Research On Several Classes Of Epidemic Models

Epidemic diseases are a major threat to public health in human history. To conquest an epidemic disease, it is important to study its etiopathogenesis, predict its trends, and then plan possible intervention measures to control the spread of the disease. Because it is impossible to study the epidemic disease by means of experimental methods, theoretical analysis and numerical simulation are needed for the study of the epidemic. Mathematical models are powerful tools for studying the dynamics of infectious diseases and evaluating the effect of various control measures.Influenza lies at the top of death toll lists in human history. Influenza viruses are associated with high morbidity and mortality in humans and continue to be a major threat to public health. Because influenza is an important global public health concern, the methods by which pandemic influenza could be contained are of widespread interest, and a variety of control measures have been implemented to contain the spread of influenza strains. This thesis applies mathematical models and methods including epidemic dynamic models, household epidemic models, and the optimal control theory and methods,to evaluate the effectiveness of several control measures for influenza pandemic. The main work is as follows:1. Early in an influenza pandemic, before effective vaccines become available, antiviral drugs are considered as the major control strategies for a pandemic influenza. However, perhaps such control strategies can be severely hindered by the low-efficacy of antiviral drugs. For this reason, the strategy of using antiviral drugs and an isolation strategy to control influenza is included in our study. Based on the SEIAR model developed by Arino et al., we use the methods of epidemic dynamics to develop an SEIQINAR compartmental model to describe the spread of influenza. This model not only considers the latent period of influenza and asymptomatic infections, but also includes antiviral prophylaxis and isolation control measures. Furthermore, the importation of exposed individuals and asymptomatic cases from other areas is further considered in this model. Numerical simulations are used to evaluate the effectiveness of control strategies via antiviral prophylaxis and isolation. Simulations show that isolation strategy plays a prominent role in containing transmission when antiviral drugs are not effective enough. Moreover, relatively few infected individuals need to be isolated per day. Because the accurate calculations of the needed numbers of antiviral drugs and the isolated infected individuals are not easily available, we give two simple expressions approximating these numbers. We also derive an estimation for the total cost of these intervention strategies. Simulation results suggest that the differences between the actual values and the approximate values of the three numbers are very small. These estimations obtained by a simple method provide a useful reference for the management department about the epidemic preparedness plans.2. In the absence of effective vaccines, antiviral drugs and nonpharmaceutical intervention measures such as voluntary self-isolation have been a part of preparedness plans for the next influenza pandemic. We used a household model to assess the effect of voluntary self-isolation on outbreak control when antiviral drugs are not provided sufficiently early. We found that the early initiation of voluntary self-isolation can overcome the negative effects caused by a delay in antiviral drug distribution when enough symptomatic individuals comply with home confinement at symptom onset. For example, for the baseline household reproduction numberRH0= 2.5, if delays of 1 or 2 days occur between clinical symptom development and the start of antiviral prophylaxis, then compliance rates of q≥0.41 and q≥0.6, respectively, are required to achieve the same level of effectiveness as starting antiviral prophylaxis at symptom onset. Moreover, this study considers the impact of a delay in the implementation of voluntary self-isolation on the effect of the voluntary self-isolation strategy. When the time to beginning voluntary self-isolation after symptom onset increases from 0 to 2 days, this strategy has a limited effect on reducing the transmission of influenza; therefore, this strategy should be implemented as soon as possible. In addition, we examine the extent to which asymptomatic infections influence the effectiveness of voluntary self-isolation. Simulation results show that the effect of voluntary self-isolation decreases substantially with the proportion of asymptomatic infections increasing.3. Although studies on using the optimal control theory to influenza are very abundant, there are little studies on the optimal strategy of antiviral prophylaxis for influenza control. Hence, based on the SEIAR model, we use the optimal control theory and methods to explore the optimal strategies of antiviral prophylaxis and treatment for influenza control. We formulate an optimal control problem by minimizing the number of infected individuals at a minimal intervention costs. We first show the existence of an optimal solution for the control problem and then derive the time-varying optimal strategies of antiviral prophylaxis and treatment using the Pontryagin’s Maximum Principle. In addition, we obtain the numerical results using an iterative approach with Runge-Kutta fourth order procedure and carry out a sensitivity analysis on some of the model parameters.