SUBHARMONIC BIFURCATION IN MALARIA-LASSA FEVER CO-INFECTION EPIDEMIC MODEL WITH OPTIMAL CONTROL APPLICATION

Abstract

Co-infection with malaria often complicate and increase the severity of disease pathogenesis. The co-infection of malaria and Lassa fever has not been fully understood. Many researcher have worked on mathematical models describing the features involved in the transmission of mono-infection of malaria and Lassa fever. However, models on co-infection that incorporate seasonal variation of vec tors needed for a full understanding and management of the co-infection in human with plasmodium falciparum and Lassa virus are sparse. Therefore, this study was designed to develop a mathematical model that incorporates seasonal variation of vectors and investigate the e ect of endemic malaria mortality rate of Lassa fever patients. A co-infection mathematical model governed by a system of ordinary di eren tial equations that incorporates seasonal variation of vectors, ξm, diagnostic factor for treatment, ηv, treatment rate, σ, biting rate, b, contact rate, w1, proportion of e ective treatment, γv and force of infection, φ0 was formulated using law of mass action. The state variables Nes, Nis and Nrs denoted the number of human pop ulation that were exposed, infected and recovered from Lassa fever, respectively, but susceptible to malaria. Moreover, Nse, Nsi and Nsr represented those exposed, infected and recovered from malaria, respectively, but susceptible to Lassa fever. The rodent population (obtained from the literature) were classi ed as Sd, Ed and Id represented those susceptible, exposed and infected, respectively. Furthermore, the mosquito population were classi ed as Sm, Em and Im denoted those suscepti ble, exposed and infected, respectively. Using next generation matrix method, the basic reproduction number, R0(a, t) of the co-infection model was computed, where a is the age and t is the time. Applying perturbation method, stable subharmonic bifurcation solutions were determined. With the aid of suitable Lyapunov function, the stability of the equilibra were explored. Using Pontryagin maximum principle, v necessary condition for the optimal control were derived. Numerical analyses of the model were carried out to investigate the parameters most responsible for disease transmission using data obtained from World Health Organization database. The established mathematical model gives a system of fourteen ordinary dif ferential equations with the rst four as: N 0 rs(t) = αγvNes + ηvσNis − µNrs + bw1φ0Nes, S 0 m(t) = ξm − λmSm(t)Nis − µmSm, E 0 m(t) = λmSm(t)Nis − µmEm, I 0 m(t) = −µmIm, where, µ and µm are human and mosquito death rate respectively;λm is the trans mission rate. The R0(a, t) was computed as R0(a, t) = √ RhmRmm, where Rmm and Rhm are the vector and human threshold parameters, respectively. An in nite num ber (n) of stable subharmonic solutions was obtained as Nss(t, ai) = Nssn (t + α), where α is the treatment rate of infected human with malaria. The disease-free and endemic equilibria were found to be globally and asymptotically stable since R0(a, t) = 0.496 and R0(a, t) = 2.684, respectively. Conditions for existence and uniqueness of optimality system, u1u2u3 were established, were u1u2u3 are the con trol functions. Vectors biting rate b and contact rate w1 among eleven positive sensitivity index parameter values: µ = 0.000038, b = 1.58, w1 = 1.38,µm = 0.054, γv = 0.00027, σ = 0.012, α = 0.50, ηv0.0053, λm = 0.26, ξm = 0.50 and φ0 = 0.24 contributed majorly to the transmission of the diseases. The formulated model captured seasonal variation of vectors and also showed that co-infection of malaria and Lassa fever increased mortality rate in infected patients.