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- International Conference Mathematical Analysis and Applications in Science and Engineering, ISEP, Porto, Portugal, June 20-22, 2024: Book of abstractsPublication . Golubitsky, Martin; Lacarnobara, Walter; Pinto, Carla M.A.; Babo, Lurdes; Mendonça, Jorge; Carvalho, Fernando; Rocha, Rui
- Power law and entropy analysis of catastrophic phenomenaPublication . Machado, J.A.Tenreiro; Pinto, Carla M.A.; Lopes, António M.Catastrophic events, such as wars and terrorist attacks, tornadoes and hurricanes, earthquakes, tsunamis, floods and landslides, are always accompanied by a large number of casualties. The size distribution of these casualties has separately been shown to follow approximate power law (PL) distributions. In this paper, we analyze the statistical distributions of the number of victims of catastrophic phenomena, in particular, terrorism, and find double PL behavior. This means that the data sets are better approximated by two PLs instead of a single one. We plot the PL parameters, corresponding to several events, and observe an interesting pattern in the charts, where the lines that connect each pair of points defining the double PLs are almost parallel to each other. A complementary data analysis is performed by means of the computation of the entropy. The results reveal relationships hidden in the data that may trigger a future comprehensive explanation of this type of phenomena.
- Effects of treatment, awareness and condom use in a coinfection model for HIV and HCV in MSMPublication . Pinto, Carla M.A.; Carvalho, Ana R.M.We develop a new a coinfection model for hepatitis C virus (HCV) and the human immunodeficiency virus (HIV). We consider treatment for both diseases, screening, unawareness and awareness of HIV infection, and the use of condoms. We study the local stability of the disease-free equilibria for the full model and for the two submodels (HCV only and HIV only submodels). We sketch bifurcation diagrams for different parameters, such as the probabilities that a contact will result in a HIV or an HCV infection. We present numerical simulations of the full model where the HIV, HCV and double endemic equilibria can be observed. We also show numerically the qualitative changes of the dynamical behavior of the full model for variation of relevant parameters. We extrapolate the results from the model for actual measures that could be implemented in order to reduce the number of infected individuals.
- Fractional model for malaria transmission under control strategiesPublication . Pinto, Carla M.A.; Machado, J.A.TenreiroWe study a fractional model for malaria transmission under control strategies.Weconsider the integer order model proposed by Chiyaka et al. (2008) in [15] and modify it to become a fractional order model. We study numerically the model for variation of the values of the fractional derivative and of the parameter that models personal protection, b. From observation of the figures we conclude that as b is increased from 0 to 1 there is a corresponding decrease in the number of infectious humans and infectious mosquitoes, for all values of α. This means that this result is invariant for variation of fractional derivative, in the values tested. These results are in agreement with those obtained in Chiyaka et al.(2008) [15] for α = 1.0 and suggest that our fractional model is epidemiologically wellposed.
- Fractional Dynamics of Computer Virus PropagationPublication . Pinto, Carla M.A.; Machado, J.A.TenreiroWe propose a fractional model for computer virus propagation. The model includes the interaction between computers and removable devices. We simulate numerically the model for distinct values of the order of the fractional derivative and for two sets of initial conditions adopted in the literature. We conclude that fractional order systems reveal richer dynamics than the classical integer order counterpart. Therefore, fractional dynamics leads to time responses with super-fast transients and super-slow evolutions towards the steady-state, effects not easily captured by the integer order models.
- A latency fractional order model for HIV dynamicsPublication . Pinto, Carla M.A.; Carvalho, Ana R.M.We study a fractional order model for HIV infection where latent T helper cells are included. We compute the reproduction number of the model and study the stability of the disease free equilibrium. We observe that the reproduction number varies with the order of the fractional derivative α. In terms of epidemics, this suggests that varying α induces a change in the patients’ epidemic status. Moreover, we simulate the variation of relevant parameters, such as the fraction of uninfected CD4+ T cells that become latently infected, and the CTLs proliferation rate due to infected CD4+ T cells. The model produces biologically reasonable results.
- Emergence of drug-resistance in HIV dynamics under distinct HAARTregimesPublication . Pinto, Carla M.A.; Carvalho, Ana R.M.In this paper we propose a model for the dynamics of HIV epidemics under distinct HAART regimes, and study the emergence of drug-resistance. The model predicts HIV dynamics of untreated HIV patients for all stages of the infection. We compute the local and the global stability of the disease-free equilibrium of the model. We simulate the model for two distinct HIV patients, the rapid progressors and the long-term non-progressors. We study the effects of equal RTI and PI efficacies, as well as distinct drug efficacies, namely RTI-based and PI-based therapeutics. Treatment is initiated when the CD4+ T cells count is less than 350 cells mm−3. The PI-based drugs seem to produce better outcomes, with respect to disease progression, than RTI-based regimes.
- Fractional complex-order model for HIV infection with drug resistance during therapyPublication . Pinto, Carla M.A.; Carvalho, Ana R.M.We propose a fractional complex-order model for drug resistance in HIV infection. We consider three distinct growth rates for the CD4+ T helper cells. We simulate the model for different values of the fractional derivative of complex order Dα±jβ, where α,β ∈ R+, and for distinct growth rates. The fractional derivative of complex order is a generalization of the integer-order derivative where α = 1 and β = 0. The fractional complex-order system reveals rich dynamics and variation of the value of the complex-order derivative sheds new light on the modeling of the intracellular delay. Additionally, fractional patterns are characterized by time responses with faster transients and slower evolutions towards the steady state.
- Persistence of low levels of plasma viremia and of the latent reservoir in patients under ART: A fractional-order approachPublication . Pinto, Carla M.A.Low levels of viral load are found in HIV-infected patients, after many years under successful suppressive anti-retroviral therapy (ART). The factors leading to this persistence are still under debate, but it is now more or less accepted that the latent reservoir may be crucial to the maintenance of this residual viremia. In this paper, we study the role of the latent reservoir in the persistence of the latent reservoir and of the plasma viremia in a fractional-order (FO) model for HIV infection. Our model assumes that (i) the latently infected cells may undergo bystander proliferation, without active viral production, (ii) the latent cell activation rate decreases with time on ART, (iii) the productively infected cells’ death rate is a function of the infected cell density. The proposed model provides new insights on the role of the latent reservoir in the persistence of the latent reservoir and of the plasma virus. Moreover, the fractional-order derivative distinguishes distinct velocities in the dynamics of the latent reservoir and of plasma virus. The later may be used to better approximations of HIV-infected patients data. To our best knowledge, this is the first FO model that deals with the role of the latent reservoir in the persistence of low levels of viremia and of the latent reservoir.
- New findings on the dynamics of HIV and TB coinfection modelsPublication . Pinto, Carla M.A.; Carvalho, Ana R.M.In this paper we study a model for HIV and TB coinfection. We consider the integer order and the fractional order versions of the model. Let α∈[0.78,1.0] be the order of the fractional derivative, then the integer order model is obtained for α=1.0. The model includes vertical transmission for HIV and treatment for both diseases. We compute the reproduction number of the integer order model and HIV and TB submodels, and the stability of the disease free equilibrium. We sketch the bifurcation diagrams of the integer order model, for variation of the average number of sexual partners per person and per unit time, and the tuberculosis transmission rate. We analyze numerical results of the fractional order model for different values of α, including α=1. The results show distinct types of transients, for variation of α. Moreover, we speculate, from observation of the numerical results, that the order of the fractional derivative may behave as a bifurcation parameter for the model. We conclude that the dynamics of the integer and the fractional order versions of the model are very rich and that together these versions may provide a better understanding of the dynamics of HIV and TB coinfection.