Mathematical modelling and control of African animal trypanosomosis with interacting populations in West Africa—Could biting flies be important in main taining the disease endemicity?

African animal trypanosomosis (AAT) is transmitted cyclically by tsetse flies and mechanically by biting flies (tabanids and stomoxyines) in West Africa. AAT caused by Trypanosoma congolense, T. vivax and T. brucei brucei is a major threat to the cattle industry. A mathematical model involving three vertebrate hosts (cattle, small ruminants and wildlife) and three vector flies (Tsetse flies, tabanids and stomoxyines) was described to identify elimination strategies. The basic reproduction number (R0) was obtained with respect to the growth rate of infected wildlife (reservoir hosts) present around the susceptible population using a next generation matrix technique. With the aid of suitable Lyapunov functions, stability analyses of disease-free and endemic equilibria were established. Simulation of the predictive model was presented by solving the system of ordinary differential equations to explore the behaviour of the model. An operational area in southwest Nigeria was simulated using generated pertinent data. The R0 < 1 in the formulated model indicates the elimination of AAT. The comprehensive use of insecticide treated targets and insecticide treated cattle (ITT/ITC) affected the feeding tsetse and other biting flies resulting in R0 < 1. The insecticide type, application timing and method, expertise and environmental conditions could affect the model stability. In areas with abundant biting flies and no tsetse flies, T. vivax showed R0 > 1 when infected wildlife hosts were present. High tsetse populations revealed R0 <1 for T. vivax when ITT and ITC were administered, either individually or together. Elimination of the transmitting vectors of AAT could cost a total of US$ 1,056,990 in southwest Nigeria. Hence, AAT in West Africa can only be controlled by strategically applying insecticides targeting all transmitting vectors, appropriate use of trypanocides, and institutionalising an appropriate barrier between the domestic and sylvatic areas.


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The symbols, values and list of references were included. Some values were estimated based on 185 the results from the field data in southwest Nigeria.
186 Since the flies will not be removed totally and permanently from the project area, the cattle 251 will still need to be examined for trypanosomosis and administered trypanocidal drugs, albeit

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Population growth of cattle is observed when there is a reduction in the infected cattle population compared to the susceptible cattle population due to the intervention strategies.
The growth can be narrowed if there is a problem with the balance such as problems of trypanocidal and insecticidal resistance (for biting flies, which could pose continuous challenge), changing climate, epidemics from other infectious diseases, ecological instability and human activities.

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Disease-free equilibrium point 285 For the disease-free equilibrium, the disease states and the left-hand side of (0.1)-(0.16) were set to zero. The resulting system is solved which is given For the ITC method of insecticidal control, vectors were targeted on the cattle rather than the 286 parasite (trypanosomes), hence, there is mortality at the point of feeding and those occurring be-

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FV −1 is called the next generation matrix.
. Therefore, the basic reproduction number, R 0 , is given by where ρ is the spectral radius of the product, FV −1 (i.e, the dominant eigenvalue of FV − 1), known as the next generation matrix.
Applying this technique to model (0.1)-(0.16), we let Then model (0.1)-(0.16) can be written as dx and the rate of individuals into and out of the compartments of the model (0 Find the derivative of F (x) and V (x) at the disease-free equilibrium point since V is a non-singular matrix, the inverse of V, V −1 can be obtained as The basic reproduction number R 01 = ρ(FV −1 , is the spectral radius of the product FV −1 . Hence, for the model (0.1)-(0.16), we arrive at In areas where tsetse flies were absent (T. vivax showed R 02 > 1 in the cattle population, when infected wildlife hosts are present), and biting flies (tabanids and stomoxyines) are abundant, the basic reproduction number of AAT increases.
Where ω 1 is the growth rate of infected wildlife introduced in the susceptible population and 301 ω 2 is the growth rate of infected small ruminants introduced in the susceptible population. It 302 is assumed that tsetse flies can also become infected when feeding on infected wildlife. This is  The field-data from southwest Nigeria entered for the mathematical model generated valuable 306 results applicable to West African countries (Fig 2).

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Global stability of disease-free equilibrium 308 The global asymptotic stability of the AAT-free equilibrium for the special case with no loss of immunity acquired by the recovered cattle and small ruminants after the treatment of try- Proof: Considering the system's Lyapunov function, acting as mechanical vectors [42]. Similarly, Anene et al. [7] observed that T. vivax was main-318 tained in the flock by tabanids in tsetse-free areas in Nigeria.

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Proof: Let R 01 > 1 and R 02 > 1 so that a unique endemic equilibrium exists and consider the following nonlinear Lyapunov function defined by we take the derivative of equation (0.18) along the solutions of the model (0.1)-(0.16) and simplify to achieve the following boundṡ further algebraic manipulation gives It is sufficient to show that f (x, y, z) ≥ 0.
Since f x = f y = f z = 0 gives rise to x = y = z and that    Remarks: It is worth mentioning that local stability of an equilibrium (situation) would imply 341 existence of that situation for a short time (depending on certain circumstances or conditions).

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Whereas global stability would imply existence of a situation forever regardless of any condition.

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In summary, global stability of system (model) implies local stability, however, local stability of 344 system does not imply global stability.

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Numerical Results

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We illustrated the theoretical results established in this study and by considering initial con- wildlife populations, were present ( Fig 3A). The size of the exposed cattle population decreases 352 with progression to the infected group (Fig 3A). The decrease in the number of infected cattle 353 contributes to the increase in number of recovered cattle (Fig 3B).  Table and S2 Table). The price of try- results. Expendables for eliminating and monitoring baits were also reported (S1 Table and S2   431   Table). To improve the success rate, the insecticidal approach (ITT) must be continuously main- In a similar manner, equation (0.23), (0.24), (0.25) and (0.26) gives Taking the limits of (0.27) Thus the following feasible region R = {S k (t), E k (t), I k (t), R k (t), S t , E t , I t , S n , S c , T n , T c , S w , I w , S r , I r , R r ∈ R 16 : Consider the following linear Lyapunov function In what follows, the time derivative of F given by (0.33) along the solutions of the model (0.1)- F ≤ 0 for R 01 ≤ 1 .Ḟ = 0 if and only if I t (t) = T c (t) = S c (t) = 0. Further, one sees that 552 (S k (t), E k (t), I k (t), R k (t), S t (t), E t (t), I t , S n , S c , T n , T c , S w , I w , S r , I r , R r ) → 553 π 0 = Λ k µ k , 0, 0, 0, Λ t µ t , 0, 0, Λ s µ s , 0, Λ b µ b , 0, Λ w µ w , 0, Λ r µ r , 0, 0 as t → ∞ since (I t , T c , S c → 0 as t → ∞. Con-554 sequently, the largest compact invariant set in {(S k (t), E k (t), I k (t), R k (t), S t (t), 555 E t (t), I t (t), S n (t), S c , T n , T c , S w , I w , S r , I r , R r ∈ R :Ḟ = 0} is a singleton {π 0 } and by LaSalle's in-556 variance principle [43], π 0 is globally asymptotically stable in R if R 01 ≤ 1. The Lyapunov derivative of (0.35) is given bẏ S t +bϕ t S t (t)(I k (t)+I w (t)+I r (t))− bϕ t S t E * t (I k (t) + I w (t) + I r (t)) E t + (γ t + µ t + σ t )E * t − (γ t + µ t + σ t )(σ t + µ t )I t γ t − (γ t + µ t + σ t )I * t E t I t + (γ t + µ t + σ t )(σ t + µ t )I * t α t +Λ w 1 − S * w S w (t) − µ w 1 − S * w S w (t) +bτe ω 1 S w (I t +T c +S c )+Λ r 1 − S * r S r (t) − µ r 1 − S * r S r (t) + bζ e ω 2 S r (I t + T c + S c ) + Λ s 1 − S * n S n (t) − µ s 1 − S * n S n (t) + bα 2 S n (I k + I w + I r ) n T n (t) + bα 1 T n (I k + I w + I r ) (0.36) At the endemic equilibrium, it is seen from (0.1)-(0.16) that Λ w = e ω 1 bτS w (I * t + S * c + T * c ) + µ w Λ r = e ω 2 bζ S r (I * t + S * c + T * c ) + µ r Λ s = bα 2 S n (I * k + I * r + I * w ) + (µ s + δ s )S * n Λ s = bα 1 T n (I * k + I * r + I * w ) We need to show that V i ≥ 0, i = 1, 2, ...12, . In order to achieve this model, considering the 568 arithmetic mean is greater than or equal to the geometric mean (AM -GM inequality), we have 569 (S * k ) 2 + (S k (t)) 2 − 2S * k S k (t) ≥ 0 so that,