Rachid Ouifki

Modeling the joint epidemics of TB and HIV in a South African township.

We present a simple mathematical model with six compartments for the interaction between HIV and TB epidemics. Using data from a township near Cape Town, South Africa, where the prevalence of HIV is above 20% and where the TB notification rate is close to 2,000 per 100,000 per year, we estimate some of the model parameters and study how various control measures might change the course of these epidemics. Condom promotion, increased TB detection and TB preventive therapy have a clear positive effect. The impact of antiretroviral therapy on the incidence of HIV is unclear and depends on the extent to which it reduces sexual transmission. However, our analysis suggests that it will greatly reduce the TB notification rate.

Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity

We derive and analyse a deterministic model for the transmission of malaria disease with mass action form of infection. Firstly, we calculate the basic reproduction number, R(0), and investigate the existence and stability of equilibria. The system is found to exhibit backward bifurcation. The implication of this occurrence is that the classical epidemiological requirement for effective eradication of malaria, R(0)<1, is no longer sufficient, even though necessary. Secondly, by using optimal control theory we derive the conditions under which it is optimal to eradicate the disease and examine the impact of a possible combined vaccination and treatment strategy on the disease transmission. When eradication is impossible, we derive the necessary conditions for optimal control of the disease using Pontryagins Maximum Principle. The results obtained from the numerical simulations of the model show that a possible vaccination combined with effective treatment regime would reduce the spread of the disease appreciably.

Modeling the Control of Trypanosomiasis Using Trypanocides or Insecticide-Treated Livestock

Background In Uganda, Rhodesian sleeping sickness, caused by Trypanosoma brucei rhodesiense, and animal trypanosomiasis caused by T. vivax and T. congolense, are being controlled by treating cattle with trypanocides and/or insecticides. We used a mathematical model to identify treatment coverages required to break transmission when host populations consisted of various proportions of wild and domestic mammals, and reptiles. Methodology/Principal Findings An Ro model for trypanosomiasis was generalized to allow tsetse to feed off multiple host species. Assuming populations of cattle and humans only, pre-intervention Ro values for T. vivax, T. congolense, and T. brucei were 388, 64 and 3, respectively. Treating cattle with trypanocides reduced R 0 for T. brucei to <1 if >65% of cattle were treated, vs 100% coverage necessary for T. vivax and T. congolense. The presence of wild mammalian hosts increased the coverage required and made control of T. vivax and T. congolense impossible. When tsetse fed only on cattle or humans, R 0 for T. brucei was <1 if 20% of cattle were treated with insecticide, compared to 55% for T. congolense. If wild mammalian hosts were also present, control of the two species was impossible if proportions of non-human bloodmeals from cattle were <40% or <70%, respectively. R 0 was <1 for T. vivax only when insecticide treatment led to reductions in the tsetse population. Under such circumstances R 0<1 for T. brucei and T. congolense if cattle make up 30% and 55%, respectively of the non-human tsetse bloodmeals, as long as all cattle are treated with insecticide. Conclusions/Significance In settled areas of Uganda with few wild hosts, control of Rhodesian sleeping sickness is likely to be much more effectively controlled by treating cattle with insecticide than with trypanocides.

Stability analysis of a model for HIV infection with RTI and three intracellular delays.

We consider a model of HIV-1 infection with a reverse transcriptase inhibitor (RTI) therapy and three delays: the first delay is defined as the time from the virus entry into the target cell to the reverse transcriptase step, the second delay represents the time from the virus entry to the production of new viruses and the third delay corresponds to the time necessary for a newly produced virus to become infectious. We analyse the stability of the steady states and determine a threshold value for the first delay at which the system exhibits a Hopf bifurcation. This might explain the clinically observed transient elevated viremia called viral blips. The frequency of the bifurcating periodic solution as well as the threshold value is approximated numerically using realistic parameter. The estimated threshold value is realistic and the frequency of the oscillations is consistent with that of the observed viral blips.

Modelling the Use of Insecticide-Treated Cattle to Control Tsetse and Trypanosoma brucei rhodesiense in a Multi-host Population

We present a mathematical model for the transmission of Trypanosoma brucei rhodesiense by tsetse vectors to a multi-host population. To control tsetse and T. b. rhodesiense, a proportion, ψ, of cattle (one of the hosts considered in the model) is taken to be kept on treatment with insecticides. Analytical expressions are obtained for the basic reproduction number, R0n in the absence, and

A general model for mortality in adult tsetse (Glossina spp.)

R_{0n}^{T}

A General HIV Incidence Inference Scheme Based on Likelihood of Individual Level Data and a Population Renewal Equation

in the presence of insecticide-treated cattle (ITC). Stability analysis of the disease-free equilibrium was carried out for the case when there is one vertebrate host untreated with insecticide. By considering three vertebrate hosts (cattle, humans and wildlife) the sensitivity analysis was carried out on the basic reproduction number (

Modelling the long-term impacts on affected children of adult HIV: benefits, challenges and a possible approach

R_{03}^{T}

An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection

) in the absence and presence of ITC. The results show that

Mathematical model of tumor-immune surveillance.

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