An　inverse　problem　to　identify　parameters　for　the　single-term　(and　multi-term)　fractional-order　system　of　an　outbreak　of　dengue　fever　is　considered.　Firstly,　we　propose　a　numerical　method　for　the　fractional-order　dengue　fever　system　based　on　the　Gorenflo-Mainardi-Moretti-Paradisi　(GMMP)　scheme　and　the　Newton　method.　Secondly,　two　methods,　the　modified　grid　approximation　method　(MGAM)　and　the　modified　hybrid　Nelder-Mead　simplex　search　and　particle　swarm　optimization　(MH-NMSS-PSO)　algorithm　are　expanded　to　estimate　the　fractional　orders　and　coefficients　for　fractional　differential　equations.　Then,　we　use　GMMP　and　MH-NMSS-PSO　to　estimate　the　parameters　of　the　fractional-order　dengue　fever　system.　With　the　new　fractional　orders　and　parameters,　our　fractional-order　dengue　fever　system　is　capable　of　providing　numerical　results　that　agree　very　well　with　the　real　data.　Furthermore,　for　searching　a　better　dengue　fever　system,　a　multi-term　fractional-order　epidemic　system　of　dengue　fever　is　proposed.　We　also　use　the　MGAM　and　MH-NMSS-PSO　to　estimate　the　fractional　orders　and　coefficients　of　the　multi-term　fractional-order　system.　To　verify　the　efficiency　and　accuracy　of　the　proposed　methods　in　dealing　with　the　fractional　inverse　problem,　a　numerical　example　with　real　data　is　investigated.　Using　the　statistics　from　the　2009　outbreak　of　the　disease　in　the　Cape　Verde　islands,　we　are　able　to　predict　the　fractional　orders　and　parameters　of　the　fractional　dengue　fever　system.　With　the　new　fractional　orders　and　parameters,　our　multi-term　fractional-order　dengue　fever　system　is　capable　of　providing　numerical　results　that　agree　better　with　the　real　data　than　other　integer-order　models.　©　2019,　Springer　Science+Business　Media,　LLC,　part　of　Springer　Nature.