Global dynamics of a general vector-borne disease model with two transmission routes

In this paper, we study the dynamics of a vector-borne disease model with two transmission paths: direct transmission through contact and indirect transmission through vector. The direct transmission is considered to be a non-monotone incidence function to describe the psychological effect of some severe diseases among the population when the number of infected hosts is large and/or the disease possesses high case fatality rate. The system has a disease-free equilibrium which is locally asymptomatically stable when the basic reproduction number (R0) is less than unity and may have up to four endemic equilibria. Analytical expression representing the epidemic growth rate is obtained for the system. Sensitivity of the two transmission pathways were compared with respect to the epidemic growth rate. We numerically find that the direct transmission coefficient is more sensitive than the indirect transmission coefficient with respect to R0 and the epidemic growth rate. Local stability of endemic equilibria is studied. Further, the global asymptotic stability of the endemic equilibrium is proved using Li and Muldowney geometric approach. The explicit condition for which the system undergoes backward bifurcation is obtained. The basic model also exhibits the hysteresis phenomenon which implies diseases will persist even when R0 < 1 although the system undergoes a forward bifurcation and this phenomenon is rarely observed in disease models. Consequently, our analysis suggests that the diseases with multiple transmission routes exhibit bi-stable dynamics. However, efficient application of temporary control in bi-stable regions will curb the disease to lower endemicity. In addition, increase in transmission heterogeneity will increase the chance of disease eradication.


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The mode of transmission of some infectious diseases may be through both direct and indirect con-9 tact. In direct transmission, a pathogen can infect a susceptible host through infected host, and in policies have been found very effective (5; 6) in decreasing the rate of infection at the late stage of SARS outbreak, even when the number of infective individuals were getting larger relatively. Additionally, the influenza pandemic of 2009 has triggered a psychological effect that caused adoption of preventive 23 measures by a significant portion of the population (4). Recent MERS-CoV outbreak in Arabian penin-24 sula also imposed a significant behaviour change in the local people (1). Thus, psychological effects 25 are necessary in modelling infectious disease transmission. The general incidence rate in this context at time t is given by N v (t) = S v (t) + I v (t). We assume all newborn hosts are fully susceptible. The susceptible host population increases at a constant rate Π h . The susceptible population decreases due to 66 getting infection from infected vectors and infected hosts and natural mortality at a rate µ h . Therefore Interpretations of all the model parameters and their biological meanings are given in the Table 1. We assume that the vector and host state variables are non-negative for all t ≥ 0. Let

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We claim the following result.
Theorem 3.1. Whenever the initial conditions are non-negative, model (2.1) has non-negative solutions and the region Ω ∈ R 5 + is positively invariant and globally attracting for the above system. 84 Proof. We can rewrite the system (2.1) in the following form 85 86 dy dt = g(y) (3. 1) with y = (y 1 , y 2 , y 3 , y 4 , y 5 ) = (S h , I h , R h , S v , I v ) and g(y) = (g 1 (y), g 2 (y), g 3 (y), g 4 (y), g 5 (y)) denote the 87 right hand side of the functions as in the following 88 89 Clearly, for every j = 1, ..., 5, g j (x) ≥ 0 if y ∈ R 5 + and y j = 0. Since the host and vector populations 90 are non-negative, thus the right hand side of the system (3.1) is locally Lipschitz in Ω. Following (12) 91 and (13), we see that the system (2.1) has a unique solution. The following equations are satisfied by 92 the total host and vector populations, Theorem 3.2. The DFE Y 0 of the system (2.1) is locally asymptotically stable, if R 0 < 1, and unstable Proof. The Jacobian of the system (2.1) at DFE Y 0 is given as follows (3.5) Then the characteristic equation of (3.5) is given by The eigenvalues of this jacobian matrix is Note that λ is the EGR for the 'SIR' sub-model of the system (2.1) and is given by If there is no indirect transmission through vector(β i → 0), then the system (2.1) becomes the 129 sub-model (3.8) and λ → λ. Additionally, from (3.6), it can be noted that The mean time spent in the infected host class is dependent on the host lifetime, the mean infectious 137 Subtracting these two expressions gives: can be determined in terms of β i as, (3.7) is involved in the expression of R SIRSI 0 given by (3.10). We will calculate the total derivative of λ: Since R 0 is fixed and using (3.13), the total derivative of (3.10) gives (3.14) (3.15) We have R 0 > 1, λ ≥ λ and λ > 0. Finally, µ v > 0. Thus, the numerator become positive and the 165 denominator is negative in (3.15), giving that dλ dβ i < 0.

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Now we calculate the relationship of dλ dβ i with ∂λ ∂β i and ∂λ ∂β d as follows: Corollary 3.1. Let R 0 > 1 and the initial EGR λ be given by (3.6).
To verify this result numerically, sensitivity index of λ with respect to the parameters β d and β i 169 are computed. The normalized forward sensitivity index of a quantity with respect to a parameter 170 is defined as the ratio of the relative change in the variable to the relative change in the parameter.
Mathematically, the normalized forward sensitivity index of a variable m that depends explicitly on a 172 parameter τ is defined as: The normalized forward sensitivity indices of λ with respect to the parameters β d and β i are found to for the parameters value Π h = 5, The fact that X β d λ = 1.6631 means that 1% increase in β d , keeping other parameters fixed, will produce that the direct transmission pathway is more sensitive to EGR and R 0 than the indirect one, which 183 indicates that the direct transmission control will be more effective in halting the early phase of the 184 epidemic.
be an arbitrary endemic equilibrium of the system(2.1). Therefore, equat-188 ing the right hand sides of the equations of system (2.1) to zero, we have and I * h comes from the equation where,  Table 2 and Table 3.
Theorem 4.1. The system (2.1) has unique endemic equilibrium Y * if R 0 > 1 and cases 1, 2, 6 and 10 202 in Table 2 are satisfied, and has unique endemic equilibrium Y * if R 0 < 1 and cases 3, 4, 5 and 14 in 203   Table 3 are satisfied. The system (2.1) could have more than one endemic equilibrium if R 0 > 1 and 204 cases 3 − −5, 7 − −9, 11 − −13, 15 and 16 in Table 2 are satisfied, and could have more than one endemic 205 equilibrium if R 0 < 1 and cases 2, 6 − −13, 15 and 16 in Table 3 are satisfied. The Jacobian matrix of the system (2.1) evaluated at endemic equilibrium Y * can be written as Case Q 1 Q 2 Q 3 Q 4 Q 5 # sign changes # possible positive real roots 1 Clearly, −µ h is one root of J(Y * ), which is negative. The remaining roots can be determined from 215 the following characteristic equation which is given by Using the RouthHurwitz criterion, the roots of equation (  (2.1) by at least one as this method is applicable to systems whose order is at most four.
as t → ∞ (20), the system (2.1) reduces to the limit system Clearly, the solutions of the limit system (4.4) with non-negative initial conditions remain nonnegative. Therefore, We can study the model in the following region However, before going to the main result, let us discuss some preliminary results and definitions. Consider the autonomous dynamical system:ẋ where f : D → R n , D ⊂ R n open set and simply connected and f ∈ C 1 (D). Let x * be the solution of locally stable and all trajectories in D converges to x * .

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Definition: The set K is absorbing subset in D for the system (4.5), if for every compact Let Q(x) be a matrix of size n 2 × n 2 and Q ∈ C 1 (D). Suppose that Q −1 exists and it is continuous 233 in K. Where K is a compact absorbing set in D. 234 We define where Q f is obtained replacing every entry q ij of Q by its directional derivatives with respect the 236 vectorial field f .

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The Lozinskii measure µ(A) with respect to the norm . in R N , N = n 2 , is defined as (18), We will apply the following (18) 238 Theorem 4.2. If K is a compact absorbing subset in the interior of D, and there exist ν > 0 such that 239 the Lozinskii measure µ(A) ≤ −ν for all x ∈ K, Then every omega limit point of system (4.4) in the 240 interior of D is an equilibrium in K.

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The system (2.1) admits exactly one endemic equilibrium Y * when R 0 > 1 for the cases 1, 2, 6, 10 in Table 2. Further, we know that the DFE Y 0 is unstable when R 0 > 1. The instability of P 0 , together with P 0 ∈ ∂Θ, which implies the uniform persistence of the state variable (21). Thus there exists a constant c > 0 such that any solution ( The uniform persistence of the system (4.4), incorporating the boundedness of Θ , suggests that the  Table 2 are satisfied and Then, the unique endemic equilibrium Y * of the system (2.1) is globally asymptotically stable in region 245 Θ.

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Proof. We show the global stability of Y * using the Muldowney's Theorem 4.2. We have to prove the 247 following to apply the theorem: 248 1. There exists a compact absorbing set K in the interior of Θ.

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The uniform persistence together with the boundedness of Θ is equivalent to the existence of an 251 absorbing compact set in the interior of Θ(22). Now we have to prove that the Lozinskii measure

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Let Q(t) be the following matrix of 6 × 6 which is invertible and C 1 We have where Q f is obtained by replacing each entry Q ij of Q with the derivative of Q ij in the direction of the 282 vector field given by the system (2.1).

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Cases 13: z 4 , z 6 > 0, z 5 < 0 and |z 5 | > |z 4 | + |z 6 | then 353 z = |z 5 | = −z 5 We have Cases 14: z 4 , z 6 > 0, z 5 < 0 and |z 5 | < |z 4 | + |z 6 | then 355 z = |z 4 | + |z 6 | = z 4 + z 6 We have Cases 15: z 5 , z 6 > 0, z 4 < 0 and |z 5 | < |z 4 | then z = |z 4 | + |z 6 | = −z 4 + z 6 We have Cases 16: z 5 , z 6 > 0, z 4 < 0 and |z 5 | > |z 4 | then 361 z = |z 5 | + |z 6 | = z 5 + z 6 We have From all cases. we obtain the following estimate where, Since by the hypothesis From (4.12), there exist ν > 0 such that Therefore the Lozinskii measure satisfies This completes the proof. phenomenon shows that if R 0 < 1, although the DFE is stable, another stable endemic equilibrium may 385 coexist simultaneously. When the multiple stable equilibrium coexist then the population will reach 386 the final equilibrium depending on the initial conditions. Backward bifurcation phenomenon for the 387 system (2.1) is shown in Fig. 1. Here, a stable DFE and a stable endemic equilibrium point coexist 388 when 0 < R c < R 0 < 1, where R c is the critical value, which is shown in Fig. 1 and R c = 0.274 for the 389 given parameters values. In Fig. 1, the green line represents the unstable equilibrium, while the blue line is established analytically. The Jacobian of the system (3.2) at diseases free equilibrium Y 0 is given as Let β i be the bifurcation parameter and using R 0 = 1, we have Here 0 is the simple eigen value and the jacobian J(Y 0 ) at β i = β * i has a right eigen vector corresponding 404 to zero eigen value is given by w = (w 1 , w 2 , w 3 , w 4 , w 5 ) and has a left eigen vector corresponding to zero 405 eigen vector is given by The following second order partial derivatives of g i at DFE Y 0 are calculated as follows: We see that the coefficient b is always positive, hence the system (2.1) undergoes backward bifurcation The backward bifurcation phenomenon would occur for those values of R 0 such that R c < R 0 < 1.

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For forward transcritical bifurcation usually a model has two locally stable branches, one is DFE 427 when R 0 < 1, which is locally asymptotically stable and another is endemic equilibrium when R 0 > 1, 428 which is stable. But there may be more than one stable endemic equilibrium for the model when R 0 > 1.

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It is possible that more than one endemic equilibrium coexist in epidemic models even when R 0 > 1.

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This leads to an unusual phenomenon of forward bifurcation with hysteresis effect, which is shown 431 numerically for our model (2.1). We find that our model exhibits a hysteresis effect where multiple 432 endemic equilibrium coexist for R 0 > 1 (see Fig. 2). The blue line represents two outer equilibrium, 433 which is stable while the green line represent the interior equilibrium, which is unstable. Further investigation reveals that the endemic equilibrium may also be stable in a region where 435 R 0 < 1 (see Fig. 3). This indicates that the disease may persist for R 0 < 1, even if the type of 436 transcritical bifurcation at R 0 = 1 is forward. Similarly, in case of hysteresis, these control measures will compel the solutions to the basin of attraction 447 of the lower endemic equilibrium. Here, we control the indirect transmission rate (β i ) for a certain period 448 of time, to force the solutions of the system to lower endemic value. We use the following to model this For backward bifurcation case, we observe that a solution of the system approaches the upper equi-  The vector-host interaction is assumed to be well-mixed in the proposed model 2.
Similarly, the transmission term from infectious hosts to susceptible vectors is given by: Here k 1 and k 2 are the levels of heterogeneity of vector-host and host-vector transmissions respec-480 tively. Since k 1 and k 2 both are the levels of heterogeneity, we assumed that k 1 = k 2 = k for simplicity.

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Therefore, replacing these heterogeneity terms into our model 2.1, we obtain the following system of Using numerical simulations, the impact of the heterogeneity parameter k have been studied. We    to switch the stability of equilibria. In case of backward bifurcation, temporary control can curb the 507 disease endemic equilibrium to disease free state. On the other hand, the system can be brought to lower 508 endemic levels by the employment of temporary control. We found that the strength of psychological 509 effect will have significant contribution in lowering the endemic infection level of the host population.

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Further investigation reveals that increase in transmission heterogeneity will ensure fast eradication of 511 the disease.

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In summary, global stability of the unique endemic equilibrium indicate that the disease will persist 513 under certain parametrization of the system. Additionally, the diseases with two transmission routes can 514 exhibit bi-stable dynamics. These bi-stable feature of the diseases may impose significant difficulties on 515 disease control strategies. Moreover, efficient application of temporary control will be helpful in bi-stable 516 regions. While investigating the effect of heterogeneity, we observe that high levels of heterogeneities 517 will result in slower disease spread.