Epidemics with multistrain interactions: The interplay between cross immunity and antibody-dependent enhancement

This paper examines the interplay of the effect of cross immunity and antibody-dependent enhancement (ADE) in multistrain diseases. Motivated by dengue fever, we study a model for the spreading of epidemics in a population with multistrain interactions mediated by both partial temporary cross immunity and ADE. Although ADE models have previously been observed to cause chaotic outbreaks, we show analytically that weak cross immunity has a stabilizing effect on the system. That is, the onset of disease fluctuations requires a larger value of ADE with small cross immunity than without. However, strong cross immunity is shown numerically to cause oscillations and chaotic outbreaks even for low values of ADE. (C) 2009 American Institute of Physics. [doi: 10.1063/1.3270261

Dynamics of a Two Serotype Disease with Antibody Dependent Enhancement

The dengue virus is a serious infectious disease that can be found in many regions of Southeast Asia. There exist four serotypes of the virus. Recovery from one serotype produces a natural immunity from that serotype. However, it also creates complexes with a second infection and will increase viral production. This process is know as antibody dependent enhancement (ADE). As a result, it is very difficult to vaccinate against the disease. An optimal vaccination would have to cover all four serotypes at once. To understand the dynamics of the disease, we will study a mathematical model for two coexisting serotypes of the dengue virus. This is done using compartmental models based on a system of differential equations. After analyzing the system, we will consider various vaccination strategies for single serotypes. The system that we study has four steady state solutions. We use various techniques of analyzing a dynamical system such as linearization about the fixed point and using the next generation matrix to determine the stability of the fixed points. The disease free equilibrium (DFE) is when both serotypes die out. Stability is determined by the basic reproduction number, R0, which is the number of secondary infections brought on by one infective in a susceptible population. When R0 is less than one, the DFE is stable, and it is unstable when R0 is greater than one. The system also has two boundary equilibrium where only one serotype will persist at a time. We use similar methods to find regions of stability for these fixed points. The fourth steady state solution is the endemic equilibrium where both of the serotypes persist. This fixed point cannot be written in a succinct closed form, so other methods are used to analyze its stability. Using symmetry, we reduce the system to four equations. Using asymptotics and numerical techniques, we approximate the stability of the endemic equilibrium and show that it goes through a Hopf bifurcation. Using the parameters for Dengue, it can been shown that the endemic equilibrium is stable and then after the Hopf bifurcation experiences oscillations. Thus, the disease is always persisting. We then model vaccination strategies to find if we can make the DFE stable. For example, if we vaccinate one hundred percent of the susceptible population against serotype one, then the system will behave as if only serotype two exists. We find a new reproduction number to determine when the existing serotype will persist and when it will die out. Another vaccination strategy is to partially vaccinate against one of the serotypes. Due to the ADE factors in the system, vaccinating partially against one serotype will never allow for the second serotype to die out. The final vaccination strategy that we consider is partial vaccination against both serotypes. We find conditions on the percentages of the population that should be vaccinated against each serotype to have the disease die out

Analysis of spatial dynamics and time delays in epidemic models

Reaction-diffusion systems and delay differential equations have been extensively used over the years to model and study the dynamics of infectious diseases. In this thesis we consider two aspects of disease dynamics: spatial dynamics in a reaction-diffusion epidemic model with nonlinear incidence rate, and a delayed epidemic model with combined effects of latency and temporary immunity. The first part of the thesis is devoted to the analysis of stability and pattern formation in an SIS-type epidemic model with nonlinear incidence rate. By considering the dynamics without spatial component, conditions for local asymptotic stability are obtained for general values of the powers of nonlinearity. We prove positivity, boundedness, invariant principle and permanence of our model. The next generation matrix method is used to derive the corresponding basic reproductive number R0, and the Routh-Hurwitz criterion is used to show that for R0 ≤ 1, the disease-free equilibrium is found to be locally asymptotically stable, for R0 > 1, a unique endemic steady state exists and is found to be locally asymptotically stable. In the presence of diffusion, Turing instability conditions are established in terms of system parameters. Numerical simulations are performed to identify the spatial regions for spots, stripes and labyrinthine patterns in the parameter space. Numerical simulations show that the system has complex and rich dynamics and can exhibit complex patterns, depending on the recovery rate r and the transmission rate β. We have discovered that whenever the transmission rate exceeds the recovery rate the system exhibits stripe patterns which correspond to a disease outbreak, and in the opposite case the system settles on spot patterns which imply the absence of disease outbreaks. Also, we find that increasing the power q can lead to epidemic outbreak even at lower values of the transmission rate β. All numerical simulations use an Implicit-Explicit (IMEX) Euler’s method, which computes diffusion terms in Fourier space and reaction terms in the real space. Numerical approximation of the model is benchmarked to prove stability of the numerical scheme, and the method is shown to converge with the correct order. Experimental order of convergence (EOC) and estimates for the error in both L2, H1 and maximum norms have also been computed. Also, we compare our results to those on infectious diseases and our model shows good predictions. In the second part of this thesis, we derive and analyse a delayed SIR model with bilinear incidence rate and two time delays which represent latency Τ1 and temporary immunity Τ2 periods. We prove both local and global stability of the system equilibria in the case when there are no time delays, i.e. both the latency and temporary immunity periods are set to zero. For the case when there is only latency (Τ1 > 1, Τ2 = 0) and the case when the two time delays are identical (Τ1 = Τ2 = Τ ), we show that the endemic steady state is always stable for any parameter values. For the case when there is only temporary immunity (Τ2 > 0, Τ1 = 0) and the case when there are both latency and temporary immunity in the system (Τ1 > 0, Τ2 > 0), we prove the existence of periodic solutions arising from the Hopf bifurcation. The endemic steady state undergoes Hopf bifurcation giving rise to stable periodic solutions. For the last two cases, we show interesting regions of (in)stability of the endemic steady state in the different parameter regimes. We find that by varying the transmission rate β, the natural death rate γ and the disease-induced death rate μ increase the regions of (in)stability. Also, we find that the dynamics of the system is richer when we have the two time delays in the model. Analytical results are supported by extensive numerical simulations, illustrating temporal behaviour of the system in different dynamical regimes. Finally, we relate our results to modelling infectious diseases and our results show good predictions of safety and epidemic outbreak

Stochastic effects in a seasonally forced epidemic model

The interplay of seasonality, the system's nonlinearities and intrinsic stochasticity is studied for a seasonally forced susceptible-exposed-infective-recovered stochastic model. The model is explored in the parameter region that corresponds to childhood infectious diseases such as measles. The power spectrum of the stochastic fluctuations around the attractors of the deterministic system that describes the model in the thermodynamic limit is computed analytically and validated by stochastic simulations for large system sizes. Size effects are studied through additional simulations. Other effects such as switching between coexisting attractors induced by stochasticity often mentioned in the literature as playing an important role in the dynamics of childhood infectious diseases are also investigated. The main conclusion is that stochastic amplification, rather than these effects, is the key ingredient to understand the observed incidence patterns.Comment: 13 pages, 9 figures, 3 table

Prey-Predator-Parasite: an Ecosystem Model With Fragile Persistence

abstract: Using a simple $SI$ infection model, I uncover the overall dynamics of the system and how they depend on the incidence function. I consider both an epidemic and endemic perspective of the model, but in both cases, three classes of incidence functions are identified. In the epidemic form, power incidences, where the infective portion $I^p$ has $p\in(0,1)$, cause unconditional host extinction, homogeneous incidences have host extinction for certain parameter constellations and host survival for others, and upper density-dependent incidences never cause host extinction. The case of non-extinction in upper density-dependent incidences extends to the case where a latent period is included. Using data from experiments with rhanavirus and salamanders, maximum likelihood estimates are applied to the data. With these estimates, I generate the corrected Akaike information criteria, which reward a low likelihood and punish the use of more parameters. This generates the Akaike weight, which is used to fit parameters to the data, and determine which incidence functions fit the data the best. From an endemic perspective, I observe that power incidences cause initial condition dependent host extinction for some parameter constellations and global stability for others, homogeneous incidences have host extinction for certain parameter constellations and host survival for others, and upper density-dependent incidences never cause host extinction. The dynamics when the incidence function is homogeneous are deeply explored. I expand the endemic considerations in the homogeneous case by adding a predator into the model. Using persistence theory, I show the conditions for the persistence of each of the predator, prey, and parasite species. Potential dynamics of the system include parasite mediated persistence of the predator, survival of the ecosystem at high initial predator levels and ecosystem collapse at low initial predator levels, persistence of all three species, and much more.Dissertation/ThesisDoctoral Dissertation Mathematics 201

Asymmetry in the Presence of Migration Stabilizes Multistrain Disease Outbreaks

We study the effect of migration between coupled populations, or patches, on the stability properties of multistrain disease dynamics. The epidemic model used in this work displays a Hopf bifurcation to oscillations in a single, well-mixed population. It is shown numerically that migration between two non-identical patches stabilizes the endemic steady state, delaying the onset of large amplitude outbreaks and reducing the total number of infections. This result is motivated by analyzing generic Hopf bifurcations with different frequencies and with diffusive coupling between them. Stabilization of the steady state is again seen, indicating that our observation in the full multistrain model is based on qualitative characteristics of the dynamics rather than on details of the disease model