I am doing everything with R and its specialized geo package (sp, rgeos, raster...). For the example you give, spplot (package sp) would do the trick.

Another possibility is to use QGIS.

Yes, QGIS can be very helpful, and it is relatively easy to learn. Depending on your goals you could also use Postgres for the epidemiological data, with PostGIS extension and access everything with QGIS. Or, make a web-application using GeoServer. 

Thank you for your replies. So basically what I have is number of cases reported in one country from different cities in past years. Based on this,  I want to color the cities with a color reference range similar to what we have in heatmap to represent intensity. Then the second is the air traffic data which I looking to map between two countries in form of lines, curved arches or something similar. So, it would be like looking for cities where maximum number of travelers arrive and then visualizing how many ppl arrived in cities which have highest number of reported cases.....is it possible with QGIS? or is there any web server to accomplish this task as I don't have much expertise with GIS softwares.

I cannot certainly called myself an expert (I just had tweaked a little bit with the matter). Actually QGIS comes in a whole bundle of software (look here http://www.osgeo.org/ for everything) allowing quite complex operations from a variety of different approaches.

Everything is open source.

The bad news is that you are almost on your own trying to figure out how to use and set-up everything for your purposes.

May be you could consider offering some thesis or grants: working soundly on GIS software (whatever software) requires a lot of expertise.

Even with other software packages!

there are several  free GIS software can be very helpful such  ( SAGA GIS ,Whitebox GAT ,GRASS GIS and Diva GIS)

I usually use fleatickrisk.com  because I work with vector diseases

It would be easier to demonstrate software capacities if you provide us with actual data.