University of Sussex
Yau,_Muhammad_Abdullahi.pdf (9.61 MB)

Analysis of spatial dynamics and time delays in epidemic models

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posted on 2023-06-08, 18:14 authored by Muhammad Abdullahi Yau
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 ß. 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 ß. 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 ?1 and temporary immunity ?2 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 (?1 > 1, ?2 = 0) and the case when the two time delays are identical (?1 = ?2 = ? ), we show that the endemic steady state is always stable for any parameter values. For the case when there is only temporary immunity (?2 > 0, ?1 = 0) and the case when there are both latency and temporary immunity in the system (?1 > 0, ?2 > 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 ß, the natural death rate ? 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.


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University of Sussex

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