Malaria is a common and life-threatening disease in many tropical and sub-tropical regions.It is caused by the protozoan parasite of the genus Plasmodium.Human malaria is caused by four different species of Plasmodium:P.falciparum,P.malariae,P.ovale and P.vivax.Among these species,P.falciparum causes the most severe form of malaria.The species P.Knowlesi infect animals but occasion-ally it can infect humans with the plasmodium parasite.The severity of malarial illness depends mostly on the immunological status of the infected person.Partial immunity develops over time through repeated infection,and without recurrent in-fection,immunity is relatively short-lived.Malaria can be prevented and treated.African swine fever(ASF)is a severe viral disease affecting domestic and wild pigs.It can be spread by live or dead pigs and pork products.Furthermore,transmission can also occur via contaminated feed,soft tick and non-living objects.Historically,outbreaks have been reported in Africa and parts of Europe,South America,and the Caribbean.More recently(August 2018),there was an outbreak of ASF in Asia.The ASF is not zoonotic,but humans can carry the ASF virus to infect pigs.In this thesis,we studied and developed mathematical models that represent the transmission,spread,and control of malaria and ASF diseases.We applied real data of malaria from Congo,DR,Sudan and Mexico.We discussed the effect of vector-bias between low and high transmission rates from three countries.Since malaria can be prevented and treated,we applied optimal control strategies to two of our models.Finally,we formulated a mathematical model that describes the transmission of ASF using quarantine as a control strategy.In Chapter 2,we developed a malaria transmission model to reduce and control the disease which is applied in DRC as a case study.The DRC is the most malaria-endemic African country according to the information provided by WHO.The basic reproduction number,R0,is calculated and the local stability as well as global stability of disease-free equilibrium(DFE)and endemic equilibrium are also studied.Furthermore,we carried out numerical simulations to confirm our analytic results.Two optimal control strategies are used in this model:the use of treated-bed net u1(t)and treatment with drugs u2(t)as a major control tool for reducing the amount of malaria infected cases.In Chapter 3,we developed a malaria transmission model with vector-bias and applied it to simulate the data of three countries,low transmission(Mexico)and high transmission(Sudan and the Congo,DR),and used it to predict the evolution of the disease for the next 14 years.We calculated the basic reproduction number,R0,using the next-generation matrix method.The existence of equilibria and corre-sponding global stability were discussed and found to be asymptotically stable.The model always has a disease-free equilibrium which is globally asymptotically stable if R0<1 and unstable if Ro>1.Also,the model has a unique endemic equilibrium which is locally asymptotically stable under certain conditions with R0>1.The endemic equilibrium is also globally asymptotically stable under specific condition-s.Furthermore,we simulated the data report confirming malaria cases of Mexico,Sudan and Congo,DR provided by WHO and also,predicted the future direction of malaria.Our simulation results showed that among the three countries,DRC is the most malaria-endemic country with the highest basic reproduction number R0=6.2047,followed by Sudan R0=1.0323 and Mexico R0=0.0168.In addition to that,malaria will die out in Mexico and persist in Sudan and Congo,DR.More-over,our sensitivity analysis of R0 showed that the mosquito biting rate,σm,is the most sensitive parameter.In Chapter 4,we studied the SEIR-SEI malaria transmission model with a stan-dard incidence rate.The stability analysis of disease-free equilibrium is investigated.The basic reproduction number,R0,is obtained using the next-generation matrix method.The existence of the feasible region where the model is well-known shows that the model exhibits the backward bifurcation phenomenon.Also,the global stability of the endemic equilibrium has been proven.Furthermore,we applied the model to fit existing data of the DRC.In addition to that,we formulated an optimal control problem with an objective function,where three controls:the preventive con-trol using Long-Lasting Insecticide Treated Net(LLITN)u1(t),the treatment with drugs of infected individuals u2(t)and the insecticide spray on the breeding grounds of mosquitoes u3(t),have been used as control measures on malaria-infected indi-viduals.Numerical simulations that were carried out to support our analytic results also suggest that using u1(t)and u2(t)at the same time is the best strategy in controlling the number of malaria-infected individuals in DRC.In Chapter 5,we formulated a new seven-dimensional mathematical model that describes the transmission of the African swine fever through direct method(infected pigs)and indirect one using humans as carriers of the virus.The basic reproduction number,R0,is obtained and the existence of the equilibria is also investigated.Moreover,our numerical simulation results with a numerical example showed that using the quarantine strategy reduces the disease transmission.In addition to that,our sensitivity analysis for R0 showed that the contact rates between infected and susceptible pigs,β2,and between exposed and susceptible human,β3,are more sensitive than the contact rate between exposed and susceptible pigs,β1.