Dynamics of a stochastic SIRS epidemic model with standard incidence and vaccination

A stochastic SIRS epidemic model with vaccination is discussed. A new stochastic threshold $ R_0^s $ is determined. When the noise is very low ($ R_0^s < 1 $), the disease becomes extinct, and if $ R_0^s > 1 $, the disease persists. Furthermore, we show that the solution of the stochastic model oscillates around the endemic equilibrium point and the intensity of the fluctuation is proportional to the intensity of the white noise. Computer simulations are used to support our findings.

Estimation and optimal control of the multiscale dynamics of Covid-19: a case study from Cameroon

This work aims at a better understanding and the optimal control of the spread of the new severe acute respiratory corona virus 2 (SARS-CoV-2). A multi-scale model giving insights on the virus population dynamics, the transmission process and the infection mechanism is proposed first. Indeed, there are human to human virus transmission, human to environment virus transmission, environment to human virus transmission and self-infection by susceptible individuals. The global stability of the disease-free equilibrium is shown when a given threshold T 0 is less or equal to 1 and the basic reproduction number R 0 is calculated. A convergence index T 1 is also defined in order to estimate the speed at which the disease extincts and an upper bound to the time of infectious extinction is given. The existence of the endemic equilibrium is conditional and its description is provided. Using Partial Rank Correlation Coefficient with a three levels fractional experimental design, the sensitivity of  R 0 , T 0 and T 1 to control parameters is evaluated. Following this study, the most significant parameter is the probability of wearing mask followed by the probability of mobility and the disinfection rate. According to a functional cost taking into account economic impacts of SARS-CoV-2, optimal fighting strategies are determined and discussed. The study is applied to real and available data from Cameroon with a model fitting. After several simulations, social distancing and the disinfection frequency appear as the main elements of the optimal control strategy against SARS-CoV-2.

The research and development process for multiscale models of infectious disease systems

Multiscale modelling of infectious disease systems falls within the domain of complexity science—the study of complex systems. However, what should be made clear is that current progress in multiscale modelling of infectious disease dynamics is still as yet insufficient to present it as a mature sub-discipline of complexity science. In this article we present a methodology for development of multiscale models of infectious disease systems. This methodology is a set of partially ordered research and development activities that result in multiscale models of infectious disease systems built from different scientific approaches. Therefore, the conclusive result of this article is a methodology to design multiscale models of infectious diseases. Although this research and development process for multiscale models cannot be claimed to be unique and final, it constitutes a good starting point, which may be found useful as a basis for further refinement in the discourse for multiscale modelling of infectious disease dynamics.

Complex systems such as infectious disease systems are inherently multilevel and multiscale systems. The study of such complex systems is called complexity science. In this article we present a methodology to design multiscale models of infectious disease systems from a complex systems perspective. Within this perspective we define complexity science as the study of the interconnected relationships of the levels and scales of organization of a complex system. We therefore, define the degree of complexity of a complex system as the number of levels and scales of organization of the complex system needed to describe it. In this work we first present a common multiscale vision of the multilevel and multiscale structure of infectious disease systems as complex systems in which the levels and scales of organization of an infectious disease system interact through different self-sustained multiscale cycles/loops (primary multiscale loops, or secondary multiscale loops, or tertiary multiscale loops) formed at different levels of organization of an infectious disease system due to ongoing reciprocal influence between the microscale and the macroscale. Guided by this multiscale vision, we propose a four-stage research and development process that result in multiscale models of infectious disease systems built from different scientific approaches.

A general method for multiscale modelling of vector-borne disease systems

The inability to develop multiscale models which can describe vector-borne disease systems in terms of the complete pathogen life cycle which represents multiple targets for control has hindered progress in our efforts to control, eliminate and even eradicate these multi-host infections. This is because it is currently not easy to determine precisely where and how in the life cycles of vector-borne disease systems the key constrains which are regarded as crucial in regulating pathogen population dynamics in both the vertebrate host and vector host operate. In this article, we present a general method for development of multiscale models of vector-borne disease systems which integrate the within-host and between-host scales for the two hosts (a vertebrate host and a vector host) that are implicated in vector-borne disease dynamics. The general multiscale modelling method is an extension of our previous work on multiscale models of infectious disease systems which established a basic science and accompanying theory of how pathogen population dynamics at within-host scale scales up to between-host scale and in turn how it scales down from between-host scale to within-host scale. Further, the general method is applied to multiscale modelling of human onchocerciasis-a vector-borne disease system which is sometimes called river blindness as a case study.

The Replication-Transmission Relativity Theory for Multiscale Modelling of Infectious Disease Systems

It is our contention that for multiscale modelling of infectious disease systems to evolve and expand in scope, it needs to be founded on a theory. Such a theory would improve our ability to describe infectious disease systems in terms of their scales and levels of organization, and their inter-relationships. In this article we present a relativistic theory for multiscale modelling of infectious disease systems, that can be considered as an extension of the relativity principle in physics, called the replication-transmission relativity theory. This replication-transmission relativity theory states that at any level of organization of an infectious disease system there is no privileged/absolute scale which would determine, disease dynamics, only interactions between the microscale and macroscale. Such a relativistic theory provides a scientific basis for a systems level description of infectious disease systems using multiscale modelling methods. The central idea of this relativistic theory is that at every level of organization of an infectious disease system, the reciprocal influence between the microscale and the macroscale establishes a pathogen replication-transmission multiscale cycle. We distinguish two kinds of reciprocal influence between the microscale and the macroscale based on systematic differences in their conditions of relevancy. Evidence for the validity of the replication-transmission relativity theory is presented using a multiscale model of hookworm infection that is developed at host level when the relationship between the microscale and the macroscale is described by one of the forms of reciprocal influence.

A coupled multiscale model to guide malaria control and elimination

In this paper, we share with the biomathematics community a new coupled multiscale model which has the potential to inform policy and guide malaria control and elimination. The formulation of this multiscale model is based on integrating four submodels which are: (i) a sub-model for the mosquito-to-human transmission of malaria parasite, (ii) a sub-model for the human-to-mosquito transmission of malaria parasite, (iii) a within-mosquito malaria parasite population dynamics sub-model and (iv) a within-human malaria parasite population dynamics sub-model. The integration of the four submodels is achieved by assuming that the transmission parameters of the sub-model for the mosquito-to-human transmission of malaria at the epidemiological scale are functions of the dependent variables of the within-mosquito sporozoite population dynamics while the transmission parameters of the sub-model for the human-to-mosquito transmission of malaria are functions of the dependent variables of the within-human gametocyte population dynamics. This establishes a unidirectionally coupled multiscale model where the within-human and within-mosquito submodels are unidirectionally coupled to the human-to-mosquito and mosquito-to-human submodels. A fast and slow time scale analysis is performed on this system. The result is a simple multiscale model which describes the mechanics of malaria transmission in terms of the major components of the complete malaria parasite life-cycle. This multiscale modelling approach may be found useful in guiding malaria control and elimination.