Mathematical modeling of viral infection dynamics in spherical organs
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Mathematical Biology
Mathematical modeling of viral infection dynamics in spherical organs Ricardo Dunia · Roger Bonnecaze
Received: 16 November 2011 / Revised: 6 September 2012 / Published online: 25 September 2012 © Springer-Verlag 2012
Abstract A general mathematical model of viral infections inside a spherical organ is presented. Transported quantities are used to represent external cells or viral particles that penetrate the organ surface to either promote or combat the infection. A diffusion mechanism is considered for the migration of transported quantities to the organ inner tissue. Cases that include the effect of penetration, diffusion and proliferation of immune system cells, the generation of latently infected cells and the delivery of antiviral treatment are analyzed. Different antiviral mechanisms are modeled in the context of spatial variation. Equilibrium conditions are also calculated to determine the radial profile after the infection progresses and antiviral therapy is delivered for a long period of time. The dynamic and equilibrium solutions obtained in this paper provide insight into the temporal and spatial evolution of viral infections. Keywords Viral infection · Organ infection · Antiviral treatment · Virus model · Infection simulation · Virus dynamics 1 Introduction The development of virus dynamic models under different infection scenarios have gained attention during recent years (Rong et al. 2007; Shi et al. 2010; Komarova and Wodarz 2010). The analysis of the progression of viral infections has provided insights into the optimization of antiviral therapies (Stengel 2008; Wodarz 2005). Nevertheless, most virus dynamic models are developed for well-mixed environments, which do not account for the migration of immune cells, virus particles and antiviral molecules to the infected areas of the host organ. In fact, models have shown that spatial structures
R. Dunia (B) · R. Bonnecaze Department of Chemical Engineering, The University of Texas at Austin, Austin, TX 78712, USA e-mail: [email protected]
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Fig. 1 Picture of heterogeneous infection in mouse kidneys, provided by mBio January/February 2012 vol. 3 no. 1. In the left picture the infection is concentrated in the organ surface, while in the right the infection appears at the core renal tissue of the kidney
such as localized populations of dead cells can adversely affect the spread of infection (Beauchemin 2006). There is also experimental evidence that the location of infected cells plays an important role in the spread of infections (Wodarz and Levy 2011). Viruses circulate in the host blood stream and their concentration may vary depending on the susceptibility of host cells to the infection. Viruses can also target solid organs made of susceptible cells, as is the case for the Hepatitis B and C. Hepatitis B virus (HBV) infection has been analyzed and simulated in a diffusion model confined to a finite domain (Wang et al. 2008). The infection traveling waves have been predicted to occur in mat
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