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We formulated and analyzed a class of coupled partial and ordinary differential equation (PDE-ODE) model to study the spread of airborne diseases. Our model describes human populations with patches and the movement of pathogens in the air with linear diffusion. The diffusing pathogens are coupled to the SIR dynamics of each population patch using an integro-differential equation. Susceptible individuals become infected at some rate whenever they are in contact with pathogens (indirect transmission), and the spread of infection in each patch depends on the density of pathogens around the patch. In the limit where the pathogens are diffusing fast, a matched asymptotic analysis is used to reduce the coupled PDE-ODE model into a nonlinear system of ODEs, which is then used to compute the basic reproduction number and final size relation for different scenarios. Numerical simulations of the reduced system of ODEs and the full PDE-ODE model are consistent, and they predict a decrease in the spread of infection as the diffusion rate of pathogens increases. Furthermore, we studied the effect of patch location on the spread of infections for the case of two population patches. Our model predicts higher infections when the patches are closer to each other.

In this work we propose a mathematical model to simulate Chikungunya spread; the spread model is implemented in a C++ cellular automata code defined on unstructured triangular grids and space visualizations are performed with Python. In order to simulate the time space spread of the Chikungunya diseases we include assumptions such as: heterogeneous human and vector densities, population mobility, geographically localized points of infection using geographical information systems, changes in the probabilities of infection, extrinsic incubation and mosquito death rate due to environmental variables. Numerical experiments reproduce the qualitative behavior of diseases spread and provide an insight to develop strategies to prevent the diseases spread.

We developed a mathematical model to study the co-interaction of HIV and syphilis infection among gay, bisexual and other men who have sex with men (gbMSM). We qualitatively analysed the model and established necessary conditions under which disease-free and endemic equilibria are asymptotically stable. We gave analytical expressions for the reproduction number, and showed that whenever the reproduction numbers of sub-models and co-interaction model are less than unity, the epidemics die out, while epidemics persist when they are greater than unity. We presented numerical simulations of the full model and showed qualitative changes of the dynamics of the full model to changes in the transmission rates. Our numerical simulations using a set of reasonable parameter values showed that: (a) both diseases die out or co-exist whenever their reproduction number is less than or exceed unity. (b) HIV infection impacts syphilis prevalence negatively and vice versa. (c) one possibility of lowering the co-infection of HIV and syphilis among gbMSM is to increase both testing and treatment rates for syphilis and HIV infection, and decrease the rate at which HIV infected individuals go off treatment.

Early in a disease outbreak, it is important to be able to estimate the final size of the epidemic in order to assess needs for treatment and to be able to compare the effects of different treatment approaches. However, it is common for epidemics, especially of diseases considered dangerous, to grow much more slowly than expected. We suggest that by assuming behavioural changes in the face of an epidemic and heterogeneity of mixing in the population it is possible to obtain reasonable early estimates.

We consider an epidemic model in which all disease transmission is through shedding of virus by infectives and acquisition by susceptibles, rather than by direct contact. This leads to an susceptible-infectious-virus-removed (SIVR) model for which we can determine the basic reproduction number and the final size relation. We extend the model to an age of infection model with virus shedding a function of the age of infection.

We give a brief outline of some of the important aspects of the development of mathematical epidemiology.

We formulate and analyze an age of infection model for epidemics of diseases transmitted by a vector, including the possibility of direct transmission as well. We show how to determine a basic reproduction number. While there is no explicit final size relation as for diseases transmitted directly, we are able to obtain estimates for the final size of the epidemic.

Vector-transmitted diseases such as dengue fever and chikungunya have been spreading rapidly in many parts of the world. The Zika virus has been known since 1947 and invaded South America in 2013. It can be transmitted not only by (mosquito) vectors but also directly through sexual contact. Zika has developed into a serious global health problem because, while most cases are asymptomatic or very light, babies born to Zika - infected mothers may develop microcephaly and other very serious birth defects.We formulate and analyze two epidemic models for vector-transmitted diseases, one appropriate for dengue and chikungunya fever outbreaks and one that includes direct transmission appropriate for Zika virus outbreaks. This is especially important because the Zika virus is the first example of a disease that can be spread both indirectly through a vector and directly (through sexual contact). In both cases, we obtain expressions for the basic reproduction number and show how to use the initial exponential growth rate to estimate the basic reproduction number. However, for the model that includes direct transmission some additional data would be needed to identify the fraction of cases transmitted directly. Data for the 2015 Zika virus outbreak in Barranquilla, Colombia has been used to fit parameters to the model developed here and to estimate the basic reproduction number.

Disease transmission models in which infectivity depends on the time since infection are of importance in studying such diseases as HIV/AIDS. They also provide a means of unifying models with exposed stages or temporary immunity. We formulate a general age of infection model and carry out a partial analysis. There are open questions in the analysis of the characteristic equation at an endemic equilibrium.

Severe acute respiratory syndrome (SARS), a new, highly contagious, viral disease, emerged in China late in 2002 and quickly spread to 32 countries and regions causing in excess of 774 deaths and 8098 infections worldwide. In the absence of a rapid diagnostic test, therapy or vaccine, isolation of individuals diagnosed with SARS and quarantine of individuals feared exposed to SARS virus were used to control the spread of infection. We examine mathematically the impact of isolation and quarantine on the control of SARS during the outbreaks in Toronto, Hong Kong, Singapore and Beijing using a deterministic model that closely mimics the data for cumulative infected cases and SARS-related deaths in the first three regions but not in Beijing until mid-April, when China started to report data more accurately. The results reveal that achieving a reduction in the contact rate between susceptible and diseased individuals by isolating the latter is a critically important strategy that can control SARS outbreaks with or without quarantine. An optimal isolation programme entails timely implementation under stringent hygienic precautions defined by a critical threshold value. Values below this threshold lead to control, but those above are associated with the incidence of new community outbreaks or nosocomial infections, a known cause for the spread of SARS in each region. Allocation of resources to implement optimal isolation is more effective than to implement sub-optimal isolation and quarantine together. A community-wide eradication of SARS is feasible if optimal isolation is combined with a highly effective screening programme at the points of entry. [ABSTRACT FROM AUTHOR]

Epidemic models with a general infective period distribution are formulated as functional differential equations, as are population models with a general life span distribution. The analysis of the local stability properties of equilibria of such models leads to a characteristic equation involving the Laplace transform of the infective period (or life span) distribution. We obtain conditions under which all roots of the characteristic equation are in the left half plane, implying asymptotic stability of equilibrium, for every infective period distribution. We also consider the converse problem of describing when instability can occur for specific infective period distributions. [ABSTRACT FROM AUTHOR]