Existence of Periodic Solutions and Stability of Zero Solution of a Mathematical Model of Schistosomiasis

A mathematical model on schistosomiasis governed by periodic differential equations with a time delay was studied. By discussing boundedness of the solutions of this model and construction of a monotonic sequence, the existence of positive periodic solution was shown. The conditions under which the model admits a periodic solution and the conditions under which the zero solution is globally stable are given, respectively. Some numerical analyses show the conditional coexistence of locally stable zero solution and periodic solutions and that it is an effective treatment by simply reducing the population of snails and enlarging the death ratio of snails for the control of schistosomiasis

The temporal patterns of disease severity and prevalence in schistosomiasis

Schistosomiasis is one of the most widespread public health problems in the world. In this work, we introduce an eco-epidemiological model for its transmission and dynamics with the purpose of explaining both intra-and inter-annual fluctuations of disease severity and prevalence. The model takes the form of a system of nonlinear differential equations that incorporate biological complexity associated with schistosome's life cycle, including a prepatent period in snails (i.e., the time between initial infection and onset of infectiousness). Nonlinear analysis is used to explore the parametric conditions that produce different temporal patterns (stationary, endemic, periodic, and chaotic). For the time-invariant model, we identify a transcritical and a Hopf bifurcation in the space of the human and snail infection parameters. The first corresponds to the occurrence of an endemic equilibrium, while the latter marks the transition to interannual periodic oscillations. We then investigate a more realistic time-varying model in which fertility of the intermediate host population is assumed to seasonally vary. We show that seasonality can give rise to a cascade of period-doubling bifurcations leading to chaos for larger, though realistic, values of the amplitude of the seasonal variation of fertility. (C) 2015 AIP Publishing LLC

The interplay between models and public health policies: Regional control for a class of spatially structured epidemics (think globally, act locally)

A review is presented here of the research carried out, by a group including the authors, on the mathematical analysis of epidemic systems. Particular attention is paid to recent analysis of optimal control problems related to spatially structured epidemics driven by environmental pollution. A relevant problem, related to the possible eradication of the epidemic, is the so called zero stabilization. In a series of papers, necessary conditions, and sufficient conditions of stabilizability have been obtained. It has been proved that it is possible to diminish exponentially the epidemic process, in the whole habitat, just by reducing the concentration of the pollutant in a nonempty and sufficiently large subset of the spatial domain. The stabilizability with a feedback control of harvesting type is related to the magnitude of the principal eigenvalue of a certain operator. The problem of finding the optimal position (by translation) of the support of the feedback stabilizing control is faced, in order to minimize both the infected population and the pollutant at a certain finite time

Advanced Nonlinear Dynamics of Population Biology and Epidemiology

abstract: Modern biology and epidemiology have become more and more driven by the need of mathematical models and theory to elucidate general phenomena arising from the complexity of interactions on the numerous spatial, temporal, and hierarchical scales at which biological systems operate and diseases spread. Epidemic modeling and study of disease spread such as gonorrhea, HIV/AIDS, BSE, foot and mouth disease, measles, and rubella have had an impact on public health policy around the world which includes the United Kingdom, The Netherlands, Canada, and the United States. A wide variety of modeling approaches are involved in building up suitable models. Ordinary differential equation models, partial differential equation models, delay differential equation models, stochastic differential equation models, difference equation models, and nonautonomous models are examples of modeling approaches that are useful and capable of providing applicable strategies for the coexistence and conservation of endangered species, to prevent the overexploitation of natural resources, to control disease’s outbreak, and to make optimal dosing polices for the drug administration, and so forth.View the article as published at https://www.hindawi.com/journals/aaa/2014/214514

Modeling and analysis of bilharzia disease

In this paper the dynamics of bilharazia disease in the humans, which represents its main host, is formulated mathematically. The proposed system is studied analytically. The local stability is investigated for all possible equilibrium points.  Using suitable Lyapunov functions the basin of attraction of each point is specified. The conditions of occurring local bifurcation in the system are established. Numerical simulations are performed to study the global dynamics of the system and specify the set of control parameters. It is observed that the system has no periodic dynamics and the disease is controlled under some conditions on the parameters. Keywords: Bilharzia; Parasite disease; Stability; Local bifurcation

Prevalence-based modeling approach of schistosomiasis : global stability analysis and integrated control assessment

A system of nonlinear differential equations is proposed to assess the effects of prevalence-dependent disease contact rate, pathogen’s shedding rates, and treatment rate on the dynamics of schistosomiasis in a general setting. The decomposition techniques by Vidyasagar and the theory of monotone systems are the main ingredients to deal completely with the global asymptotic analysis of the system. Precisely, a threshold quantity for the analysis is derived and the existence of a unique endemic equilibrium is shown. Irrespective of the initial conditions, we prove that the solutions converge either to the disease-free equilibrium or to the endemic equilibrium, depending on whether the derived threshold quantity is less or greater than one. We assess the role of an integrated control strategy driven by human behavior changes through the incorporation of prevalence-dependent increasing the prophylactic treatment and decreasing the contact rate functions, as well as the mechanical water sanitation and the biological elimination of snails. Because schistosomiasis is endemic, the aim is to mitigate the endemic level of the disease. In this regard, we show both theoretically and numerically that: the reduction of contact rate through avoidance of contaminated water, the enhancement of prophylactic treatment, the water sanitation, and the removal of snails can reduce the endemic level and, to an ideal extent, drive schistosomiasis to elimination.The University of Pretoria Senior Postdoctoral Program Grant.https://www.springer.com/journal/403142022-01-20hj2021Mathematics and Applied Mathematic

Treating cofactors can reverse the expansion of a primary disease epidemic

<p>Abstract</p> <p>Background</p> <p>Cofactors, "nuisance" conditions or pathogens that affect the spread of a primary disease, are likely to be the norm rather than the exception in disease dynamics. Here we present a "simplest possible" demographic model that incorporates two distinct effects of cofactors: that on the transmission of the primary disease from an infected host bearing the cofactor, and that on the acquisition of the primary disease by an individual that is not infected with the primary disease but carries the cofactor.</p> <p>Methods</p> <p>We constructed and analyzed a four-patch compartment model that accommodates a cofactor. We applied the model to HIV spread in the presence of the causal agent of genital schistosomiasis, <it>Schistosoma hematobium</it>, a pathogen commonly co-occurring with HIV in sub-Saharan Africa.</p> <p>Results</p> <p>We found that cofactors can have a range of effects on primary disease dynamics, including shifting the primary disease from non-endemic to endemic, increasing the prevalence of the primary disease, and reversing demographic growth when the host population bears only the primary disease to demographic decline. We show that under parameter values based on the biology of the HIV/<it>S. haematobium </it>system, reduction of the schistosome-bearing subpopulations (e.g. through periodic use of antihelminths) can slow and even reverse the spread of HIV through the host population.</p> <p>Conclusions</p> <p>Typical single-disease models provide estimates of future conditions and guidance for direct intervention efforts relating only to the modeled primary disease. Our results suggest that, in circumstances under which a cofactor affects the disease dynamics, the most effective intervention effort might not be one focused on direct treatment of the primary disease alone. The cofactor model presented here can be used to estimate the impact of the cofactor in a particular disease/cofactor system without requiring the development of a more complicated model which incorporates many other specific aspects of the chosen disease/cofactor pair. Simulation results for the HIV/<it>S. haematobium </it>system have profound implications for disease management in developing areas, in that they provide evidence that in some cases treating cofactors may be the most successful and cost-effective way to slow the spread of primary diseases.</p