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- 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.
- 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.
- 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.
- The role of synaptic transmission in a HIV model with memoryPublication . Pinto, Carla; Carvalho, Ana R.M.We propose a mathematical model with memory for the dynamics of HIV epidemics, where two transmission modes, cell-to-cell and virus-to-cell, and drug resistance are considered. Systems with memory, or fractional order systems, have largely been applied to the modeling of several real life phenomena. Here, we consider a fractional model where the order of the non-integer derivative takes values in the interval [0.5, 1.0]. We prove the local and global stability of the disease-free equilibrium. We study the role of the cell-to-cell transmission probability on the dynamics of the model, and on the value of the reproduction number, R0, for distinct values of the fractional order derivative, α. Moreover, we show evidence of an improvement of HIV infected patients quality of life, due to the increase of the drug efficacy. In the end, important inferences are drawn.
- Mathematical model for HIV dynamics in HIV-specific helper cellsPublication . Pinto, Carla M.A.; Carvalho, Ana R.M.In this paper we study a delay mathematical model for the dynamics of HIV in HIV-specific CD4 + T helper cells. We modify the model presented by Roy and Wodarz in 2012, where the HIV dynamics is studied, considering a single CD4 + T cell population. Non-specific helper cells are included as alternative target cell population, to account for macrophages and dendritic cells. In this paper, we include two types of delay: (1) a latent period between the time target cells are contacted by the virus particles and the time the virions enter the cells and; (2) virus production period for new virions to be produced within and released from the infected cells. We compute the reproduction number of the model, R0, and the local stability of the disease free equilibrium and of the endemic equilibrium. We find that for values of R0<1, the model approaches asymptotically the disease free equilibrium. For values of R0>1, the model approximates asymptotically the endemic equilibrium. We observe numerically the phenomenon of backward bifurcation for values of R0⪅1. This statement will be proved in future work. We also vary the values of the latent period and the production period of infected cells and free virus. We conclude that increasing these values translates in a decrease of the reproduction number. Thus, a good strategy to control the HIV virus should focus on drugs to prolong the latent period and/or slow down the virus production. These results suggest that the model is mathematically and epidemiologically well-posed.
- Strange patterns in one ring of Chen oscillators coupled to a ‘buffer’ cellPublication . Pinto, Carla M.A.; Carvalho, Ana R.M.We study curious dynamical patterns appearing in networks of one ring of cells coupled to a ‘buffer’ cell. Depending on how the cells in the ring are coupled to the ‘buffer’ cell, the full network may have a nontrivial group of symmetries or a nontrivial group of ‘interior’ symmetries. This group is Z3 in the unidirectional case and D3 in the bidirectional case. We simulate the coupled cell systems associated with the networks and obtain steady states, rotating waves, quasiperiodic behavior, and chaos. The different patterns seem to arise through a sequence of Hopf, period-doubling, and period-halving bifurcations. The behavior of the systems with exact symmetry are similar to the ones with ‘interior’ symmetry. The network architecture appears to explain some features, whereas the properties of the Chen oscillator, used to model cells’ internal dynamics, may explain others. We use XPPAUT and MATLAB to numerically compute the relevant states.
- Within-host and synaptic transmissions: contributions to the spread of HIV infectionPublication . Carvalho, Ana R.M.; Pinto, Carla M.A.We study the contributions of within-host (virus-to-cell) and synaptic (cell-to-cell) transmissions in a mathematical model for human immunodeficiency virus epidemics. The model also includes drug resistance. We prove the local and global stability of the disease-free equilibrium and the local stability of the endemic equilibrium. We analyse the effect of the cell-to-cell transmission rate on the value of the reproduction number, R0. Moreover, we show evidence of a qualitative change in the models’ dynamics, subjected to the value of the drug efficacy. In the end, important inferences are drawn.
- A coinfection model for HIV and HCVPublication . Pinto, Carla M.A.; Carvalho, Ana R.M.We study a mathematical model for the human immunodeficiency virus (HIV) and hepatites C virus (HCV) coinfection. The model predicts four distinct equilibria: the disease free, the HIV endemic, the HCV endemic, and the full endemic equilibria. The local and global stability of the disease free equilibrium was calculated for the full model and the HIV and HCV submodels. We present numerical simulations of the full model where the distinct equilibria can be observed. We show simulations of the qualitative changes of the dynamical behavior of the full model for variation of relevant parameters. From the results of the model, we infer possible measures that could be implemented in order to reduce the number of infected individuals.