The Dynamics of Epidemic Model with Two Types of Infectious Diseases and Vertical Transmission

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2016, Article ID 4907964, 16 pages http://dx.doi.org/10.1155/2016/4907964 Research Article The Dynamics of Epidemic Model with Two Types of Infectious Diseases and Vertical Transmission Raid Kamel Naji and Reem Mudar Hussien Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq Correspondence should be addressed to Reem Mudar Hussien; Received 27 August 2015; Accepted 10 December 2015 Academic Editor: Zhen Jin Copyright © 2016 R. K. Naji and R. M. Hussien. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An epidemic model that describes the dynamics of the spread of infectious diseases is proposed. Two different types of infectious diseases that spread through both horizontal and vertical transmission in the host population are considered. The basic reproduction number 𝑅0 is determined. The local and the global stability of all possible equilibrium points are achieved. The local bifurcation analysis and Hopf bifurcation analysis for the four-dimensional epidemic model are studied. Numerical simulations are used to confirm our obtained analytical results. 1. Introduction Mathematical models can be defined as a method of emulating real life situations with mathematical equations to expect their future behavior. In epidemiology, mathematical models play role as a tool in analyzing the spread and control of infectious diseases. Although one of the most famous principles of ecology is the competitive exclusion principle that stipulates “two species competing for the same resources cannot coexist indefinitely with the same ecological niche” [1, 2], Volterra was the first scientist who used the mathematical modeling and showed that the indefinite coexistence of two or more species limited by the same resource is impossible [3]. Moreover, Ackleh and Allen [4] were the first who used the competitive exclusion principle of the infectious disease with different levels in single host population. It is well known that one of the most useful parameters concerning infectious diseases is called basic reproduction number. It can be specific to each strain of an epidemic model. In fact the basic reproduction number of the model is defined as the maximum reproduction numbers of other strains [5– 7]. Diekmann et al. [8] had studied epidemic models with one strain, while Martcheva in [9] studied the 𝑆𝐼𝑆-type of disease with multistrain. However, Ackleh and Allen [10] studied 𝑆𝐼𝑅-type of disease with n strain and vertical transmission. Keeping the above in view, in our proposed model two strains with two different types of infectious diseases are considered. Accordingly two different reproduction numbers are obtained and then competitive exclusion principle is presented. It is assumed that two different types of diseases transmission, say horizontal and vertical transmission, are used too. The horizontal transmission occurs by direct contact between infected and susceptible individuals, while vertical transmission occurs when the parasite is transmitted from parent to offspring [11–13]. The incidence of an epidemiological model is defined as the rate at which susceptible becomes infectious. Different types of incidence rates are introduced into literatures [14–17]. Finally two types of incidence rates, say bilinear mass action and nonlinear type, are used with the horizontal and vertical transmission, respectively. The local and global stability for all possible equilibria are carried out with the help of Lyapunov function and LaSalle’s invariant principle [18]. An application of Sotomayor theorem [19, 20] for local bifurcations is used to study the occurrence of local bifurcations near the equilibria. The Hopf bifurcation [21, 22] conditions are derived. Finally, numerical simulations are used to confirm our obtained analytical results and specify the control set of parameters. 2. Model Formulation Consider a real world system consisting of a host population 𝑁(𝑡) that is divided into four compartments: 𝑆(𝑡) which 2 Journal of Applied Mathematics represents the number of susceptible individuals at time 𝑡; 𝐼1 (𝑡) and 𝐼2 (𝑡) that represent the number of infected individuals at time 𝑡 for 𝑆𝐼𝑅𝑆-type of disease and 𝑆𝐼𝑆-type of disease, respectively; finally 𝑅(𝑡) that represents the number of recovered individuals at time 𝑡, thus 𝑁(𝑡) = 𝑆(𝑡) + 𝐼1 (𝑡) + 𝐼2 (𝑡) + 𝑅(𝑡). Now in order to formulate the dynamics of the above system mathematically, the following assumptions have been adopted: (1) There is a constant number of the host populations entering to the system with recruitment rate Λ > 0. (2) There is a vertical transmission of both of the diseases; that is, the infectious host gives birth to a new infected host of rates 0 ≤ 𝑝1 ≤ 1 and 0 ≤ 𝑝2 ≤ 1 for the diseases 𝐼1 and 𝐼2 , respectively. Consequently 𝑝1 𝐼1 and 𝑝2 𝐼2 individuals enter into infected compartments 𝐼1 and 𝐼2 , respectively, and the same quantities are disappearing from recruitment in the susceptible compartment. (3) The diseases are transmitted by contact, according to the mass action law, between the individuals in the 𝑆compartment and those in 𝐼𝑖 (𝑖 = 1, 2) compartments with nonlinear incidence rate for 𝐼1 that is given by 𝛽1 𝑆𝐼1 /(1 + 𝐼1 ), in which 𝛽1 > 0 represents the infection force rate while 1/(1 + 𝐼1 ) represents the inhibition effect of the crowding effect of the infected individuals, and linear incidence rate for 𝐼2 that is given by 𝛽2 𝑆𝐼2 , where 𝛽2 > 0 represents the infection rate. 𝑑𝐼2 = 𝛽2 𝑆𝐼2 − (𝜇 + 𝛼2 + 𝛾 − 𝑝2 ) 𝐼2 , 𝑑𝑡 𝑑𝑅 = 𝛿𝐼1 − (𝜂 + 𝜇) 𝑅 𝑑𝑡 (1) with the initial condition 𝑆(0) > 0, 𝐼1 (0) > 0, 𝐼2 (0) > 0, and 𝑅(0) > 0. Moreover to insure that the recruitment Λ in the susceptible compartment is always positive the following hypotheses are assumed to be holding always: 𝛿 ≥ 𝑝1 , 𝛾 ≥ 𝑝2 . (2) Theorem 1. The closed set Ω = {(𝑆, 𝐼1 , 𝐼2 , 𝑅) ∈ R4+ : 𝑁 ≤ Λ/𝜇} is positively invariant and attracting with respect to model (1). Proof. Let (𝑆(𝑡), 𝐼1 (𝑡), 𝐼2 (𝑡), 𝑅(𝑡)) be any solution of system (1) with any given initial condition. Then by adding all the equations in system (1) we obtain that 𝑑𝑁 = Λ − 𝜇𝑆 − (𝜇 + 𝛼1 ) 𝐼1 − (𝜇 + 𝛼2 ) 𝐼2 − 𝜇𝑅 𝑑𝑡 (3) ≤ Λ − 𝜇𝑁. Thus, from standard comparison theorem [20], we obtain 𝑁 (𝑡) ≤ 𝑁 (0) 𝑒−𝜇𝑡 + Λ (1 − 𝑒−𝜇𝑡 ) . 𝜇 (4) (4) The individuals in the 𝐼1 compartment are facing death due to the disease with infection death rate 𝛼1 ≥ 0. They recover from disease and get immunity with a recovery rate 𝛿 > 0. Consequently it is easy to verify that (5) The individuals in the 𝐼2 compartment are facing death due to the disease with infection death rate 𝛼2 ≥ 0. They also recover from the disease but return back to be susceptible with recovery rate 𝛾 > 0. Thus, Ω is positively invariant. Further, when 𝑁(0) > Λ/𝜇, then either the (...truncated)