysteresis early in our new time-series.</p> / Master of Science (MSc)